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Motion & mechanics (7)Waves, light & the very small (8)Electricity (6)Heat, fluids & HVAC (10)Chemistry (10)Water & environment (9)Materials, ground & site (13)Mathematics (7)

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Pythagorean Theorem

a2+b2=c2a^{2} + b^{2} = c^{2}

GeometryRelates the three sides of a right triangle.

Area of a Circle

A=πr2A = \pi r^{2}

GeometryArea enclosed by a circle of radius r.

Circumference of a Circle

C=2πrC = 2 \pi r

GeometryDistance around a circle of radius r.

Area of a Triangle

A=12bhA = \tfrac{1}{2} b h

GeometryArea from base and perpendicular height.

Rectangle Area

A=l⋅wA = l \cdot w

GeometryArea of a rectangle as length times width.

Parallelogram Area

A=b⋅hA = b \cdot h

GeometryArea of a parallelogram from its base and perpendicular height.

Trapezoid Area

A=a+b2⋅hA = \frac{a + b}{2} \cdot h

GeometryArea of a trapezoid as the average of its two parallel sides times the perpendicular height.

Ellipse Area

A=πabA = \pi a b

GeometryArea of an ellipse from its semi-major and semi-minor axes, with π ≈ 3.14159265.

Circular Sector Area

A=12r2θA = \frac{1}{2} r^{2} \theta

GeometryTrigonometryArea of a pie-slice sector of a circle from its radius and central angle in radians.

Arc Length

s=rθs = r \theta

GeometryTrigonometryLength of a circular arc as radius times central angle in radians.

Distance Formula (2D)

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

GeometryAlgebraStraight-line distance between two points in the coordinate plane.

Sphere Volume

V=43πr3V = \frac{4}{3} \pi r^{3}

GeometryVolume enclosed by a sphere of radius r, using π ≈ 3.14159265.

Sphere Surface Area

S=4πr2S = 4 \pi r^{2}

GeometrySurface area of a sphere of radius r, using π ≈ 3.14159265.

Cylinder Volume

V=πr2hV = \pi r^{2} h

GeometryWater TreatmentVolume of a right circular cylinder from its radius and height, using π ≈ 3.14159265.

Cylinder Surface Area

S=2πr2+2πrhS = 2 \pi r^{2} + 2 \pi r h

GeometryTotal surface area of a closed cylinder — two end caps plus the wrapped side — using π ≈ 3.14159265.

Cone Volume

V=13πr2hV = \frac{1}{3} \pi r^{2} h

GeometryVolume of a right circular cone — one-third of the matching cylinder — using π ≈ 3.14159265.

Pyramid Volume

V=13BhV = \frac{1}{3} B h

GeometryVolume of any pyramid as one-third of its base area times its perpendicular height.

Rectangular Prism Volume

V=l⋅w⋅hV = l \cdot w \cdot h

GeometryVolume of a box as the product of its length, width, and height.

Cube Volume

V=a3V = a^{3}

GeometryVolume of a cube as its edge length raised to the third power.

Cube Surface Area

S=6a2S = 6 a^{2}

GeometrySurface area of a cube as six times the area of one square face.

Cube Space Diagonal

d=a3d = a \sqrt{3}

GeometryLength of the interior diagonal joining opposite corners of a cube.

Right-Triangle Sine Ratio (SOH)

sin⁡θ=oh\sin\theta = \frac{o}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its opposite side and the hypotenuse.

Right-Triangle Cosine Ratio (CAH)

cos⁡θ=ah\cos\theta = \frac{a}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its adjacent side and the hypotenuse.

Right-Triangle Tangent Ratio (TOA)

tan⁡θ=oa\tan\theta = \frac{o}{a}

TrigonometryGeometryRelates an acute angle of a right triangle to the ratio of its opposite and adjacent legs.

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

TrigonometryGeometryFinds the third side of any triangle from two sides and their included angle, or the angle from all three sides.

Law of Sines

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

TrigonometryGeometryIn any triangle, each side is proportional to the sine of its opposite angle.

Triangle Area (Two Sides and Included Angle)

A=12 absin⁡CA = \tfrac{1}{2}\,ab\sin C

TrigonometryGeometryComputes a triangle's area from two sides and the angle between them, with no height needed.

Slope Between Two Points

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

AlgebraGeometrySteepness of a line as rise over run between two points.

Slope-Intercept Form of a Line

y=mx+by = mx + b

AlgebraGeometryEquation of a straight line from its slope and y-intercept.

Exponential Growth

A=A0(1+r)tA = A_0 (1 + r)^{t}

AlgebraAmount after t periods of compound growth at rate r per period.

Exponential Decay

A=A0(1−r)tA = A_0 (1 - r)^{t}

AlgebraAmount remaining after t periods of losing a fixed fraction r per period.

Exponential Growth by Doubling Time

N=N0⋅2t/TN = N_0 \cdot 2^{t/T}

AlgebraGrowth of a quantity that doubles every fixed interval T.

Arithmetic Sequence nth Term

an=a1+(n−1)da_n = a_1 + (n - 1) d

AlgebraThe nth term of a sequence that steps by a fixed common difference.

Arithmetic Series Sum

Sn=n2(2a1+(n−1)d)S_n = \frac{n}{2} \left( 2 a_1 + (n - 1) d \right)

AlgebraSum of the first n terms of an arithmetic sequence.

Geometric Sequence nth Term

an=a1 r n−1a_n = a_1 \, r^{\,n-1}

AlgebraThe nth term of a sequence that multiplies by a fixed ratio each step.

Geometric Series Sum

Sn=a1(1−rn)1−rS_n = \frac{a_1 (1 - r^{n})}{1 - r}

AlgebraSum of the first n terms of a geometric sequence.

Infinite Geometric Series

S=a11−rS = \frac{a_1}{1 - r}

AlgebraLimit of a geometric series with ratio between −1 and 1, summed forever.

Simple Interest

I=P r tI = P \, r \, t

AlgebraInterest earned on a principal at a flat rate per period, without compounding.

Compound Interest (Periodic)

A=P(1+rn)ntA = P \left( 1 + \frac{r}{n} \right)^{n t}

AlgebraAmount after t periods when interest compounds n times per period at rate r.

Continuous Compounding

A=P ertA = P \, e^{r t}

AlgebraAmount after t periods when interest compounds continuously at rate r.

Present Value

PV=FV(1+r)t\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}

AlgebraToday's value of a future amount, discounted at rate r per period.

Percent Change

c=xnew−xoldxoldc = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}

AlgebraRelative change from an old value to a new one, as a fraction of the old value.

Distance Formula (3D)

d=Δx2+Δy2+Δz2d = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}

AlgebraStraight-line distance between two points from their separations along each axis.

Speed, Distance & Time

v=dtv = \tfrac{d}{t}

PhysicsAverage speed from distance travelled and elapsed time.

Newton's Second Law

F=maF = m a

PhysicsForce equals mass times acceleration.

Density

ρ=mV\rho = \tfrac{m}{V}

PhysicsChemistryWater TreatmentMass per unit volume.

Kinetic Energy

Ek=12mv2E_k = \tfrac{1}{2} m v^{2}

PhysicsEnergy of a mass in motion.

Final Velocity (Uniform Acceleration)

v=v0+atv = v_0 + a t

MechanicsPhysicsFinal velocity after accelerating uniformly from an initial velocity for a given time.

Displacement (Uniform Acceleration)

d=v0t+12at2d = v_0 t + \tfrac{1}{2} a t^2

MechanicsPhysicsDistance travelled under constant acceleration, starting from an initial velocity, over a time t.

Velocity-Displacement Relation (v² = v₀² + 2ad)

v2=v02+2adv^2 = v_0^2 + 2 a d

MechanicsPhysicsLinks initial and final speeds to acceleration and displacement without involving time.

Displacement from Average Velocity

d=v0+v2 td = \frac{v_0 + v}{2} \, t

MechanicsPhysicsDisplacement as the average of initial and final velocities multiplied by the elapsed time, valid for uniform acceleration.

Displacement from Final Velocity (d = vt − ½at²)

d=vt−12at2d = v t - \tfrac{1}{2} a t^2

MechanicsPhysicsThe fifth kinematic equation: displacement from the FINAL velocity and the time, for when the starting speed is the unknown.

Linear Momentum (p = mv)

p=mvp = m v

MechanicsPhysicsMomentum as the product of an object's mass and velocity.

Impulse (J = FΔt)

J=F ΔtJ = F \, \Delta t

MechanicsPhysicsImpulse delivered by an average force acting over a contact time, equal to the change in momentum.

Centripetal Acceleration (a = v²/r)

ac=v2ra_c = \frac{v^2}{r}

MechanicsPhysicsInward acceleration of an object moving in a circle at constant speed.

Centripetal Force (F = mv²/r)

Fc=mv2rF_c = \frac{m v^2}{r}

MechanicsPhysicsNet inward force required to keep a mass moving in a circle at constant speed.

Speed in Circular Motion (v = 2πr/T)

v=2πrTv = \frac{2\pi r}{T}

MechanicsPhysicsSpeed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per period.

Work (W = Fd cos θ)

W=Fdcos⁡θW = F d \cos\theta

MechanicsPhysicsWork done by a constant force acting at an angle to the displacement.

Power (P = W/t)

P=WtP = \frac{W}{t}

MechanicsPhysicsAverage power as work or energy delivered per unit time.

Power from Force and Velocity (P = Fv)

P=FvP = F v

MechanicsPhysicsInstantaneous power delivered by a force parallel to the velocity.

Newton's Law of Universal Gravitation

F=Gm1m2r2F = \frac{G m_1 m_2}{r^2}

MechanicsPhysicsAttractive force between two masses, with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2018).

Weight (W = mg)

W=mgW = m g

MechanicsPhysicsWeight of a mass at Earth's surface, using standard gravity g = 9.80665 m/s² (exact by definition).

Gravitational Potential Energy (U = mgh)

U=mghU = m g h

MechanicsPhysicsEnergy stored by raising a mass to height h near Earth's surface, with g = 9.80665 m/s².

Pressure (P = F/A)

P=FAP = \frac{F}{A}

MechanicsPhysicsPressure as perpendicular force spread over an area.

Hydrostatic Pressure (P = ρgh)

P=ρghP = \rho g h

MechanicsPhysicsWater TreatmentGauge pressure at depth h in a fluid of density ρ, using g = 9.80665 m/s².

Hooke's Law

F=kxF = k x

MechanicsPhysicsRestoring force of an ideal spring, proportional to its displacement from rest.

Elastic Potential Energy

U=12kx2U = \tfrac{1}{2} k x^{2}

MechanicsPhysicsEnergy stored in an ideal spring displaced x from its rest length.

Torque

τ=rFsin⁡θ\tau = r F \sin\theta

MechanicsPhysicsTurning effect of a force applied at distance r from a pivot, at angle θ to the lever arm.

Angular Velocity (ω = θ/t)

ω=θt\omega = \frac{\theta}{t}

MechanicsPhysicsAverage angular velocity: the angle swept divided by the time taken.

Angular Velocity from Period

ω=2πT\omega = \frac{2\pi}{T}

MechanicsWaves & OscillationsPhysicsOne full revolution is 2π radians, so angular velocity is 2π divided by the period.

Linear Speed from Rotation (v = ωr)

v=ωrv = \omega r

MechanicsPhysicsA point at radius r on a rotating body moves with linear speed ωr.

Centripetal Acceleration (a = ω²r)

ac=ω2ra_c = \omega^{2} r

MechanicsPhysicsCentripetal acceleration written in terms of angular velocity rather than linear speed.

Angular Acceleration

α=ω−ω0t\alpha = \frac{\omega - \omega_0}{t}

MechanicsPhysicsAverage angular acceleration: the change in angular velocity divided by the time taken.

Angular Displacement (θ = ω₀t + ½αt²)

θ=ω0t+12αt2\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

MechanicsPhysicsAngle turned under constant angular acceleration, the rotational twin of x = v₀t + ½at².

Torque with a Lever Arm (τ = rF sin θ)

τ=rFsin⁡θ\tau = r F \sin\theta

MechanicsPhysicsTorque produced by a force applied at distance r from the pivot, at angle θ to the lever.

Newton's Second Law for Rotation (τ = Iα)

τ=Iα\tau = I \alpha

MechanicsPhysicsNet torque equals moment of inertia times angular acceleration — F = ma for spinning things.

Rotational Kinetic Energy

KErot=12Iω2KE_{rot} = \tfrac{1}{2} I \omega^{2}

MechanicsPhysicsKinetic energy stored in rotation: half the moment of inertia times angular velocity squared.

Angular Momentum (L = Iω)

L=IωL = I \omega

MechanicsPhysicsAngular momentum of a rotating body: moment of inertia times angular velocity.

Moment of Inertia: Point Mass

I=mr2I = m r^{2}

MechanicsPhysicsRotational inertia of a compact mass circling at radius r from the axis.

Moment of Inertia: Solid Disk

I=12mr2I = \tfrac{1}{2} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid disk or cylinder about its central axis.

Moment of Inertia: Solid Sphere

I=25mr2I = \tfrac{2}{5} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid sphere about an axis through its center.

Rotational Power (P = τω)

P=τωP = \tau \omega

MechanicsPhysicsMechanical power delivered by a torque turning at angular velocity ω.

Volumetric Flow Rate (Q = Av)

Q=AvQ = A v

Fluid MechanicsWater TreatmentPhysicsFlow through a duct or pipe: cross-sectional area times average flow velocity.

Continuity Equation (A₁v₁ = A₂v₂)

A1v1=A2v2A_1 v_1 = A_2 v_2

Fluid MechanicsWater TreatmentPhysicsFor incompressible flow, the same volume per second passes every cross-section of the pipe.

Dynamic Pressure (q = ½ρv²)

q=12ρv2q = \tfrac{1}{2} \rho v^{2}

Fluid MechanicsPhysicsThe kinetic energy per unit volume of a moving fluid — the pressure of motion itself.

Buoyant Force (Archimedes' Principle)

Fb=ρVgF_b = \rho V g

Fluid MechanicsPhysicsThe upward force on a submerged body equals the weight of the fluid it displaces, with g = 9.80665 m/s².

Torricelli's Law (v = √(2gh))

v=2ghv = \sqrt{2 g h}

Fluid MechanicsWater TreatmentPhysicsSpeed of fluid jetting from an opening a depth h below the free surface, with g = 9.80665 m/s².

Pressure Head (h = P/ρg)

h=Pρgh = \frac{P}{\rho g}

Fluid MechanicsWater TreatmentPhysicsConverts a pressure into the equivalent height of a fluid column, with g = 9.80665 m/s².

Velocity Head (h = v²/2g)

hv=v22gh_v = \frac{v^{2}}{2g}

Fluid MechanicsWater TreatmentPhysicsThe kinetic energy of a flow expressed as an equivalent column height, with g = 9.80665 m/s².

Reynolds Number

Re=ρvDμRe = \frac{\rho v D}{\mu}

Fluid MechanicsPhysicsThe dimensionless ratio of inertial to viscous forces that decides laminar versus turbulent flow.

Poiseuille's Law

Q=π ΔP r48μLQ = \frac{\pi \, \Delta P \, r^{4}}{8 \mu L}

Fluid MechanicsPhysicsLaminar flow rate through a round pipe — proportional to the fourth power of the radius.

Stokes' Drag (F = 6πμrv)

F=6πμrvF = 6\pi \mu r v

Fluid MechanicsPhysicsViscous drag on a small sphere creeping through a fluid at low Reynolds number.

Specific Gravity

SG=ρρwaterSG = \frac{\rho}{\rho_{water}}

Fluid MechanicsWater TreatmentChemistryDensity expressed as a multiple of water's 1000 kg/m³.

Hydraulic Power (P = ρgQh)

P=ρgQhP = \rho g Q h

Fluid MechanicsWater TreatmentPhysicsPower needed to lift a flow Q through a head h, with g = 9.80665 m/s².

Gauge and Absolute Pressure

Pabs=Pgauge+PatmP_{abs} = P_{gauge} + P_{atm}

Fluid MechanicsThermodynamicsPhysicsAbsolute pressure is the gauge reading plus the surrounding atmospheric pressure.

Sensible Heat (Q = mcΔT)

Q=mcΔTQ = m c \Delta T

ThermodynamicsPhysicsChemistryHeat needed to change a mass's temperature: mass times specific heat times the temperature change.

Boyle's Law

P1V1=P2V2P_1 V_1 = P_2 V_2

ThermodynamicsChemistryPhysicsAt constant temperature, pressure times volume stays constant for a fixed amount of gas.

Charles's Law

V1T1=V2T2\frac{V_1}{T_1} = \frac{V_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant pressure, gas volume is directly proportional to absolute temperature.

Gay-Lussac's Law

P1T1=P2T2\frac{P_1}{T_1} = \frac{P_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant volume, gas pressure is directly proportional to absolute temperature.

Combined Gas Law

P1V1T1=P2V2T2\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

ThermodynamicsChemistryPhysicsFor a fixed amount of gas, pressure times volume over absolute temperature stays constant between any two states.

Thermal Efficiency

η=WQh\eta = \frac{W}{Q_h}

ThermodynamicsPhysicsFraction of heat input that a heat engine converts into useful work.

Carnot Efficiency

η=1−TcTh\eta = 1 - \frac{T_c}{T_h}

ThermodynamicsPhysicsThe maximum possible efficiency of any heat engine operating between two absolute temperatures.

Stefan-Boltzmann Law

P=εσAT4P = \varepsilon \sigma A T^4

ThermodynamicsPhysicsPower radiated by a hot surface, using the Stefan–Boltzmann constant σ = 5.670374419×10⁻⁸ W/(m²·K⁴).

Latent Heat

Q=mLQ = m L

ThermodynamicsPhysicsHeat absorbed or released when a mass changes phase at constant temperature.

Thermal Linear Expansion

ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

ThermodynamicsPhysicsLength change of a solid caused by a temperature change, via the linear expansion coefficient.

Heat Conduction Rate

P=kAΔTdP = \tfrac{k A \Delta T}{d}

ThermodynamicsPhysicsSteady-state heat flow through a slab by Fourier's law of conduction.

Wave Speed (v = fλ)

v=fλv = f \lambda

Waves & OscillationsPhysicsThe universal wave equation: a wave's speed equals its frequency times its wavelength.

Period-Frequency Relation

T=1fT = \frac{1}{f}

Waves & OscillationsPhysicsPeriod and frequency are reciprocals: seconds per cycle versus cycles per second.

Simple Pendulum Period

T=2πLgT = 2\pi \sqrt{\frac{L}{g}}

Waves & OscillationsMechanicsPhysicsPeriod of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).

Fundamental Frequency of a String

f=v2Lf = \frac{v}{2L}

Waves & OscillationsPhysicsLowest standing-wave frequency of a string fixed at both ends.

Period of a Spring-Mass Oscillator

T=2πmkT = 2\pi \sqrt{\tfrac{m}{k}}

Waves & OscillationsMechanicsPhysicsPeriod of a mass bouncing on a spring, set only by the mass and the spring stiffness.

Sound Intensity (I = P/A)

I=PAI = \frac{P}{A}

Waves & OscillationsPhysicsSound intensity is the acoustic power passing through each square meter of surface.

Inverse-Square Law for Sound

I=P4πr2I = \frac{P}{4\pi r^{2}}

Waves & OscillationsPhysicsA point source's intensity falls with the square of distance as its power spreads over an expanding sphere.

Decibel Sound Level

β=10log⁡10 ⁣(II0)\beta = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Waves & OscillationsPhysicsSound level in decibels compares an intensity to the threshold of hearing, I₀ = 10⁻¹² W/m².

Doppler Effect (Approaching Source)

f′=fvv−vsf' = \frac{f v}{v - v_s}

Waves & OscillationsPhysicsObserved frequency rises when a sound source approaches, as each wavefront is emitted closer to the listener.

Doppler Effect (Approaching Observer)

f′=f(v+vo)vf' = \frac{f (v + v_o)}{v}

Waves & OscillationsPhysicsA listener moving toward a stationary source meets wavefronts more often and hears a higher frequency.

Wave Speed on a String

v=Fμv = \sqrt{\frac{F}{\mu}}

Waves & OscillationsPhysicsWaves travel faster on a tighter, lighter string: speed is the square root of tension over linear density.

Fundamental of a Closed Pipe

f=v4Lf = \frac{v}{4L}

Waves & OscillationsPhysicsA pipe closed at one end resonates with a quarter wavelength inside, giving a fundamental of v/4L.

Harmonic Frequencies

fn=nf1f_n = n f_1

Waves & OscillationsPhysicsThe nth harmonic of a vibrating system is n times its fundamental frequency.

Beat Frequency

fbeat=f1−f2f_{\text{beat}} = f_1 - f_2

Waves & OscillationsPhysicsTwo nearby tones interfere to produce a loudness pulse at their difference frequency, with f₁ the higher of the pair.

Speed of Sound in Air

v=331.3+0.606 TCv = 331.3 + 0.606\, T_C

Waves & OscillationsPhysicsThe speed of sound in dry air grows about 0.6 m/s for every degree Celsius above freezing.

Thin Lens Equation

1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

OpticsPhysicsRelates a thin lens's focal length to its object and image distances.

Lens Magnification (m = −d_i/d_o)

m=−didom = -\frac{d_i}{d_o}

OpticsPhysicsImage magnification from the ratio of image to object distance; the minus sign tracks inversion.

Magnification from Heights (m = h_i/h_o)

m=hihom = \frac{h_i}{h_o}

OpticsPhysicsMagnification as the ratio of image height to object height.

Focal Length of a Spherical Mirror

f=R2f = \frac{R}{2}

OpticsPhysicsA spherical mirror focuses parallel rays at half its radius of curvature.

Snell's Law of Refraction

n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2

OpticsPhysicsLight bends at an interface so that n sin θ stays the same on both sides.

Index of Refraction (n = c/v)

n=cvn = \frac{c}{v}

OpticsPhysicsHow much a medium slows light: the ratio of c to the speed of light in the medium.

Critical Angle for Total Internal Reflection

sin⁡θc=n2n1\sin\theta_c = \frac{n_2}{n_1}

OpticsPhysicsBeyond this angle of incidence, light in the denser medium reflects totally instead of refracting.

Brewster's Angle

tan⁡θB=n2n1\tan\theta_B = \frac{n_2}{n_1}

OpticsPhysicsThe incidence angle at which reflected light is completely polarized.

Lens Power in Diopters

P=1fP = \frac{1}{f}

OpticsPhysicsThe optician's unit: lens power in diopters is the reciprocal of focal length in meters.

Double-Slit Fringe Spacing

Δy=λLd\Delta y = \frac{\lambda L}{d}

OpticsWaves & OscillationsPhysicsSpacing between adjacent bright fringes on a screen a distance L behind two slits d apart.

Diffraction Grating Equation

mλ=dsin⁡θm \lambda = d \sin\theta

OpticsWaves & OscillationsPhysicsBright beams leave a grating at angles where the path difference d sin θ is a whole number of wavelengths.

Apparent Depth

d′=dnd' = \frac{d}{n}

OpticsPhysicsViewed from straight above, an object under water appears at depth d/n.

Thin-Film Constructive Interference (Bright Reflection)

2nt=(m+12)λ2 n t = \left(m + \tfrac{1}{2}\right)\lambda

OpticsWaves & OscillationsPhysicsBright-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a half-odd number of wavelengths.

Thin-Film Destructive Interference (Dark Reflection)

2nt=mλ2 n t = m \lambda

OpticsWaves & OscillationsPhysicsDark-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a whole number of wavelengths.

Ohm's Law

V=IRV = I R

Electricity & MagnetismPhysicsRelates the voltage across a conductor to the current through it and its resistance.

Electrical Power (P = VI)

P=VIP = V I

Electricity & MagnetismPhysicsPower delivered to a component as the product of the voltage across it and the current through it.

Electrical Power (P = I²R)

P=I2RP = I^{2} R

Electricity & MagnetismPhysicsPower dissipated as heat in a resistance carrying a current (Joule heating).

Electrical Power (P = V²/R)

P=V2RP = \frac{V^{2}}{R}

Electricity & MagnetismPhysicsPower dissipated in a resistance held at a fixed voltage.

Two Resistors in Series

Rt=R1+R2R_{t} = R_{1} + R_{2}

Electricity & MagnetismPhysicsTotal resistance of two resistors connected end to end is simply their sum.

Two Resistors in Parallel

Rt=R1R2R1+R2R_{t} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}

Electricity & MagnetismPhysicsTotal resistance of two resistors connected side by side: product over sum, always less than either branch.

Electric Charge (Q = It)

Q=ItQ = I t

Electricity & MagnetismPhysicsTotal charge transferred by a steady current flowing for a given time.

RC Time Constant

τ=RC\tau = R C

Electricity & MagnetismPhysicsCharacteristic charging/discharging time of a resistor-capacitor circuit.

RC Capacitor Discharge

V=V0 e−t/τV = V_{0} \, e^{-t/\tau}

Electricity & MagnetismPhysicsExponential decay of the voltage on a capacitor discharging through a resistor.

Electrical Energy (E = Pt)

E=PtE = P t

Electricity & MagnetismPhysicsEnergy consumed by a device drawing constant power over a period of time.

Coulomb's Law

F=ke q1q2r2F = \frac{k_e \, q_{1} q_{2}}{r^{2}}

Electricity & MagnetismPhysicsElectrostatic force between two point charges, with kₑ = 8.9875517923×10⁹ N·m²/C².

Capacitance (C = Q/V)

C=QVC = \frac{Q}{V}

Electricity & MagnetismPhysicsDefines capacitance as the charge stored per volt applied across a capacitor.

Energy Stored in a Capacitor

E=12CV2E = \tfrac{1}{2} C V^{2}

Electricity & MagnetismPhysicsEnergy banked in a capacitor's electric field from its capacitance and voltage.

Transformer Voltage Ratio

VsVp=NsNp\frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}}

Electricity & MagnetismPhysicsRelates the primary and secondary voltages of an ideal transformer to its turns ratio.

Magnetic Force on a Moving Charge

F=qvBsin⁡θF = q v B \sin\theta

Electricity & MagnetismPhysicsMagnitude of the magnetic (Lorentz) force on a charge moving through a magnetic field.

Magnetic Force on a Current-Carrying Wire

F=BILsin⁡θF = B I L \sin\theta

Electricity & MagnetismPhysicsMagnitude of the force on a straight current-carrying wire in a uniform magnetic field.

Two Capacitors in Series

Ct=C1C2C1+C2C_{t} = \frac{C_{1} C_{2}}{C_{1} + C_{2}}

Electricity & MagnetismPhysicsCombines two series capacitors into a total that is smaller than either one.

Two Capacitors in Parallel

Ct=C1+C2C_{t} = C_{1} + C_{2}

Electricity & MagnetismPhysicsCombines two parallel capacitors by simply adding their capacitances.

Magnetic Flux (Φ = BA cos θ)

Φ=BAcos⁡θ\Phi = B A \cos\theta

Electricity & MagnetismPhysicsThe magnetic field threading a surface: field times area times the cosine of the tilt angle.

Faraday's Law of Induction

ε=NΔΦΔt\varepsilon = N \frac{\Delta\Phi}{\Delta t}

Electricity & MagnetismPhysicsThe EMF induced in a coil of N turns by flux changing at ΔΦ/Δt (magnitude form).

Motional EMF (ε = BLv)

ε=BLv\varepsilon = B L v

Electricity & MagnetismPhysicsVoltage generated across a conductor of length L moving at speed v through a field B.

Magnetic Field of a Solenoid

B=μ0NILB = \frac{\mu_0 N I}{L}

Electricity & MagnetismPhysicsThe uniform field inside a long coil of N turns and length L carrying current I.

Energy Stored in an Inductor

E=12LI2E = \tfrac{1}{2} L I^{2}

Electricity & MagnetismPhysicsEnergy held in an inductor's magnetic field: half the inductance times current squared.

Inductive Reactance (X_L = 2πfL)

XL=2πfLX_L = 2\pi f L

Electricity & MagnetismPhysicsAn inductor's opposition to AC current, rising in proportion to frequency.

Capacitive Reactance (X_C = 1/2πfC)

XC=12πfCX_C = \frac{1}{2\pi f C}

Electricity & MagnetismPhysicsA capacitor's opposition to AC current, falling as frequency rises.

RL Time Constant (τ = L/R)

τ=LR\tau = \frac{L}{R}

Electricity & MagnetismPhysicsHow quickly current builds or decays in an inductor-resistor circuit.

LC Resonant Frequency

f=12πLCf = \frac{1}{2\pi\sqrt{LC}}

Electricity & MagnetismWaves & OscillationsPhysicsThe natural oscillation frequency of an inductor-capacitor pair.

RMS and Peak Voltage

Vrms=Vpeak2V_{rms} = \frac{V_{peak}}{\sqrt{2}}

Electricity & MagnetismPhysicsFor a sine wave, the effective (heating-equivalent) voltage is the peak divided by √2.

Resistance of a Wire (R = ρL/A)

R=ρLAR = \frac{\rho L}{A}

Electricity & MagnetismPhysicsResistance from material resistivity, length, and cross-sectional area.

Force Between Parallel Wires

F=μ0I1I2ℓ2πdF = \frac{\mu_0 I_1 I_2 \ell}{2\pi d}

Electricity & MagnetismPhysicsMagnetic force between two parallel current-carrying wires separated by distance d.

Photon Energy (E = hf)

E=hfE = h f

Modern PhysicsPhysicsThe energy of a single photon: Planck's constant times the light's frequency.

Photon Energy from Wavelength (E = hc/λ)

E=hcλE = \frac{h c}{\lambda}

Modern PhysicsPhysicsPhoton energy written in terms of wavelength: shorter waves, more energetic photons.

Photon Momentum (p = h/λ)

p=hλp = \frac{h}{\lambda}

Modern PhysicsPhysicsMassless but not momentum-less: a photon carries h divided by its wavelength.

De Broglie Wavelength

λ=hmv\lambda = \frac{h}{m v}

Modern PhysicsPhysicsEvery moving particle has a wavelength: Planck's constant over its momentum mv.

Mass-Energy Equivalence (E = mc²)

E=mc2E = m c^{2}

Modern PhysicsPhysicsThe rest energy locked in mass: multiply by the speed of light squared.

Photoelectric Effect

KEmax=Ephoton−ϕKE_{max} = E_{photon} - \phi

Modern PhysicsPhysicsMaximum kinetic energy of an ejected electron: photon energy (hf) minus the work function.

Lorentz Factor

γ=11−β2\gamma = \frac{1}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsThe relativistic stretch factor, written in terms of β = v/c.

Time Dilation

Δt=Δt01−β2\Delta t = \frac{\Delta t_0}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsA moving clock's proper time Δt₀ stretches to Δt for a stationary observer; β = v/c.

Length Contraction

L=L01−β2L = L_0 \sqrt{1 - \beta^{2}}

Modern PhysicsPhysicsA moving object's rest length L₀ contracts to L along its direction of motion; β = v/c.

Radioactive Activity (A = λN)

A=λNA = \lambda N

Modern PhysicsChemistryPhysicsDecays per second: the decay constant times the number of undecayed nuclei.

Wien's Displacement Law

λmax=bT\lambda_{max} = \frac{b}{T}

Modern PhysicsThermodynamicsPhysicsThe peak wavelength of thermal radiation, with b = 2.8978 × 10⁻³ m·K.

Bohr Model Energy Levels

En=−13.606 eVn2E_n = -\frac{13.606\ \mathrm{eV}}{n^{2}}

Modern PhysicsChemistryPhysicsEnergy of the hydrogen atom's nth level: −13.606 eV divided by n².

Orbital Velocity

v=GMrv = \sqrt{\frac{GM}{r}}

Astronomy & GravitationMechanicsPhysicsSpeed of a body in a circular orbit of radius r around a central mass M.

Orbital Period

T=2πr3GMT = 2\pi \sqrt{\frac{r^{3}}{GM}}

Astronomy & GravitationMechanicsPhysicsTime for one circular orbit of radius r around a central mass M — Kepler's third law in Newtonian form.

Escape Velocity

v=2GMrv = \sqrt{\frac{2GM}{r}}

Astronomy & GravitationMechanicsPhysicsMinimum launch speed needed to escape the gravity of a mass M starting from distance r, with no further propulsion.

Gravitational Field Strength

g=GMr2g = \frac{GM}{r^{2}}

Astronomy & GravitationMechanicsPhysicsGravitational acceleration produced by a mass M at distance r from its center.

Kepler's Third Law (Ratio Form)

T12T22=a13a23\frac{T_1^{2}}{T_2^{2}} = \frac{a_1^{3}}{a_2^{3}}

Astronomy & GravitationMechanicsPhysicsFor two bodies orbiting the same central mass, the squares of their periods are in the same ratio as the cubes of their orbital sizes.

Gravitational Potential Energy (Orbital)

U=−GMmrU = -\frac{GMm}{r}

Astronomy & GravitationMechanicsPhysicsGravitational potential energy of a mass m at distance r from a central mass M, taking zero at infinite separation.

Schwarzschild Radius

rs=2GMc2r_s = \frac{2GM}{c^{2}}

Astronomy & GravitationPhysicsRadius of the event horizon of a non-rotating black hole of mass M.

Apparent Brightness (Inverse-Square Law)

b=L4πd2b = \frac{L}{4\pi d^{2}}

Astronomy & GravitationPhysicsReceived flux from a source of luminosity L spread over a sphere of radius d — brightness falls with the square of distance.

Stellar Parallax Distance

d=1 AUpd = \frac{1\,\text{AU}}{p}

Astronomy & GravitationPhysicsDistance to a star from its annual parallax angle p, using the small-angle form of d = AU/tan p.

Ideal Gas Law

PV=nRTP V = n R T

ChemistryPhysicsRelates pressure, volume, amount, and temperature of a gas. R = 8.314 J/(mol·K).

Moles from Mass (n = m/M)

n=mMn = \frac{m}{M}

ChemistryConverts a measured mass into an amount of substance by dividing by the molar mass.

Molarity (C = n/V)

C=nVC = \frac{n}{V}

ChemistryDefines molar concentration as moles of solute per volume of solution.

Percent Yield

% yield=mactualmtheoretical×100%\%\,\text{yield} = \frac{m_{\text{actual}}}{m_{\text{theoretical}}} \times 100\%

ChemistryCompares the mass actually isolated from a reaction to the maximum mass stoichiometry predicts.

Percent Composition of an Element

%X=a MXMcompound×100%\%X = \frac{a\,M_X}{M_{\text{compound}}} \times 100\%

ChemistryGives the mass percent an element contributes to a compound from the formula subscript and molar masses.

Gas Density from Molar Mass

ρ=PMRT\rho = \frac{PM}{RT}

ChemistryThermodynamicsPhysicsGives an ideal gas's density from its molar mass, pressure, and absolute temperature using R = 8.314462618 J/(mol·K).

Graham's Law of Effusion

r1r2=M2M1\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}

ChemistryPhysicsRelates the effusion rates of two gases to the inverse square root of their molar masses.

Half-Life Decay

N=N0(12)t/t1/2N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}

ChemistryPhysicsGives the quantity remaining after repeated halvings over an elapsed time measured in half-lives.

Half-Life and Decay Constant

t1/2=ln⁡2λt_{1/2} = \frac{\ln 2}{\lambda}

ChemistryPhysicsConverts between a half-life and the exponential decay constant via the factor ln 2 = 0.6931471806.

Gas Volume at STP

V=n VmV = n\,V_m

ChemistryConverts between moles of an ideal gas and its volume at STP using the molar volume Vm = 22.414 L/mol.

pH from Hydrogen Ion Concentration

pH=−log⁡10 [H+]\mathrm{pH} = -\log_{10}\,[\mathrm{H^+}]

ChemistryWater TreatmentExpresses acidity as the negative base-10 logarithm of the hydrogen ion concentration in mol/L.

pH and pOH Relation

pH+pOH=14\mathrm{pH} + \mathrm{pOH} = 14

ChemistryLinks the acidity and basicity scales of any aqueous solution at 25 °C, where pKw = 14.

Dilution Equation (C1V1 = C2V2)

C1V1=C2V2C_1 V_1 = C_2 V_2

ChemistryWater TreatmentStates that concentration times volume is conserved when a solution is diluted, since the moles of solute do not change.

Particles from Moles (Avogadro's Number)

N=n NAN = n\,N_A

ChemistryConverts an amount in moles into an actual particle count using the Avogadro constant NA = 6.02214076 × 10²³ per mole.

Heat of Reaction

q=nΔHq = n \Delta H

ChemistryThermodynamicsScales a reaction's molar enthalpy change by the amount reacted to give the total heat released or absorbed.

Molality (b = n/m)

b=nmsolventb = \frac{n}{m_{\text{solvent}}}

ChemistryDefines molality as moles of solute per kilogram of solvent.

Mole Fraction

x1=n1n1+n2x_1 = \frac{n_1}{n_1 + n_2}

ChemistryGives the fraction of all moles in a two-component mixture contributed by the solute.

Osmotic Pressure (Π = MRT)

Π=MRT\Pi = M R T

ChemistryGives the osmotic pressure of a dilute solution from its molar concentration and absolute temperature using the van 't Hoff equation.

Raoult's Law

P=x P0P = x \, P^{0}

ChemistryGives the vapor pressure of a solvent above an ideal solution as its mole fraction times the pure solvent's vapor pressure.

Boiling-Point Elevation

ΔTb=Kb b\Delta T_b = K_b \, b

ChemistryGives how far a dissolved solute raises a solvent's boiling point from the molality and the solvent's ebullioscopic constant.

Freezing-Point Depression

ΔTf=Kf b\Delta T_f = K_f \, b

ChemistryGives how far a dissolved solute lowers a solvent's freezing point from the molality and the solvent's cryoscopic constant.

Mass Percent of a Solution

c=msolutemsolution×100%c = \frac{m_{\text{solute}}}{m_{\text{solution}}} \times 100\%

ChemistryExpresses a solution's concentration as the solute's share of the total solution mass in percent.

Beer–Lambert Law (A = εbc)

A=ε b cA = \varepsilon\, b\, c

ChemistryRelates a sample's absorbance to its molar absorptivity, path length, and concentration — the working equation of UV-Vis spectrophotometry.

Absorbance and Transmittance

A=−log⁡10 ⁣(%T100)A = -\log_{10}\!\left(\frac{\%T}{100}\right)

ChemistryConverts a spectrophotometer's percent transmittance into absorbance and back, the logarithmic bridge between what a detector sees and what Beer's law needs.

Henderson–Hasselbalch Equation (Weak Acid Buffer)

pH=pKa+log⁡10 ⁣[A−][HA]\mathrm{pH} = \mathrm{p}K_a + \log_{10}\!\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}

ChemistryGives the pH of a buffer from the acid's pKa and the ratio of conjugate base to undissociated acid, the master equation of buffer preparation.

Henderson–Hasselbalch Equation (Weak Base Buffer)

pOH=pKb+log⁡10 ⁣[BH+][B]\mathrm{pOH} = \mathrm{p}K_b + \log_{10}\!\frac{[\mathrm{BH^+}]}{[\mathrm{B}]}

ChemistryGives the pOH of a weak base buffer from the base's pKb and the ratio of conjugate acid to free base, the mirror image of the acid form.

pKa from Acid Dissociation Constant

pKa=−log⁡10Ka\mathrm{p}K_a = -\log_{10} K_a

ChemistryConverts an acid dissociation constant into its logarithmic pKa form and back, compressing a huge range of acid strengths onto one readable scale.

pKb from Base Dissociation Constant

pKb=−log⁡10Kb\mathrm{p}K_b = -\log_{10} K_b

ChemistryConverts a base dissociation constant into its logarithmic pKb form and back, the basic-side counterpart of the pKa scale.

Ka and Kb Relation through Kw

Ka Kb=KwK_a \, K_b = K_w

ChemistryLinks a weak acid's dissociation constant to that of its conjugate base through the ion product of water, Kw = 1.0 × 10⁻¹⁴ at 25 °C.

pH of a Weak Acid from Ka

pH=−log⁡10Ka C\mathrm{pH} = -\log_{10}\sqrt{K_a\,C}

ChemistryEstimates the pH of a weak monoprotic acid solution from its dissociation constant and formal concentration using the standard x-is-small approximation.

Percent Ionization of a Weak Acid

% ion=[H+]C×100%\%\,\text{ion} = \frac{[\mathrm{H^+}]}{C} \times 100\%

ChemistryExpresses what fraction of a weak acid has actually donated its proton, comparing the equilibrium hydrogen ion concentration to the formal concentration.

Solubility Product of a 1:1 Salt

Ksp=s2K_{sp} = s^{2}

ChemistryLinks the solubility product of an AB salt such as AgCl or BaSO4 to its molar solubility, since each formula unit releases one cation and one anion.

Solubility Product of an AB₂ Salt

Ksp=4s3K_{sp} = 4s^{3}

ChemistryLinks the solubility product of an AB2 or A2B salt such as CaF2 or Mg(OH)2 to its molar solubility, with the factor 4 from the doubled ion.

Kp from Kc (Kp = Kc(RT)^Δn)

Kp=Kc(RT)ΔnK_p = K_c (RT)^{\Delta n}

ChemistryThermodynamicsConverts a gas-phase equilibrium constant between pressure and concentration bases using the change in moles of gas, with R = 0.08206 L·atm/(mol·K).

Equilibrium Constant Kc (A + B ⇌ C + D)

Kc=[C][D][A][B]K_c = \frac{[\mathrm{C}][\mathrm{D}]}{[\mathrm{A}][\mathrm{B}]}

ChemistryComputes the equilibrium constant or reaction quotient for a one-to-one reaction from the four species concentrations, the law of mass action in its simplest form.

Nernst Equation

E=E∘−RTnFln⁡QE = E^{\circ} - \frac{RT}{nF}\ln Q

ChemistryCorrects a cell's standard potential for non-standard concentrations, the equation behind every pH meter, ion-selective probe, and battery voltage curve.

Standard Cell Potential from Half-Cells

Ecell∘=Ecathode∘−Eanode∘E^{\circ}_{\text{cell}} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}}

ChemistryBuilds a galvanic cell's standard voltage by subtracting the anode's standard reduction potential from the cathode's, using tabulated half-cell values.

Gibbs Free Energy Change (ΔG = ΔH − TΔS)

ΔG=ΔH−T ΔS\Delta G = \Delta H - T\,\Delta S

ChemistryThermodynamicsCombines a reaction's enthalpy and entropy changes at a given temperature to decide whether it can happen spontaneously.

Gibbs Free Energy and the Equilibrium Constant

ΔG∘=−RTln⁡K\Delta G^{\circ} = -RT\ln K

ChemistryThermodynamicsConverts between a reaction's standard free energy change and its equilibrium constant, the bridge joining thermodynamics to equilibrium tables.

Arrhenius Equation

k=A e−Ea/RTk = A\,e^{-E_a/RT}

ChemistryGives a reaction's rate constant from its activation energy, pre-exponential factor, and absolute temperature — the core law of chemical kinetics.

Arrhenius Two-Temperature Form

ln⁡k2k1=EaR(1T1−1T2)\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

ChemistryExtracts an activation energy from two rate constants measured at two temperatures, eliminating the pre-exponential factor entirely.

Activation Energy from an Arrhenius Plot

Ea=−R×slopeE_a = -R \times \text{slope}

ChemistryConverts the slope of a ln k versus 1/T Arrhenius plot into an activation energy, the standard graphical method for kinetics data.

Zero-Order Integrated Rate Law

[A]=[A]0−kt[\mathrm{A}] = [\mathrm{A}]_0 - kt

ChemistryGives the concentration remaining in a zero-order reaction, where the rate is constant and concentration falls in a straight line with time.

First-Order Integrated Rate Law

[A]=[A]0 e−kt[\mathrm{A}] = [\mathrm{A}]_0\,e^{-kt}

ChemistryGives the concentration remaining in a first-order reaction, the exponential decay that governs radioactive decay and most drug clearance.

Second-Order Integrated Rate Law

1[A]=1[A]0+kt\frac{1}{[\mathrm{A}]} = \frac{1}{[\mathrm{A}]_0} + kt

ChemistryGives the concentration remaining in a second-order reaction, where the reciprocal of concentration rises linearly with time.

Half-Life of a Second-Order Reaction

t1/2=1k [A]0t_{1/2} = \frac{1}{k\,[\mathrm{A}]_0}

ChemistryGives the half-life of a second-order reaction, which unlike a first-order half-life depends on the starting concentration.

Titration: Concentration of an Unknown

Ca=n CbVbVaC_a = \frac{n\,C_b V_b}{V_a}

ChemistryFinds an analyte's concentration from the titre volume, titrant concentration, and the reaction's mole ratio at the equivalence point.

Normality from Molarity

N=M×neqN = M \times n_{\text{eq}}

ChemistryConverts molarity into normality by multiplying by the number of reactive equivalents each mole of solute supplies.

Partial Pressure from Mole Fraction

Pi=xi PtotalP_i = x_i \, P_{\text{total}}

ChemistryThermodynamicsGives a gas component's partial pressure as its mole fraction times the total pressure, the practical form of Dalton's law of partial pressures.

Hess's Law (Three-Step Sum)

ΔHrxn=ΔH1+ΔH2+ΔH3\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2 + \Delta H_3

ChemistryThermodynamicsHess's law: the enthalpy change of a target reaction is the sum of the enthalpy changes of the steps you route it through.

Standard Enthalpy of Reaction from Formation Enthalpies

ΔHrxn∘=∑ΔHf,prod∘−∑ΔHf,react∘\Delta H^{\circ}_{\text{rxn}} = \sum \Delta H^{\circ}_{f,\text{prod}} - \sum \Delta H^{\circ}_{f,\text{react}}

ChemistryThermodynamicsThe tabulated form of Hess's law: standard enthalpy of reaction equals the summed formation enthalpies of the products minus those of the reactants.

Reaction Quotient Q (aA + bB ⇌ cC)

Q=[C]c[A]a [B]bQ = \frac{[\mathrm{C}]^{c}}{[\mathrm{A}]^{a}\,[\mathrm{B}]^{b}}

ChemistryReaction quotient Q from any concentrations, raised to stoichiometric powers — the number you compare with K to predict which way a mixture shifts.

Faraday's Law of Electrolysis (m = QM/nF)

m=QMnFm = \frac{Q M}{n F}

ChemistryFaraday's law of electrolysis: the mass plated out at an electrode from the charge passed, the molar mass, and the electrons transferred per ion.

Quadratic Formula (Positive Root)

x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

AlgebraThe larger of the two solutions of ax² + bx + c = 0, taking the plus branch of the square root in the quadratic formula.

Quadratic Formula (Negative Root)

x=−b−b2−4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

AlgebraThe smaller of the two solutions of ax² + bx + c = 0, taking the minus branch of the square root in the quadratic formula.

Discriminant of a Quadratic

Δ=b2−4ac\Delta = b^2 - 4ac

AlgebraThe quantity b² − 4ac under the square root, which reveals how many real roots a quadratic has before you solve it.

Sum of the Roots of a Quadratic

S=−baS = -\frac{b}{a}

AlgebraVieta's relation giving the sum of a quadratic's two roots directly from its coefficients, without solving the equation.

Product of the Roots of a Quadratic

P=caP = \frac{c}{a}

AlgebraVieta's relation giving the product of a quadratic's two roots directly from its coefficients, without solving the equation.

Vertex x-Coordinate of a Parabola

h=−b2ah = -\frac{b}{2a}

AlgebraGeometryThe x-coordinate of a parabola's vertex, sitting midway between the two roots and marking the axis of symmetry.

Vertex y-Coordinate of a Parabola

k=c−b24ak = c - \frac{b^2}{4a}

AlgebraGeometryThe y-coordinate of a parabola's vertex — its minimum value when a is positive, and its maximum when a is negative.

Midpoint Formula

xm=x1+x22x_m = \frac{x_1 + x_2}{2}

AlgebraGeometryThe coordinate halfway between two endpoints along one axis; apply it to x and again to y for the midpoint of a segment.

Circle Radius from Centre and a Point

r=(x−h)2+(y−k)2r = \sqrt{(x - h)^2 + (y - k)^2}

AlgebraGeometryRadius of a circle passing through a given point, measured from its centre using the distance formula inside the circle equation.

Distance from a Point to a Line

d=∣Ax0+By0+C∣A2+B2d = \frac{\left| A x_0 + B y_0 + C \right|}{\sqrt{A^2 + B^2}}

AlgebraGeometryPerpendicular distance from a point to a line written in general form Ax + By + C = 0, the shortest gap between the two.

Point-Slope Form of a Line

y=y1+m(x−x1)y = y_1 + m(x - x_1)

AlgebraGeometryEquation of a straight line through one known point with a known slope, giving the y-value at any x you care to choose.

Slope from Standard Form of a Line

m=−ABm = -\frac{A}{B}

AlgebraGeometrySlope of a line written in standard form Ax + By = C, read straight off the two coefficients without rearranging anything.

x-Intercept of a Line

xint=−bmx_{\text{int}} = -\frac{b}{m}

AlgebraGeometryWhere a line in slope-intercept form crosses the x-axis, found by setting y to zero and solving the remaining equation for x.

Perpendicular Slope Relation

m2=−1m1m_2 = -\frac{1}{m_1}

AlgebraGeometrySlope of a line perpendicular to a given line — the negative reciprocal, so that the two slopes always multiply to −1.

Logarithm Change of Base

log⁡bx=ln⁡xln⁡b\log_b x = \frac{\ln x}{\ln b}

AlgebraValue of a logarithm in any base, rewritten with natural logarithms so an ordinary calculator can evaluate it directly.

Logarithm of a Product

log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y

AlgebraThe logarithm of a product equals the sum of the logarithms, the identity that once turned multiplication into simple addition.

Logarithm of a Quotient

log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y

AlgebraThe logarithm of a quotient equals the difference of the logarithms, turning a division problem into a simple subtraction.

Logarithm of a Power

log⁡b ⁣(xn)=nlog⁡bx\log_b\!\left(x^{n}\right) = n \log_b x

AlgebraThe logarithm of a power pulls the exponent out in front, the identity that unlocks any equation with the unknown in an exponent.

Solve an Exponential Equation for the Exponent

x=ln⁡(y/a)ln⁡bx = \frac{\ln (y / a)}{\ln b}

AlgebraSolves a·bˣ = y for the unknown exponent x using logarithms, the standard move behind every growth and decay problem.

Power of a Power

(bm)n=bmn\left( b^{m} \right)^{n} = b^{mn}

AlgebraRaising a power to another power multiplies the two exponents, the exponent law behind every nested exponential expression.

nth Root as a Fractional Exponent

y=xn=x1/ny = \sqrt[n]{x} = x^{1/n}

AlgebraThe nth root of a number written as a fractional exponent, so that roots and powers obey exactly the same set of rules.

Binomial Coefficient (n Choose k)

(nk)=n!k! (n−k)!\binom{n}{k} = \frac{n!}{k! \, (n-k)!}

AlgebraThe number of ways to choose k items from n, which is also the coefficient of a single term in a binomial expansion.

Completing the Square: the Constant Needed

k=(b2)2k = \left( \frac{b}{2} \right)^{2}

AlgebraThe constant you add to x² + bx to turn it into a perfect square trinomial, the pivotal step in completing the square.

Arithmetic Mean of Two Numbers

A=a+b2A = \frac{a + b}{2}

AlgebraThe ordinary average of two numbers, the balance point sitting exactly halfway between them on the number line.

Geometric Mean of Two Numbers

G=abG = \sqrt{ab}

AlgebraGeometryThe square root of the product of two numbers, the average that suits growth rates, ratios and scale factors of any kind.

Harmonic Mean of Two Numbers

H=2aba+bH = \frac{2ab}{a + b}

AlgebraThe reciprocal-based average of two numbers, the right average for rates such as speeds covering equal distances.

Inverse Variation

y=kxy = \frac{k}{x}

AlgebraRelation in which y falls as x rises so that their product stays at a constant k, the shape of Boyle's law and gear ratios.

Cramer's Rule for x (2×2 System)

x=c1b2−b1c2a1b2−b1a2x = \frac{c_1 b_2 - b_1 c_2}{a_1 b_2 - b_1 a_2}

AlgebraCramer's rule for the x that solves a pair of simultaneous linear equations, written as a ratio of two determinants.

Cramer's Rule for y (2×2 System)

y=a1c2−c1a2a1b2−b1a2y = \frac{a_1 c_2 - c_1 a_2}{a_1 b_2 - b_1 a_2}

AlgebraCramer's rule for the y that solves a pair of simultaneous linear equations, written as a ratio of two determinants.

Magnitude of a 2D Vector

∣v⃗∣=vx2+vy2|\vec{v}| = \sqrt{v_x^2 + v_y^2}

Vectors & MatricesAlgebraGeometryFinds the length of a two-dimensional vector from its x- and y-components, the Pythagorean theorem written for arrows instead of triangles.

Magnitude of a 3D Vector

∣v⃗∣=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Vectors & MatricesAlgebraGeometryFinds the length of a three-dimensional vector from its x-, y-, and z-components, extending Pythagoras into space.

Direction Angle of a 2D Vector

θ=atan2⁡(vy,  vx)\theta = \operatorname{atan2}(v_y,\; v_x)

Vectors & MatricesTrigonometryAlgebraGives the direction a two-dimensional vector points, measured counterclockwise from the positive x-axis, from its two components.

x-Component from Magnitude and Angle

vx=∣v⃗∣cos⁡θv_x = |\vec{v}| \cos\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its horizontal part from the vector's length and the angle it makes with the positive x-axis.

y-Component from Magnitude and Angle

vy=∣v⃗∣sin⁡θv_y = |\vec{v}| \sin\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its vertical part from the vector's length and the angle it makes with the positive x-axis.

Unit Vector Component (Normalization)

u^x=vx∣v⃗∣\hat{u}_x = \frac{v_x}{|\vec{v}|}

Vectors & MatricesAlgebraScales one component of a vector down to the matching component of the unit vector that points the same way, by dividing by the magnitude.

Dot Product of Two 2D Vectors (Components)

a⃗⋅b⃗=axbx+ayby\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y

Vectors & MatricesAlgebraMultiplies two plane vectors component by component and adds the results, giving the scalar that measures how much they share a direction.

Dot Product of Two 3D Vectors (Components)

a⃗⋅b⃗=axbx+ayby+azbz\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z

Vectors & MatricesAlgebraMultiplies two space vectors component by component and sums the three products, the standard test for perpendicularity in three dimensions.

Dot Product from Magnitudes and Included Angle

a⃗⋅b⃗=∣a⃗∣ ∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta

Vectors & MatricesTrigonometryPhysicsGives the scalar product of two vectors from their lengths and the angle between them, the geometric face of the dot product.

Angle Between Two 2D Vectors (Components)

θ=arccos⁡(axbx+aybyax2+ay2 bx2+by2)\theta = \arccos\left(\frac{a_x b_x + a_y b_y}{\sqrt{a_x^2+a_y^2}\,\sqrt{b_x^2+b_y^2}}\right)

Vectors & MatricesTrigonometryAlgebraFinds the angle separating two plane vectors directly from their four components, by way of the normalized dot product.

Cross Product Magnitude

∣a⃗×b⃗∣=∣a⃗∣ ∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta

Vectors & MatricesTrigonometryPhysicsGives the length of the cross product of two vectors from their magnitudes and the angle between them, equal to the area they span.

Cross Product z-Component of Two 2D Vectors

(a⃗×b⃗)z=axby−aybx(\vec{a}\times\vec{b})_z = a_x b_y - a_y b_x

Vectors & MatricesAlgebraGeometryComputes the signed out-of-plane cross product of two plane vectors from their components, whose sign reveals their turning direction.

Scalar Projection of One Vector onto Another

compb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Vectors & MatricesAlgebraPhysicsMeasures how far a vector reaches along the direction of another, the length of the shadow it casts on that second vector.

Resultant of Two Vectors at an Angle

R=A2+B2+2ABcos⁡θR = \sqrt{A^2 + B^2 + 2AB\cos\theta}

Vectors & MatricesTrigonometryPhysicsFinds the magnitude of the sum of two vectors from their lengths and the angle between them, the parallelogram rule in one equation.

Vector Subtraction (Component Form)

cx=ax−bxc_x = a_x - b_x

Vectors & MatricesAlgebraPhysicsSubtracts one vector from another one component at a time, the operation behind every relative-velocity and displacement-change problem.

Magnitude of a Scaled Vector

∣kv⃗∣=∣k∣ ∣v⃗∣|k\vec{v}| = |k|\,|\vec{v}|

Vectors & MatricesAlgebraGives the length of a vector after multiplying it by a scalar, which stretches or shrinks the arrow without ever bending it.

Determinant of a 2×2 Matrix

D=ad−bcD = ad - bc

Vectors & MatricesAlgebraComputes the determinant of a two-by-two matrix as the difference of its diagonal products, the number that decides invertibility.

Determinant of a 3×3 Matrix

D=a11(a22a33−a23a32)−a12(a21a33−a23a31)+a13(a21a32−a22a31)D = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

Vectors & MatricesAlgebraComputes the determinant of a three-by-three matrix by cofactor expansion along the first row, using all nine entries.

Parallelogram Area from Two Vectors

A=∣axby−aybx∣A = \left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the parallelogram spanned by two plane vectors directly from their four components, no angle or height required.

Triangle Area from Two Vectors

A=12∣axby−aybx∣A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the triangle formed by two plane vectors from a shared vertex, using only their components — the shoelace formula.

Scalar Triple Product (Parallelepiped Volume)

V=∣a⃗⋅(b⃗×c⃗)∣V = \left|\vec{a}\cdot(\vec{b}\times\vec{c})\right|

Vectors & MatricesGeometryAlgebraFinds the volume of the parallelepiped spanned by three space vectors, from the nine components, as the size of their scalar triple product.

Inverse of a 2×2 Matrix (Top-Left Element)

(M−1)11=dad−bc(M^{-1})_{11} = \frac{d}{ad - bc}

Vectors & MatricesAlgebraGives the top-left entry of the inverse of a two-by-two matrix: the opposite diagonal entry divided by the determinant.

Trace of a 2×2 Matrix

tr⁡(M)=a11+a22\operatorname{tr}(M) = a_{11} + a_{22}

Vectors & MatricesAlgebraAdds the two entries on the main diagonal of a square matrix, a quantity unchanged by any change of basis.

Work from Force and Displacement Components

W=Fxdx+FydyW = F_x d_x + F_y d_y

Vectors & MatricesPhysicsMechanicsComputes the work done by a force from the components of the force and the displacement, without needing the angle between them.

Z-Score (Standard Score)

z=x−μσz = \frac{x - \mu}{\sigma}

StatisticsAlgebraHow many standard deviations a value sits above or below the mean, turning any measurement into a comparable standard score.

Variance and Standard Deviation

σ2=σ⋅σ\sigma^2 = \sigma \cdot \sigma

StatisticsAlgebraConverts between variance and standard deviation: the variance is the square of the standard deviation, and the deviation is its square root.

Coefficient of Variation

CV=sxˉCV = \frac{s}{\bar{x}}

StatisticsAlgebraRelative variability: the standard deviation expressed as a fraction of the mean, so spreads measured on different scales can be compared.

Standard Error of the Mean

SE=σnSE = \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraHow much a sample mean typically wanders from the true mean, shrinking with the square root of the sample size.

Margin of Error for a Mean

E=zσnE = z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraHalf-width of a confidence interval for a mean, built from the critical z-value, the standard deviation and the sample size.

Confidence Interval Lower Limit

L=xˉ−zσnL = \bar{x} - z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraThe lower bound of a confidence interval for a mean, pulling the critical z-value and standard error back from the sample mean.

Confidence Interval Upper Limit

U=xˉ+zσnU = \bar{x} + z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraThe upper bound of a confidence interval for a mean, adding the critical z-value times the standard error to the sample mean.

Sample Size for a Mean

n=(zσE)2n = \left( \frac{z \sigma}{E} \right)^{2}

StatisticsAlgebraHow many observations a study needs to estimate a mean within a target margin of error at a chosen confidence level.

Sample Size for a Proportion

n=z2 p(1−p)E2n = \frac{z^{2} \, p (1 - p)}{E^{2}}

StatisticsAlgebraHow many respondents a survey needs to estimate a percentage within a target margin of error at a chosen confidence level.

Standard Error of a Proportion

SE=p(1−p)nSE = \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraHow much a sample percentage typically wanders from the true population proportion, given the proportion and the sample size.

Margin of Error for a Proportion

E=zp(1−p)nE = z \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraThe plus-or-minus quoted with a poll result, built from the critical z-value, the sample proportion and the number of respondents.

One-Sample Z-Test Statistic

z=xˉ−μσ/nz = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed population mean when the population standard deviation is known, in standard-error units.

One-Sample T-Test Statistic

t=xˉ−μs/nt = \frac{\bar{x} - \mu}{s / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed value when the standard deviation is estimated from the sample itself rather than known.

Two-Sample Z-Test Statistic

z=xˉ1−xˉ2σ12n1+σ22n2z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{\sigma_1^{2}}{n_1} + \dfrac{\sigma_2^{2}}{n_2}}}

StatisticsAlgebraCompares two independent sample means when both population standard deviations are known, scaled by the combined standard error.

Pooled Standard Deviation

sp=(n1−1)s12+(n2−1)s22n1+n2−2s_p = \sqrt{\frac{(n_1 - 1) s_1^{2} + (n_2 - 1) s_2^{2}}{n_1 + n_2 - 2}}

StatisticsAlgebraCombines two sample standard deviations into a single estimate of common spread, weighting each by its degrees of freedom.

Degrees of Freedom (One-Sample t)

df=n−1df = n - 1

StatisticsAlgebraThe degrees of freedom used to look up a critical value for a one-sample t-test or confidence interval, one fewer than the sample size.

Cohen's d (Effect Size)

d=xˉ1−xˉ2spd = \frac{\bar{x}_1 - \bar{x}_2}{s_p}

StatisticsAlgebraStandardised effect size: the gap between two group means measured in pooled standard deviations rather than raw units.

Range (Max minus Min)

R=xmax⁡−xmin⁡R = x_{\max} - x_{\min}

StatisticsAlgebraThe simplest measure of spread in a data set: the distance from the smallest observation to the largest.

Midrange

M=xmax⁡+xmin⁡2M = \frac{x_{\max} + x_{\min}}{2}

StatisticsAlgebraThe midpoint between the largest and smallest observations, a quick centre estimate computed from just the two extremes.

Interquartile Range (IQR)

IQR=Q3−Q1IQR = Q_3 - Q_1

StatisticsAlgebraThe width of the middle half of a data set, from the first quartile to the third, and the spread measure box plots are built on.

Outlier Lower Fence

LF=Q1−1.5 IQRLF = Q_1 - 1.5 \, IQR

StatisticsAlgebraTukey's lower cutoff for outliers: any observation below one and a half interquartile ranges under the first quartile is flagged.

Outlier Upper Fence

UF=Q3+1.5 IQRUF = Q_3 + 1.5 \, IQR

StatisticsAlgebraTukey's upper cutoff for outliers: any observation above one and a half interquartile ranges beyond the third quartile is flagged.

Regression Slope from Correlation

b=rsysxb = r \frac{s_y}{s_x}

StatisticsAlgebraThe least-squares slope of a regression line, recovered from the correlation coefficient and the two standard deviations.

Regression Line Intercept

a=yˉ−bxˉa = \bar{y} - b \bar{x}

StatisticsAlgebraThe y-intercept of a least-squares line, fixed by the requirement that the line pass through the point of averages.

Predicted Value from a Regression Line

y^=a+bx\hat{y} = a + b x

StatisticsAlgebraReads a prediction off a fitted least-squares line for any chosen value of the predictor, given the intercept and slope.

Coefficient of Determination (R²)

R2=r2R^{2} = r^{2}

StatisticsAlgebraThe share of variation in y explained by a simple linear regression, obtained by squaring the correlation coefficient.

Chi-Square Contribution of One Cell

χcell2=(O−E)2E\chi^{2}_{\text{cell}} = \frac{(O - E)^{2}}{E}

StatisticsAlgebraHow much a single cell of a contingency or goodness-of-fit table adds to the chi-square statistic, from its observed and expected counts.

Normal Probability Density

f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}

StatisticsAlgebraThe height of the normal bell curve at a given value, set by the distance from the mean in standard deviations.

Percent Error

PE=∣xmeas−xacc∣xaccPE = \frac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}

StatisticsAlgebraHow far a measurement strays from the accepted value, expressed as a fraction of that accepted value for lab reports and calibration checks.

Percent Difference

PD=∣x1−x2∣(x1+x2)/2PD = \frac{|x_1 - x_2|}{(x_1 + x_2)/2}

StatisticsAlgebraCompares two measurements of equal standing by dividing their gap by their average, when neither counts as the accepted value.

Weighted Mean of Two Groups

xˉ=n1xˉ1+n2xˉ2n1+n2\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

StatisticsAlgebraCombines the averages of two groups into one overall mean, weighting each group by how many observations it contains.

Classical Probability

P=fnP = \frac{f}{n}

ProbabilityStatisticsProbability of an event as the number of favourable outcomes divided by the total number of equally likely outcomes.

Complement Rule

P(Ac)=1−P(A)P(A^{c}) = 1 - P(A)

ProbabilityStatisticsThe chance an event does not happen is one minus the chance it does, because every trial must end in one case or the other.

Addition Rule (Mutually Exclusive Events)

P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

ProbabilityStatisticsFor events that cannot both happen, the chance that either one occurs is simply the sum of their separate probabilities.

General Addition Rule

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

ProbabilityStatisticsThe chance that either of two events happens, correcting the simple sum by subtracting the overlap that would be counted twice.

Multiplication Rule (Independent Events)

P(A∩B)=P(A) P(B)P(A \cap B) = P(A) \, P(B)

ProbabilityStatisticsWhen one event has no influence on the other, the chance that both occur is the product of their separate probabilities.

Conditional Probability

P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}

ProbabilityStatisticsThe chance of A once B is known to have happened, found by rescaling the overlap to the reduced sample space B.

General Multiplication Rule

P(A∩B)=P(A) P(B∣A)P(A \cap B) = P(A) \, P(B \mid A)

ProbabilityStatisticsChance that both events happen when the second depends on the first, as in drawing two cards without replacement from a deck.

Bayes' Theorem (Two Hypotheses)

P(A∣B)=P(B∣A) P(A)P(B∣A) P(A)+P(B∣Ac) (1−P(A))P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B \mid A) \, P(A) + P(B \mid A^{c}) \, \left(1 - P(A)\right)}

ProbabilityStatisticsUpdates a prior belief into a posterior after evidence arrives, weighing the true-positive rate against the false-positive rate.

Odds and Probability

O=P1−PO = \frac{P}{1 - P}

ProbabilityStatisticsConverts between a probability and odds in favour, the ratio of the chance it happens to the chance it does not.

Factorial

n!=n×(n−1)×⋯×2×1n! = n \times (n-1) \times \cdots \times 2 \times 1

ProbabilityAlgebraThe product of every whole number from n down to 1, counting the ways that n distinct objects can be arranged in order.

Permutations (nPr)

nPr=n!(n−r)!{}_{n}P_{r} = \frac{n!}{(n-r)!}

ProbabilityAlgebraNumber of ways to choose r items from n distinct items when the order of the selection matters, as with podium finishes.

Combinations (nCr)

nCr=(nr)=n!r! (n−r)!{}_{n}C_{r} = \binom{n}{r} = \frac{n!}{r! \, (n-r)!}

ProbabilityAlgebraNumber of ways to choose r items from n distinct items when the order does not matter, as with poker hands or lottery tickets.

Arrangements with Repetition

N=nrN = n^{r}

ProbabilityAlgebraNumber of sequences of length r drawn from n available choices when repeats are allowed, as with PIN codes and passwords.

Binomial Probability

P(X=k)=(nk)pk(1−p) n−kP(X = k) = \binom{n}{k} p^{k} (1 - p)^{\,n-k}

ProbabilityStatisticsChance of exactly k successes in n independent trials that each succeed with the same fixed probability p, as with coin tosses.

Binomial Distribution Mean

μ=np\mu = n p

ProbabilityStatisticsExpected number of successes across n independent trials that each succeed with probability p — the mean of the binomial distribution.

Binomial Distribution Variance

σ2=np(1−p)\sigma^{2} = n p (1 - p)

ProbabilityStatisticsSpread of the number of successes across n independent trials, at its largest when the per-trial chance p sits at one half.

Geometric Distribution (First Success)

P(X=k)=(1−p) k−1pP(X = k) = (1 - p)^{\,k-1} p

ProbabilityStatisticsChance that the first success in a run of repeated independent trials arrives exactly on trial number k, after k - 1 failures.

Expected Trials Until First Success

E[X]=1pE[X] = \frac{1}{p}

ProbabilityStatisticsAverage number of independent attempts needed before the first success when each attempt succeeds with probability p.

Poisson Probability

P(X=k)=λke−λk!P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}

ProbabilityStatisticsChance of exactly k events in a fixed interval when events occur independently at a constant average rate lambda.

Expected Value of a Bet

E=p W−(1−p) LE = p \, W - (1 - p) \, L

ProbabilityStatisticsAverage profit per play of a two-outcome wager that pays W with probability p and costs L the rest of the time, over many plays.

Probability of At Least One Success

P=1−(1−p)nP = 1 - (1 - p)^{n}

ProbabilityStatisticsChance that at least one of n independent attempts succeeds, found as one minus the chance that every single attempt fails.

Birthday Problem (All Distinct)

P=N!(N−n)!  N nP = \frac{N!}{(N - n)! \; N^{\,n}}

ProbabilityStatisticsChance that n independent picks from N equally likely options are all different, the engine behind the birthday paradox.

Hypergeometric Probability

P(X=k)=(Kk)(N−Kn−k)(Nn)P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}

ProbabilityStatisticsChance of drawing exactly k successes in a sample of n taken without replacement from a population of N holding K successes.

Projectile Range on Level Ground

R=v02sin⁡2θgR = \frac{v_0^{2} \sin 2\theta}{g}

MechanicsPhysicsHorizontal distance a projectile covers over level ground, from its launch speed and angle, ignoring air resistance.

Projectile Maximum Height

H=v02sin⁡2θ2gH = \frac{v_0^{2} \sin^{2}\theta}{2g}

MechanicsPhysicsPeak height reached by a projectile launched at a given speed and angle above level ground, ignoring air resistance.

Projectile Time of Flight

T=2v0sin⁡θgT = \frac{2 v_0 \sin\theta}{g}

MechanicsPhysicsTotal time a projectile stays airborne before returning to its launch height, set by launch speed and angle with g = 9.80665 m/s².

Horizontal Velocity Component

vx=vcos⁡θv_x = v \cos\theta

MechanicsPhysicsHorizontal component of a projectile's launch velocity — the part of the speed that carries it downrange at a constant rate.

Vertical Velocity Component

vy=vsin⁡θv_y = v \sin\theta

MechanicsPhysicsVertical component of a projectile's launch velocity — the part of the speed that fights gravity and sets the time aloft.

Drop Height of a Horizontally Launched Projectile

y=12gt2y = \tfrac{1}{2} g t^{2}

MechanicsPhysicsDistance a horizontally launched projectile falls in a given time, independent of how fast it was thrown sideways.

Kinetic Friction Force (f = μₖN)

fk=μkNf_k = \mu_k N

MechanicsPhysicsFriction force resisting a sliding surface, equal to the coefficient of kinetic friction times the normal force.

Maximum Static Friction (f = μₛN)

fs,max⁡=μsNf_{s,\max} = \mu_s N

MechanicsPhysicsLargest static friction force available before an object breaks loose and slides, from the static coefficient and normal force.

Angle of Repose (μ = tan θ)

μs=tan⁡θ\mu_s = \tan\theta

MechanicsPhysicsSteepest angle a surface can be tilted before an object slides, where the coefficient of static friction equals tan θ.

Normal Force on an Incline (N = mg cos θ)

N=mgcos⁡θN = m g \cos\theta

MechanicsPhysicsNormal force pressing a resting mass into an incline, equal to the component of its weight perpendicular to the slope.

Weight Component Along an Incline (mg sin θ)

F∥=mgsin⁡θF_{\parallel} = m g \sin\theta

MechanicsPhysicsComponent of an object's weight acting down the slope of an incline — the force that drives it toward the bottom.

Acceleration Down a Frictionless Incline

a=gsin⁡θa = g \sin\theta

MechanicsPhysicsAcceleration of an object sliding freely down a frictionless incline, set only by gravity and the slope angle.

Acceleration Down an Incline with Friction

a=g(sin⁡θ−μkcos⁡θ)a = g\left(\sin\theta - \mu_k \cos\theta\right)

MechanicsPhysicsAcceleration of an object sliding down an incline once kinetic friction opposes the motion, from the slope angle and μₖ.

Terminal Velocity

vt=2mgρACdv_t = \sqrt{\frac{2 m g}{\rho A C_d}}

MechanicsPhysicsSteady falling speed at which drag balances weight, from mass, air density, frontal area, and the drag coefficient.

Drag Force (F = ½CdρAv²)

FD=12CdρAv2F_D = \tfrac{1}{2} C_d \rho A v^{2}

MechanicsPhysicsAerodynamic drag on a body moving through a fluid, growing with the square of speed and with frontal area and density.

Banked Curve Angle

θ=arctan⁡ ⁣(v2rg)\theta = \arctan\!\left(\frac{v^{2}}{r g}\right)

MechanicsPhysicsBank angle that lets a vehicle round a curve of a given radius at a given speed with no reliance on sideways friction.

Maximum Speed on a Flat Curve

vmax⁡=μsgrv_{\max} = \sqrt{\mu_s g r}

MechanicsPhysicsFastest a vehicle can round a flat, unbanked curve before friction can no longer supply the centripetal force.

Work–Energy Theorem

W=12m(v2−v02)W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)

MechanicsPhysicsNet work done on an object equals its change in kinetic energy, linking force and distance to a change in speed.

Conservation of Momentum (Two Bodies)

m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

MechanicsPhysicsConservation of linear momentum in a two-body collision, solving any one mass or velocity from the other five.

Perfectly Inelastic Collision

v=m1u1+m2u2m1+m2v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

MechanicsPhysicsCommon velocity of two bodies that stick together after a perfectly inelastic collision, from conservation of momentum.

Elastic Collision — Final Velocity of Body 1

v1=(m1−m2)u1+2m2u2m1+m2v_1 = \frac{\left(m_1 - m_2\right) u_1 + 2 m_2 u_2}{m_1 + m_2}

MechanicsPhysicsFinal velocity of the first body in a one-dimensional elastic collision, where both momentum and kinetic energy survive.

Coefficient of Restitution

e=v2−v1u1−u2e = \frac{v_2 - v_1}{u_1 - u_2}

MechanicsPhysicsRatio of separation speed to approach speed in a collision, measuring how much of the relative motion survives impact.

Bounce Height from Coefficient of Restitution

h2=e2h1h_2 = e^{2} h_1

MechanicsPhysicsHeight a dropped ball rebounds to, from the drop height and the coefficient of restitution of the bounce.

SHM Displacement at Time t

x=Acos⁡(ωt)x = A \cos\left(\omega t\right)

MechanicsPhysicsDisplacement of a simple harmonic oscillator at time t, a cosine of amplitude A and angular frequency ω released from full stretch.

SHM Maximum Velocity

vmax⁡=Aωv_{\max} = A \omega

MechanicsPhysicsMaximum speed of a simple harmonic oscillator, reached at the equilibrium point, equal to amplitude times angular frequency.

SHM Maximum Acceleration

amax⁡=Aω2a_{\max} = A \omega^{2}

MechanicsPhysicsMaximum acceleration of a simple harmonic oscillator, reached at the turning points where the restoring force is largest.

Mechanical Advantage of a Lever

MA=dedlMA = \frac{d_e}{d_l}

MechanicsPhysicsMechanical advantage of a lever as the ratio of effort arm to load arm, showing how much the lever multiplies force.

Pulley System Effort Force

F=WnF = \frac{W}{n}

MechanicsPhysicsEffort force needed to lift a load with a pulley system, divided down by the number of rope sections supporting the load.

Gear Ratio

GR=NoutNinGR = \frac{N_{out}}{N_{in}}

MechanicsPhysicsGear ratio of a meshing pair as the driven gear's tooth count divided by the driver's, setting the torque and speed trade.

Machine Efficiency

η=WoutWin\eta = \frac{W_{out}}{W_{in}}

MechanicsPhysicsEfficiency of a machine as useful work out divided by work in, with the shortfall lost to friction, heat, and noise.

Rope Tension When Lifting a Mass

T=m(g+a)T = m\left(g + a\right)

MechanicsPhysicsTension in a rope lifting a mass with an upward acceleration, exceeding the static weight by the factor (g + a).

Atwood Machine Acceleration

a=(m1−m2)gm1+m2a = \frac{\left(m_1 - m_2\right) g}{m_1 + m_2}

MechanicsPhysicsAcceleration of an Atwood machine — two masses joined by a rope over a frictionless pulley, driven by their difference.

Normal (Axial) Stress

σ=PA\sigma = \frac{P}{A}

Strength of MaterialsMechanicsPhysicsAxial stress in a bar or hanger rod — the internal force divided by the cross-sectional area that carries it, in Pa or psi.

Normal Strain (ε = δ/L)

ε=δL\varepsilon = \frac{\delta}{L}

Strength of MaterialsMechanicsPhysicsNormal strain as the change in length divided by the original length, a dimensionless ratio usually quoted in microstrain.

Young's Modulus (E = σ/ε)

E=σεE = \frac{\sigma}{\varepsilon}

Strength of MaterialsMechanicsPhysicsYoung's modulus as the ratio of normal stress to normal strain, the stiffness constant of a material in its elastic range.

Axial Deformation (δ = PL/AE)

δ=PLAE\delta = \frac{P L}{A E}

Strength of MaterialsMechanicsPhysicsElongation of an axially loaded bar from load, length, area and Young's modulus — the workhorse δ = PL/AE of hanger design.

Average Shear Stress (τ = V/A)

τ=VA\tau = \frac{V}{A}

Strength of MaterialsMechanicsPhysicsAverage shear stress on a bolt, pin or weld throat: the transverse force divided by the area resisting it, in Pa or psi.

Shear Modulus (G = τ/γ)

G=τγG = \frac{\tau}{\gamma}

Strength of MaterialsMechanicsPhysicsShear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

Poisson's Ratio

ν=εlatεax\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Strength of MaterialsMechanicsPhysicsPoisson's ratio, the lateral contraction per unit of axial extension — close to 0.30 for steel and 0.33 for aluminium.

Relation Between E, G and ν

E=2G(1+ν)E = 2G(1 + \nu)

Strength of MaterialsMechanicsPhysicsThe isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and Poisson's ratio in one step.

Bulk Modulus (K = ΔP·V₀/ΔV)

K=ΔP V0ΔVK = \frac{\Delta P \, V_0}{\Delta V}

Strength of MaterialsMechanicsPhysicsBulk modulus from the pressure rise and the volume change it produces; water sits near 2.2 GPa and hydraulic oil near 1.5 GPa.

Factor of Safety

FS=σuσallowFS = \frac{\sigma_{u}}{\sigma_{allow}}

Strength of MaterialsMechanicsPhysicsFactor of safety as ultimate or yield strength divided by the allowable working stress, the engineer's declared margin of ignorance.

Thermal Stress in a Restrained Member

σ=EαΔT\sigma = E \alpha \Delta T

Strength of MaterialsMechanicsPhysicsStress raised in a fully restrained member that is heated or cooled — the cause of rail sun kinks and cracked pipe anchors.

Area Moment of Inertia — Rectangle

I=bh312I = \frac{b h^{3}}{12}

Strength of MaterialsMechanicsGeometryArea moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and entered as a plain number.

Area Moment of Inertia — Solid Round Bar

I=πd464I = \frac{\pi d^{4}}{64}

Strength of MaterialsMechanicsGeometryArea moment of inertia of a solid round bar about a diameter, I = πd⁴/64, returned in m⁴ and entered as a plain number.

Elastic Section Modulus (S = I/c)

S=IcS = \frac{I}{c}

Strength of MaterialsMechanicsGeometryElastic section modulus S = I/c, in m³, the single number that turns a bending moment straight into a bending stress.

Bending Stress (σ = Mc/I)

σ=McI\sigma = \frac{M c}{I}

Strength of MaterialsMechanicsPhysicsBending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I entered in m⁴.

Bending Stress from Section Modulus (σ = M/S)

σ=MS\sigma = \frac{M}{S}

Strength of MaterialsMechanicsPhysicsBending stress straight from the moment and a tabulated section modulus S in m³, the everyday form used with steel tables.

Max Bending Moment — Centre Point Load

M=PL4M = \frac{P L}{4}

Strength of MaterialsMechanicsPhysicsMaximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4, occurring under the load.

Max Bending Moment — Uniform Load

M=wL28M = \frac{w L^{2}}{8}

Strength of MaterialsMechanicsPhysicsMaximum bending moment at midspan of a simply supported beam under a uniformly distributed load, the classic M = wL²/8.

Beam Deflection — Simply Supported, Centre Load

δ=PL348EI\delta = \frac{P L^{3}}{48 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with I entered in m⁴.

Beam Deflection — Simply Supported, Uniform Load

δ=5wL4384EI\delta = \frac{5 w L^{4}}{384 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I entered in m⁴.

Cantilever Deflection — End Load

δ=PL33EI\delta = \frac{P L^{3}}{3 E I}

Strength of MaterialsMechanicsPhysicsTip deflection of a cantilever carrying a point load at its free end, δ = PL³/3EI, with I entered in m⁴ as a plain number.

Polar Moment of Inertia — Solid Shaft

J=πd432J = \frac{\pi d^{4}}{32}

Strength of MaterialsMechanicsGeometryPolar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴ — exactly twice the diametral moment of inertia.

Torsional Shear Stress (τ = Tr/J)

τ=TrJ\tau = \frac{T r}{J}

Strength of MaterialsMechanicsPhysicsTorsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J entered in m⁴ as a plain number.

Angle of Twist (φ = TL/JG)

φ=TLJG\varphi = \frac{T L}{J G}

Strength of MaterialsMechanicsPhysicsAngle of twist of a round shaft under torque, φ = TL/JG, the stiffness check that governs long drive and torque shafts.

Euler Critical Buckling Load

Pcr=π2EI(KL)2P_{cr} = \frac{\pi^{2} E I}{(K L)^{2}}

Strength of MaterialsMechanicsPhysicsEuler's critical buckling load for a slender column, using the end-condition factor K and the area moment of inertia in m⁴.

Radius of Gyration (r = √(I/A))

r=IAr = \sqrt{\frac{I}{A}}

Strength of MaterialsMechanicsGeometryRadius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a column really is.

Slenderness Ratio (KL/r)

λ=KLr\lambda = \frac{K L}{r}

Strength of MaterialsMechanicsPhysicsSlenderness ratio KL/r of a compression member, the single number that decides whether a column crushes or buckles.

Hoop Stress in a Thin-Walled Cylinder

σh=pd2t\sigma_{h} = \frac{p d}{2 t}

Strength of MaterialsMechanicsPhysicsHoop (circumferential) stress in a thin-walled pipe or pressure vessel, σ = pd/2t — exactly twice the longitudinal stress.

Longitudinal Stress in a Thin-Walled Cylinder

σl=pd4t\sigma_{l} = \frac{p d}{4 t}

Strength of MaterialsMechanicsPhysicsLongitudinal (axial) stress in a thin-walled cylinder under internal pressure, σ = pd/4t — exactly half the hoop stress.

Bolt Preload from Torque (T = KDF)

T=KDFT = K D F

Strength of MaterialsMechanicsPhysicsBolt preload from tightening torque using the nut factor K, T = KDF, the field method behind every published torque spec.

Hydronic Heat Transfer (Water)

Q˙=ρwcwV˙ ΔT\dot{Q} = \rho_w c_w \dot{V} \, \Delta T

HVAC & HydronicsThermodynamicsFluid MechanicsHeat carried by a water loop from flow rate and supply-to-return ΔT — the SI form of the trade rule BTU/hr = 500 × GPM × ΔT.

Glycol Loop Heat Transfer (Capacity Derate)

Q˙=ρcV˙ ΔT\dot{Q} = \rho c \dot{V} \, \Delta T

HVAC & HydronicsThermodynamicsWater TreatmentHeat carried by a glycol loop using the actual mix density and specific heat, which is how the 500 constant derates for antifreeze.

Air Sensible Heat (1.08 Rule)

Q˙s=ρacaV˙ ΔT\dot{Q}_s = \rho_a c_a \dot{V} \, \Delta T

HVAC & HydronicsThermodynamicsSensible heat carried by an airstream from CFM and dry-bulb ΔT, assuming standard air — the trade rule BTU/hr = 1.08 × CFM × ΔT.

Air Latent Heat (0.68 Rule)

Q˙l=ρaV˙hfg ΔW\dot{Q}_l = \rho_a \dot{V} h_{fg} \, \Delta W

HVAC & HydronicsThermodynamicsLatent heat from dehumidifying an airstream, set by airflow and the change in humidity ratio — the trade rule BTU/hr = 0.68 × CFM × Δgrains.

Air Total Heat (4.5 Rule)

Q˙t=ρaV˙ Δh\dot{Q}_t = \rho_a \dot{V} \, \Delta h

HVAC & HydronicsThermodynamicsTotal (sensible plus latent) heat carried by an airstream from airflow and enthalpy change — the trade rule BTU/hr = 4.5 × CFM × Δh.

Tons of Refrigeration from BTU/hr

T=Q˙12,000 BTU/hrT = \frac{\dot{Q}}{12{,}000\ \text{BTU/hr}}

HVAC & HydronicsThermodynamicsConverts a cooling load in BTU/hr (or kW) to tons of refrigeration, where one ton is 12,000 BTU/hr or 3.5169 kW.

Coefficient of Performance (COP)

COP=Q˙W˙\mathrm{COP} = \frac{\dot{Q}}{\dot{W}}

HVAC & HydronicsThermodynamicsEfficiency of a heat pump or chiller: useful heating or cooling delivered divided by the electrical power drawn to deliver it.

Energy Efficiency Ratio (EER)

EER=Q˙ [BTU/hr]W˙ [W]\mathrm{EER} = \frac{\dot{Q}\ [\text{BTU/hr}]}{\dot{W}\ [\text{W}]}

HVAC & HydronicsThermodynamicsCooling efficiency as BTU/hr of capacity per watt of electrical input, a deliberately mixed-unit ratio equal to 3.412 times the COP.

EER to COP Conversion

EER=3.412×COP\mathrm{EER} = 3.412 \times \mathrm{COP}

HVAC & HydronicsThermodynamicsConverts between the two efficiency scales, since one watt of input equals 3.412 BTU/hr and both ratios describe the same machine.

Chiller Efficiency (kW per Ton)

kW/ton=W˙ [kW]Q˙ [tons]\mathrm{kW/ton} = \frac{\dot{W}\ [\text{kW}]}{\dot{Q}\ [\text{tons}]}

HVAC & HydronicsThermodynamicsThe chiller-plant efficiency metric: kilowatts drawn per ton of cooling produced, where lower is better and 0.5 kW/ton is excellent.

Condenser Water Flow Rate

V˙=Q˙⋅HRFρwcw ΔT\dot{V} = \frac{\dot{Q} \cdot \mathrm{HRF}}{\rho_w c_w \, \Delta T}

HVAC & HydronicsFluid MechanicsWater TreatmentTower water flow needed to reject a chiller's load plus compressor heat, the physics behind the 3 gpm per ton at 10 °F rule of thumb.

Round Duct Air Velocity

v=4V˙πd2v = \frac{4 \dot{V}}{\pi d^{2}}

HVAC & HydronicsFluid MechanicsAir velocity in a round duct from the volume flow and the duct diameter, the check that keeps branches quiet and mains efficient.

Equivalent Round Duct Diameter

De=1.30(ab)0.625(a+b)0.25D_e = 1.30 \frac{(ab)^{0.625}}{(a+b)^{0.25}}

HVAC & HydronicsFluid MechanicsHuebscher's equation for the round duct that has the same friction loss and airflow as a given rectangular duct of sides a and b.

Air Changes per Hour (ACH)

ACH=3600 V˙Vroom\mathrm{ACH} = \frac{3600 \, \dot{V}}{V_{room}}

HVAC & HydronicsFluid MechanicsHow many times per hour a ventilation rate replaces the air in a room, the ventilation yardstick behind BTU/hr = 60 × CFM ÷ room volume.

Sensible Heat Ratio (SHR)

SHR=Q˙sQ˙s+Q˙l\mathrm{SHR} = \frac{\dot{Q}_s}{\dot{Q}_s + \dot{Q}_l}

HVAC & HydronicsThermodynamicsThe fraction of a cooling coil's total load that is sensible, the number that decides whether a room ends up cool or merely cold and clammy.

Mixed Air Temperature

Tm=f Toa+(1−f) TraT_m = f \, T_{oa} + (1-f) \, T_{ra}

HVAC & HydronicsThermodynamicsTemperature of the blend leaving a mixing box, weighted by the outdoor air fraction — the reading that verifies an economizer's damper position.

Expansion Tank Acceptance Volume

Vt=Vs e1−P1P2V_t = \frac{V_s \, e}{1 - \dfrac{P_1}{P_2}}

HVAC & HydronicsFluid MechanicsDiaphragm expansion tank size for a closed hydronic loop from system volume, water expansion and the absolute fill and relief pressures.

Loop Water Expansion Volume

ΔV=V0 β ΔT\Delta V = V_0 \, \beta \, \Delta T

HVAC & HydronicsFluid MechanicsThermodynamicsVolume a hydronic loop's water gains when heated, from the starting volume, the volumetric expansion coefficient and the temperature rise.

Hydronic Static Fill Pressure

P=ρwgH+PmarginP = \rho_w g H + P_{margin}

HVAC & HydronicsFluid MechanicsCold fill pressure a closed loop needs to lift water to its highest point plus a safety margin, the SI form of the 2.31 ft per psi rule.

Radiator Output at Non-Rated Temperature

Q˙=Q˙r(ΔTΔTr)n\dot{Q} = \dot{Q}_r \left(\frac{\Delta T}{\Delta T_r}\right)^{n}

HVAC & HydronicsThermodynamicsCorrects a radiator or baseboard's catalogue output to the actual water-to-air temperature difference using the emitter exponent n.

Seasonal Heating Energy (Degree-Day Method)

E=Q˙d ΔTm tΔTd ηE = \frac{\dot{Q}_d \, \Delta T_m \, t}{\Delta T_d \, \eta}

HVAC & HydronicsThermodynamicsEstimates seasonal fuel energy by scaling the design heat loss with the average temperature deficit, season length and equipment efficiency.

Boiler or Furnace Output from Input

Q˙out=Q˙in η\dot{Q}_{out} = \dot{Q}_{in} \, \eta

HVAC & HydronicsThermodynamicsUsable heat delivered by a boiler or furnace from its fuel input rate and its efficiency rating, the nameplate arithmetic behind AFUE.

Combustion (Stack) Efficiency — Siegert Formula

η=100−A ΔTCO2\eta = 100 - A \, \frac{\Delta T}{\mathrm{CO_2}}

HVAC & HydronicsThermodynamicsClassic flue-gas efficiency estimate from net stack temperature and flue CO₂ percentage, with a fuel constant A of about 0.66 for gas.

Refrigerant Superheat

SH=Tsuction−Tsat\mathrm{SH} = T_{suction} - T_{sat}

HVAC & HydronicsThermodynamicsDegrees of superheat at the compressor suction: how much warmer the vapour is than its saturation temperature at the same pressure.

Refrigerant Subcooling

SC=Tsat−Tliquid\mathrm{SC} = T_{sat} - T_{liquid}

HVAC & HydronicsThermodynamicsDegrees of subcooling at the condenser outlet: how much colder the liquid is than its saturation temperature at the same pressure.

Refrigerant Mass Flow Rate

m˙=Q˙Δh\dot{m} = \frac{\dot{Q}}{\Delta h}

HVAC & HydronicsThermodynamicsRefrigerant circulated per unit time from the cooling capacity and the enthalpy change across the evaporator, the basis of compressor sizing.

Pump Affinity Law — Flow vs Speed

Q2Q1=N2N1\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsWater TreatmentFirst affinity law: a centrifugal pump's capacity changes in direct proportion to shaft speed when the impeller diameter is unchanged.

Pump Affinity Law — Head vs Speed

H2H1=(N2N1)2\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentSecond affinity law: pump head varies with the square of shaft speed, so a 20% speed cut costs 36% of the developed head.

Pump Affinity Law — Power vs Speed

P2P1=(N2N1)3\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsWater TreatmentThird affinity law: absorbed power varies with the cube of shaft speed — the single relation that pays for every variable-frequency drive.

Pump Affinity Law — Flow vs Impeller Diameter

Q2Q1=D2D1\frac{Q_2}{Q_1} = \frac{D_2}{D_1}

HVAC & HydronicsFluid MechanicsWater TreatmentCapacity scales directly with trimmed impeller diameter at constant speed, the classic way to de-rate an oversized centrifugal pump permanently.

Pump Affinity Law — Head vs Impeller Diameter

H2H1=(D2D1)2\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentDeveloped head falls with the square of the trimmed impeller diameter, so a 10% trim sheds about 19% of the head at constant speed.

Fan Affinity Law — Airflow vs Speed

Q2Q1=N2N1\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsPhysicsFan airflow in CFM changes in direct proportion to wheel speed, the first law used when re-sheaving a belt-driven air handler.

Fan Affinity Law — Static Pressure vs Speed

SP2SP1=(N2N1)2\frac{SP_2}{SP_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsPhysicsFan static pressure rises with the square of wheel speed, the reason a modest re-sheave can overpressurise ductwork and blow out flex connections.

Fan Affinity Law — Power vs Speed

P2P1=(N2N1)3\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsPhysicsFan brake power varies with the cube of wheel speed — the law behind variable-air-volume energy savings and behind burnt-out re-sheaved motors.

Pump Water Horsepower

WHP=Q H SG3960WHP = \frac{Q \, H \, SG}{3960}

HVAC & HydronicsFluid MechanicsWater TreatmentUseful power delivered to the liquid; the 3960 divisor assumes US gallons per minute, feet of head and horsepower output.

Pump Brake Horsepower

BHP=Q H SG3960 ηBHP = \frac{Q \, H \, SG}{3960 \, \eta}

HVAC & HydronicsFluid MechanicsWater TreatmentShaft power the motor must actually supply; the 3960 constant assumes gpm, feet of head and horsepower, with efficiency as a fraction.

Pump Efficiency from Hydraulic and Shaft Power

η=PhydPshaft\eta = \frac{P_{hyd}}{P_{shaft}}

HVAC & HydronicsFluid MechanicsWater TreatmentPump efficiency is the ratio of hydraulic power delivered to the liquid over the mechanical power absorbed at the shaft.

Fan Brake Horsepower

BHP=Q⋅SP6356 ηBHP = \frac{Q \cdot SP}{6356 \, \eta}

HVAC & HydronicsFluid MechanicsPhysicsShaft power a fan absorbs; the 6356 divisor assumes cubic feet per minute, inches of water gauge and horsepower at the given efficiency.

Total Dynamic Head

TDH=hs+hf+hvTDH = h_s + h_f + h_v

HVAC & HydronicsFluid MechanicsWater TreatmentThe head a pump must develop: static lift plus friction losses plus velocity head, all expressed in feet or metres of the pumped liquid.

Net Positive Suction Head Available (NPSHa)

NPSHa=hatm+hs−hf−hvpNPSH_a = h_{atm} + h_s - h_f - h_{vp}

HVAC & HydronicsFluid MechanicsWater TreatmentAbsolute head available at the pump suction above the liquid's vapour pressure — the margin that keeps a pump from cavitating.

Darcy–Weisbach Head Loss

hf=f LD v22gh_f = f \, \frac{L}{D} \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsPhysicsThe rigorous pipe friction equation: head loss from friction factor, length-to-diameter ratio and velocity head, with g = 9.80665 m/s².

Laminar Friction Factor (f = 64/Re)

f=64Ref = \frac{64}{Re}

HVAC & HydronicsFluid MechanicsPhysicsIn laminar pipe flow the Darcy friction factor depends only on Reynolds number — roughness plays no part below about Re = 2300.

Swamee–Jain Friction Factor

f=0.25[log⁡10 ⁣(ε3.7D+5.74Re0.9)]2f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}}\right)\right]^{2}}

HVAC & HydronicsFluid MechanicsPhysicsAn explicit turbulent friction factor within about 1% of the implicit Colebrook–White equation, valid for Re from 5000 to 10⁸.

Hazen–Williams Head Loss

hf=10.67 L Q1.852C1.852D4.8704h_f = \frac{10.67 \, L \, Q^{1.852}}{C^{1.852} D^{4.8704}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe waterworks head-loss equation in SI form, with Q in m³/s and D in m; the 10.67 constant is 4.727 when working in feet and cubic feet per second.

Hazen–Williams Velocity

v=0.849 C R0.63S0.54v = 0.849 \, C \, R^{0.63} S^{0.54}

HVAC & HydronicsFluid MechanicsWater TreatmentMean water velocity from hydraulic radius and hydraulic gradient; the 0.849 SI constant becomes 1.318 when R is in feet and v in feet per second.

Minor Loss from K Factor

hL=K v22gh_L = K \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsWater TreatmentHead lost through a valve or fitting as a multiple of velocity head, with g = 9.80665 m/s² and K taken from a fitting table.

Equivalent Length of a Fitting

Leq=KDfL_{eq} = \frac{K D}{f}

HVAC & HydronicsFluid MechanicsWater TreatmentConverts a fitting's K factor into the length of straight pipe that would cause the same friction loss at the same friction factor.

Valve Flow Coefficient (Cv)

Q=CvΔPSGQ = C_v \sqrt{\frac{\Delta P}{SG}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe US valve-sizing relation: Cv is the gpm of 60 °F water a valve passes at 1 psi drop, so Q is in gpm and ΔP in psi.

Valve Flow Coefficient (Kv, metric)

Q=KvΔpSGQ = K_v \sqrt{\frac{\Delta p}{SG}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe metric valve-sizing relation: Kv is the m³/h of water a valve passes at 1 bar drop, related to Cv by Cv ≈ 1.156 Kv.

Pipe Internal Volume

V=πD24LV = \frac{\pi D^{2}}{4} L

HVAC & HydronicsFluid MechanicsWater TreatmentThe liquid a run of pipe holds, from inside diameter and developed length — the starting point for every flush, fill or chemical dose.

Partially Filled Horizontal Cylindrical Tank

V=L[r2cos⁡−1 ⁣(r−hr)−(r−h)2rh−h2]V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]

HVAC & HydronicsFluid MechanicsGeometryLiquid volume in a horizontal cylinder from the wetted depth, using the circular segment area times the tank length.

Pipe Velocity from Flow and Diameter

v=4QπD2v = \frac{4Q}{\pi D^{2}}

HVAC & HydronicsFluid MechanicsWater TreatmentAverage velocity in a full round pipe from volumetric flow and inside diameter — the first check on any piping design.

Barlow's Formula (Pipe Pressure Rating)

P=2StDP = \frac{2 S t}{D}

HVAC & HydronicsFluid MechanicsMechanicsInternal pressure a pipe can hold from wall stress, wall thickness and outside diameter — the thin-wall hoop-stress relation used by pipeline codes.

Expansion Loop Leg Length (Guided Cantilever)

L=3ED ΔSaL = \sqrt{\frac{3 E D \, \Delta}{S_a}}

HVAC & HydronicsFluid MechanicsMechanicsLeg length an expansion loop or offset needs to absorb a given thermal movement without exceeding the pipe's allowable stress.

Water Hammer Surge (Joukowsky Equation)

ΔP=ρ a Δv\Delta P = \rho \, a \, \Delta v

HVAC & HydronicsFluid MechanicsPhysicsPeak pressure surge from a sudden change in flow velocity: fluid density times pressure-wave celerity times the velocity change.

Pump Specific Speed (Ns)

Ns=NQH0.75N_s = \frac{N \sqrt{Q}}{H^{0.75}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe dimensional index that classifies impeller type, evaluated in US units with N in rpm, Q in gpm and H in feet at the best efficiency point.

Three-Phase Real Power

P=3 VLIL PFP = \sqrt{3} \, V_{L} I_{L} \, \text{PF}

Electrical TradeElectricity & MagnetismReal power drawn by a balanced three-phase load from its line-to-line voltage, line current, and power factor.

Single-Phase Real Power with Power Factor

P=VI PFP = V I \, \text{PF}

Electrical TradeElectricity & MagnetismTrue power of a single-phase AC load: volts times amps times power factor, the fraction of the current doing real work.

Three-Phase Apparent Power

S=3 VLILS = \sqrt{3} \, V_{L} I_{L}

Electrical TradeElectricity & MagnetismApparent power in volt-amperes for a balanced three-phase load — the quantity that sizes transformers, cables and breakers.

Power Factor from Real and Apparent Power

PF=PS\text{PF} = \frac{P}{S}

Electrical TradeElectricity & MagnetismPower factor as the ratio of real power in watts to apparent power in volt-amperes, the share of supplied capacity doing work.

Reactive Power (Power Triangle)

Q=S2−P2Q = \sqrt{S^{2} - P^{2}}

Electrical TradeElectricity & MagnetismReactive power in vars from the power triangle, where apparent power is the hypotenuse over real and reactive legs.

Power-Factor Correction kvar

Qc=P(tan⁡φ1−tan⁡φ2)Q_{c} = P \left( \tan\varphi_{1} - \tan\varphi_{2} \right)

Electrical TradeElectricity & MagnetismSize of the capacitor bank in vars needed to lift a load from its existing power factor up to a chosen target power factor.

Power-Factor Correction Capacitance

C=Qc2πfV2C = \frac{Q_{c}}{2\pi f V^{2}}

Electrical TradeElectricity & MagnetismCapacitance needed across a single-phase load to supply a given number of vars at the supply voltage and frequency.

Phase Angle from Power Factor

φ=arccos⁡(PF)\varphi = \arccos(\text{PF})

Electrical TradeElectricity & MagnetismThe angle by which current lags or leads voltage, found from the power factor — the link between the ratio and the triangle.

Wye Line and Phase Voltage

VL=3 VφV_{L} = \sqrt{3} \, V_{\varphi}

Electrical TradeElectricity & MagnetismIn a wye (star) connection the line-to-line voltage is √3 times the line-to-neutral phase voltage of each winding.

Delta Line and Phase Current

IL=3 IφI_{L} = \sqrt{3} \, I_{\varphi}

Electrical TradeElectricity & MagnetismIn a delta connection the line current is √3 times the current in each winding, while line and phase voltages are equal.

Voltage Drop, Single Phase

Vd=2ρLIAV_{d} = \frac{2 \rho L I}{A}

Electrical TradeElectricity & MagnetismVoltage lost in a single-phase run from conductor resistivity, one-way length, current and area — the 2 counts both conductors.

Voltage Drop, Three Phase

Vd=3 ρLIAV_{d} = \frac{\sqrt{3} \, \rho L I}{A}

Electrical TradeElectricity & MagnetismLine-to-line voltage drop on a balanced three-phase run, using √3 rather than 2 because there is no return conductor.

Percent Voltage Drop

%Vd=100 VdVs\%V_{d} = \frac{100 \, V_{d}}{V_{s}}

Electrical TradeElectricity & MagnetismVoltage drop expressed as a percentage of the supply voltage, the form codes use for the 3% branch and 5% total limits.

Three-Phase Motor Full-Load Current

I=Pout3 V PF ηI = \frac{P_{out}}{\sqrt{3} \, V \, \text{PF} \, \eta}

Electrical TradeElectricity & MagnetismLine current of a three-phase motor from its shaft output power, voltage, power factor and nameplate efficiency in percent.

Motor Efficiency

η=100 PoutPin\eta = \frac{100 \, P_{out}}{P_{in}}

Electrical TradeElectricity & MagnetismPercentage efficiency of a motor or drive as mechanical output power divided by electrical input power, times one hundred.

Induction Motor Slip

s=100 (Ns−Nr)Nss = \frac{100 \, (N_{s} - N_{r})}{N_{s}}

Electrical TradeElectricity & MagnetismPercent slip of an induction motor: how far the rotor falls behind the rotating magnetic field, as a share of synchronous speed.

Synchronous Speed from Frequency and Poles

Ns=2fpN_{s} = \frac{2f}{p}

Electrical TradeElectricity & MagnetismSpeed of an AC machine's rotating field from supply frequency and pole count — the familiar 120f/p when read out in rpm.

Conductor Resistance Temperature Correction

R2=R1[1+α(T2−T1)]R_{2} = R_{1} \left[ 1 + \alpha (T_{2} - T_{1}) \right]

Electrical TradeElectricity & MagnetismCorrects a conductor's resistance from one temperature to another using the material's temperature coefficient of resistance.

Voltage Divider

Vout=VinR2R1+R2V_{out} = V_{in} \frac{R_{2}}{R_{1} + R_{2}}

Electrical TradeElectricity & MagnetismOutput of two resistors in series across a source: the input voltage split in proportion to the lower resistor's share.

Current Divider

I1=ItR2R1+R2I_{1} = I_{t} \frac{R_{2}}{R_{1} + R_{2}}

Electrical TradeElectricity & MagnetismHow a total current splits between two parallel resistors — each branch takes the share set by the opposite resistance.

Series RLC Impedance

Z=R2+(XL−XC)2Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}

Electrical TradeElectricity & MagnetismMagnitude of impedance in a series RLC circuit, combining resistance with the net reactance left after XL and XC cancel.

Series RL or RC Impedance

Z=R2+X2Z = \sqrt{R^{2} + X^{2}}

Electrical TradeElectricity & MagnetismImpedance magnitude of a resistance in series with a single reactance, whether that reactance is inductive or capacitive.

Q Factor of a Series Resonant Circuit

Q=1RLCQ = \frac{1}{R} \sqrt{\frac{L}{C}}

Electrical TradeElectricity & MagnetismQuality factor of a series RLC circuit, measuring how sharply it resonates and how much it magnifies voltage at resonance.

Decibel Power Gain

GdB=10log⁡10 ⁣(P2P1)G_{dB} = 10 \log_{10}\!\left(\frac{P_{2}}{P_{1}}\right)

Electrical TradeElectricity & MagnetismGain or loss in decibels between two power levels, the logarithmic ratio used throughout audio, RF and communications work.

Decibel Voltage Gain

GdB=20log⁡10 ⁣(V2V1)G_{dB} = 20 \log_{10}\!\left(\frac{V_{2}}{V_{1}}\right)

Electrical TradeElectricity & MagnetismGain or loss in decibels computed from a voltage ratio, which carries a factor of 20 because power varies with the square of voltage.

LED Series Resistor

R=Vs−VfIR = \frac{V_{s} - V_{f}}{I}

Electrical TradeElectricity & MagnetismResistor needed to run an LED at a chosen current from a given supply, absorbing the difference above the forward voltage.

Inverting Op-Amp Gain

G=−RfRinG = -\frac{R_{f}}{R_{in}}

Electrical TradeElectricity & MagnetismClosed-loop gain of an inverting operational amplifier, set purely by the ratio of feedback resistor to input resistor.

Non-Inverting Op-Amp Gain

G=1+RfRinG = 1 + \frac{R_{f}}{R_{in}}

Electrical TradeElectricity & MagnetismClosed-loop gain of a non-inverting operational amplifier, always one more than the feedback-to-ground resistor ratio.

555 Astable Frequency

f=1ln⁡2 (R1+2R2)Cf = \frac{1}{\ln 2 \, (R_{1} + 2R_{2}) C}

Electrical TradeElectricity & MagnetismOutput frequency of a 555 timer in astable mode; datasheets round the exact 1/ln2 factor in the numerator to 1.44.

Glycol Dilution

C1V1=C2V2C_1 V_1 = C_2 V_2

Water TreatmentChemistryDilution equation for mixing glycol (or any concentrate) to a target strength.

Cycles of Concentration (COC = M/B)

COC=MB\text{COC} = \frac{M}{B}

Water TreatmentFluid MechanicsCycles of concentration for a cooling tower from the makeup and blowdown flows — the master number every treatment program is built around.

Cycles of Concentration from Conductivity

COC=σtσm\text{COC} = \frac{\sigma_t}{\sigma_m}

Water TreatmentChemistryCycles of concentration read straight off a conductivity meter: tower water conductivity divided by makeup water conductivity.

Cycles of Concentration from Chloride

COC=CltClm\text{COC} = \frac{\mathrm{Cl}_t}{\mathrm{Cl}_m}

Water TreatmentChemistryCycles of concentration from a chloride titration — the conservative tracer that neither precipitates nor gets dosed into the system.

Cooling Tower Evaporation Rate

E=0.001 R ΔTE = 0.001 \, R \, \Delta T

Water TreatmentFluid MechanicsThermodynamicsEvaporation loss from a cooling tower using the industry rule of 0.1% of recirculation per degree Fahrenheit of range.

Blowdown Rate from Cycles

B=ECOC−1B = \frac{E}{\text{COC} - 1}

Water TreatmentFluid MechanicsBlowdown a cooling tower must bleed to hold a target cycles of concentration, given its evaporation rate.

Cooling Tower Makeup Water Rate

M=E+B+DM = E + B + D

Water TreatmentFluid MechanicsTotal makeup water a cooling tower needs: the sum of evaporation, blowdown to drain, and drift carried out in the air stream.

Cooling Tower Drift Loss

D=d100 RD = \frac{d}{100} \, R

Water TreatmentFluid MechanicsDrift (windage) loss from a cooling tower as a percentage of the recirculation rate, the fraction of basin water blown out as droplets.

Percent Blowdown

%B=BM×100\%B = \frac{B}{M} \times 100

Water TreatmentFluid MechanicsBlowdown expressed as a percentage of makeup water — the share of purchased water that goes straight to the sewer.

Cooling Tower Range

ΔT=Th−Tc\Delta T = T_h - T_c

Water TreatmentThermodynamicsCooling tower range: the temperature drop the tower achieves between the hot water returning from the plant and the cold basin water.

Cooling Tower Approach

A=Tc−TwbA = T_c - T_{wb}

Water TreatmentThermodynamicsCooling tower approach: how many degrees the cold basin water sits above the ambient wet-bulb temperature, the true measure of tower performance.

Cooling Tower Heat Rejection

Q=500 R ΔTQ = 500 \, R \, \Delta T

Water TreatmentThermodynamicsFluid MechanicsHeat a cooling tower rejects from flow and range using the trade constant 500 = 8.34 lb/gal × 60 min/h × 1 BTU/(lb·°F).

Chemical Feed Rate from Dose

W=C Q ρwW = C \, Q \, \rho_w

Water TreatmentChemistryChemical feed rate in pounds per day from a ppm dose and a water flow, using the trade density of 8.34 lb per US gallon.

Dose Achieved from Chemical Added

C=mV ρwC = \frac{m}{V \, \rho_w}

Water TreatmentChemistryConcentration reached in a system of known volume after adding a measured mass of chemical, at 8.34 lb of water per US gallon.

Closed Loop Slug Dose Volume

Vp=C Vs ρwρpV_p = \frac{C \, V_s \, \rho_w}{\rho_p}

Water TreatmentChemistryVolume of liquid product needed for a one-shot slug dose to hit a target ppm in a closed loop of known volume and product density.

Product Dose from Active Strength

Dp=100 DaAD_p = \frac{100 \, D_a}{A}

Water TreatmentChemistryConverts a required active-ingredient dose into the as-supplied product dose when the drum is only a given percent active.

Holding Time Index

HTI=ln⁡2  VB\text{HTI} = \frac{\ln 2 \; V}{B}

Water TreatmentChemistryHolding time index: the half-life of a chemical in a bled system, the time for half the treatment to wash out at a given blowdown rate.

System Volume from Turnover Time

V=R tV = R \, t

Water TreatmentFluid MechanicsSystem water volume estimated from the recirculation rate and the measured turnover time — the field method when no drawings exist.

Boiler Cycles of Concentration

COC=TDSbTDSfw\text{COC} = \frac{\text{TDS}_b}{\text{TDS}_{fw}}

Water TreatmentChemistryCycles of concentration in a steam boiler: boiler water TDS divided by feedwater TDS, since steam leaves the dissolved solids behind.

Boiler Blowdown Percent

%B=TDSfwTDSb×100\%B = \frac{\text{TDS}_{fw}}{\text{TDS}_b} \times 100

Water TreatmentChemistryBoiler blowdown as a percentage of feedwater, set by the ratio of feedwater TDS to the maximum TDS allowed in the drum.

Boiler Blowdown Rate from Steam Rate

B=SCOC−1B = \frac{S}{\text{COC} - 1}

Water TreatmentThermodynamicsContinuous blowdown a steam boiler must carry, in pounds per hour, from its steam production rate and target cycles of concentration.

Condensate Return Percentage

%CR=ScS×100\%CR = \frac{S_c}{S} \times 100

Water TreatmentThermodynamicsPercentage of generated steam that comes back to the boiler house as condensate — the headline efficiency number for any steam plant.

Boiler Makeup from Condensate Return

M=S(1−%CR100)M = S\left(1 - \frac{\%CR}{100}\right)

Water TreatmentThermodynamicsFresh makeup water a steam plant must treat, from the steam production rate and the fraction of condensate that comes back.

Boiler Horsepower to Heat Output

Q=33,475  BHPQ = 33{,}475 \; \text{BHP}

Water TreatmentThermodynamicsConverts boiler horsepower to heat output using the ASME definition of 33,475 BTU per hour per boiler horsepower.

Boiler Horsepower to Steam Rate

S=34.5  BHPS = 34.5 \; \text{BHP}

Water TreatmentThermodynamicsSteam output of a boiler from its horsepower rating, at the ASME definition of 34.5 lb/h of steam from and at 212 °F per BHP.

Flash Steam Percentage

%F=hf1−hf2hfg2×100\%F = \frac{h_{f1} - h_{f2}}{h_{fg2}} \times 100

Water TreatmentThermodynamicsPercentage of hot condensate that flashes to steam when let down to a lower pressure, from the saturated liquid and latent enthalpies.

Corrosion Rate from Coupon Weight Loss

P=m Yρ A tP = \frac{m \, Y}{\rho \, A \, t}

Water TreatmentChemistryUniform corrosion rate as metal thickness lost per year, from a coupon's weight loss, density, exposed area and exposure time.

Volume of Water Over a Period

V=Q tV = Q \, t

Water TreatmentFluid MechanicsWater a flow delivers over a period — the step that turns a makeup or blowdown rate into the daily or annual volume a customer is billed for.

Chemical Mass Consumed Over a Period

m=W tm = W \, t

Water TreatmentChemistryTreatment chemical a feed rate consumes over a period — the step that turns pounds per day into the annual tonnage a contract is priced on.

Chiller Heat Rejection

Qr=Qe HRFQ_r = Q_e \, \mathrm{HRF}

Water TreatmentThermodynamicsHVAC & HydronicsHeat a chiller's tower has to reject: the evaporator load times the heat rejection factor that adds the compressor's own work to the load.

Chemical-Consuming Loss Rate

L=B+DL = B + D

Water TreatmentChemistryFluid MechanicsFlow that actually carries treatment out of a cooling tower — blowdown plus drift, because evaporation leaves every molecule of inhibitor behind.

Cost of Water Over a Period

Cw=V pwC_w = V \, p_w

Water TreatmentFluid MechanicsWater & WastewaterCost of the water a system buys over a period: the metered volume times the utility's rate, in whatever currency that rate was in.

Sewer Credit for Evaporated Water

Cc=Ve psC_c = V_e \, p_s

Water TreatmentFluid MechanicsWater & WastewaterSewer credit for water a tower evaporates: the volume that never reaches the drain, valued at the municipal sewer rate.

Chemical Treatment Cost

Cp=m pcC_p = m \, p_c

Water TreatmentChemistryInvoice for a treatment chemical: the mass of product consumed over a period times its unit price, the second line of a water budget.

Energy Cost from a Utility Rate

Ce=E peC_e = E \, p_e

Water TreatmentThermodynamicsHVAC & HydronicsCost of the energy a system consumes: kilowatt-hours or fuel BTUs times the utility rate, for tower fans, pumps and boiler gas.

Net Water and Sewer Cost of a Cooling Tower

C=Vm pw+(Vm−Ve) psC = V_m \, p_w + (V_m - V_e) \, p_s

Water TreatmentFluid MechanicsWater & WastewaterFull water and sewer bill for a cooling tower: makeup charged at the water rate, plus only the volume actually discharged at the sewer rate.

Total Water Treatment Program Cost

C=Cw+Cp+CeC = C_w + C_p + C_e

Water TreatmentChemistryFluid MechanicsTotal operating cost of a treated cooling system: the water and sewer bill, the chemical invoice and the energy bill added together.

Langelier Saturation Index (LSI)

LSI=pH−pHs\mathrm{LSI} = \mathrm{pH} - \mathrm{pH_s}

Water TreatmentChemistryThe classic calcium-carbonate scaling index: measured pH minus the saturation pH, positive for scaling and negative for corrosive water.

Saturation pH (pHs) for Langelier's Index

pHs=(9.3+A+B)−(C+D)\mathrm{pH_s} = (9.3 + A + B) - (C + D)

Water TreatmentChemistryComputes the pH at which water is saturated with calcium carbonate from its TDS, temperature, calcium hardness and total alkalinity as CaCO₃.

Ryznar Stability Index (RSI)

RSI=2 pHs−pH\mathrm{RSI} = 2\,\mathrm{pH_s} - \mathrm{pH}

Water TreatmentChemistryAn empirical scaling index scaled so that values below about 6 predict scale and values above about 7 predict corrosion in the same water.

Puckorius (Practical) Scaling Index

PSI=2 pHs−pHeq,pHeq=1.465 log⁡10Alk+4.54\mathrm{PSI} = 2\,\mathrm{pH_s} - \mathrm{pH_{eq}}, \quad \mathrm{pH_{eq}} = 1.465\,\log_{10}\mathrm{Alk} + 4.54

Water TreatmentChemistryA Ryznar variant that replaces measured pH with an equilibrium pH derived from alkalinity, giving a truer scaling call in poorly buffered cooling water.

Larson–Skold Index

LS=Cl35.45+SO448.03Alk50.04\mathrm{LS} = \dfrac{\frac{\mathrm{Cl}}{35.45} + \frac{\mathrm{SO_4}}{48.03}}{\frac{\mathrm{Alk}}{50.04}}

Water TreatmentChemistryRatio of aggressive chloride and sulphate equivalents to protective bicarbonate alkalinity, predicting pitting of mild steel in distribution water.

Total Hardness as CaCO₃

TH=2.497 Ca+4.118 Mg\mathrm{TH} = 2.497\,\mathrm{Ca} + 4.118\,\mathrm{Mg}

Water TreatmentChemistryConverts a lab report's calcium and magnesium ion results into a single total hardness expressed as milligrams per litre of CaCO₃.

Grains per Gallon ↔ ppm Hardness

H=17.118 GH = 17.118\,G

Water TreatmentChemistryConverts water hardness between grains per US gallon and mg/L as CaCO₃ using the exact 17.118 mg/L per grain trade factor.

Ion Concentration as CaCO₃ Equivalent

CCaCO3=Cion×50.04EWC_{\mathrm{CaCO_3}} = C_{\mathrm{ion}} \times \frac{50.04}{\mathrm{EW}}

Water TreatmentChemistryRestates any ion's concentration in the trade's common currency of CaCO₃ by scaling with the ratio of equivalent weights, 50.04 over the ion's own.

Equivalent Weight from Molar Mass and Valence

EW=Mz\mathrm{EW} = \frac{M}{z}

Water TreatmentChemistryGives the gram-equivalent weight of an ion or compound as its molar mass divided by the number of charges or replaceable hydrogens it carries.

Total Alkalinity as CaCO₃ from Species

TA=0.8202 HCO3+1.6679 CO3+2.9425 OH\mathrm{TA} = 0.8202\,\mathrm{HCO_3} + 1.6679\,\mathrm{CO_3} + 2.9425\,\mathrm{OH}

Water TreatmentChemistrySums bicarbonate, carbonate and hydroxide reported as their own ions into one total alkalinity expressed as milligrams per litre of CaCO₃.

TDS Estimated from Conductivity (TDS = k × EC)

TDS=k×EC\mathrm{TDS} = k \times \mathrm{EC}

Water TreatmentChemistryEstimates total dissolved solids in mg/L from a conductivity reading in µS/cm using an empirical factor k, typically 0.55 to 0.70 for natural waters.

Water Resistivity and Conductivity

ρ=1σ\rho = \frac{1}{\sigma}

Water TreatmentChemistryRelates the resistivity and conductivity of ultrapure and deionized water as exact reciprocals, the pair of scales used on DI and RO polishing loops.

Hardness Load Removed per Regeneration

m=C Vm = C \, V

Water TreatmentChemistryGives the mass of hardness as CaCO₃ a softener must remove between regenerations from the raw hardness and the volume of water treated.

Days Between Softener Regenerations

t=mcapC Qt = \frac{m_{\text{cap}}}{C \, Q}

Water TreatmentChemistryGives the run time a softener achieves between regenerations from its rated hardness capacity, the raw water hardness and the average water usage rate.

Softener Resin Volume Required

V=mcapqV = \frac{m_{\text{cap}}}{q}

Water TreatmentChemistrySizes the ion-exchange resin bed from the hardness capacity required per regeneration and the resin's rated capacity per unit volume.

Salt Dose per Regeneration

msalt=D Vm_{\text{salt}} = D \, V

Water TreatmentChemistryGives the mass of sodium chloride a softener draws each regeneration from the resin volume and the programmed salt dosage per unit volume.

Hardness Removal Efficiency and Leakage

R=Cin−CoutCinR = \frac{C_{\text{in}} - C_{\text{out}}}{C_{\text{in}}}

Water TreatmentChemistryGives the fraction of hardness a softener or membrane removes from the inlet and outlet concentrations, with the remainder being leakage to service.

Chlorine Dose, Demand and Residual

D=Cdemand+CresD = C_{\text{demand}} + C_{\text{res}}

Water TreatmentChemistryThe fundamental chlorination balance: the dose applied equals the chlorine consumed by the water's demand plus the residual left for disinfection.

Chemical Feed Rate (lb/day = mg/L × MGD × 8.34)

m˙=C Q\dot m = C \, Q

Water TreatmentChemistryConverts a target chemical dose and a plant flow into a mass feed rate, the classic pounds-per-day formula built on water weighing 8.34 lb per gallon.

Hypochlorite Product Mass from Available Chlorine

mprod=mCl×100%pm_{\text{prod}} = \frac{m_{\mathrm{Cl}} \times 100\%}{p}

Water TreatmentChemistryConverts a required mass of available chlorine into the mass of hypochlorite product to weigh out, given the product's percent available chlorine.

Chlorine Dose from a Weight of Product

C=m pVC = \frac{m \, p}{V}

Water TreatmentChemistryGives the free chlorine concentration produced by dissolving a known weight of hypochlorite product of known strength in a known volume of water.

Breakpoint Chlorine-to-Ammonia Ratio

R=Cl2NH3-NR = \frac{\mathrm{Cl_2}}{\mathrm{NH_3\text{-}N}}

Water TreatmentChemistryThe weight ratio of chlorine dose to ammonia nitrogen, which must reach about 7.6 to 1 to pass breakpoint and produce a free chlorine residual.

Pounds of Active Chemical in a Tank

m=V×SG×ρw×pm = V \times \mathrm{SG} \times \rho_w \times p

Water TreatmentChemistryGives the mass of active chemical held in a storage tank from its volume, the solution's specific gravity and the percent active ingredient.

Acid Feed to Reduce Alkalinity

m˙=ΔAlk Q×EW50.04p/100%\dot m = \frac{\Delta \mathrm{Alk} \, Q \times \frac{\mathrm{EW}}{50.04}}{p/100\%}

Water TreatmentChemistrySizes the acid feed needed to knock a target amount of alkalinity out of a stream, correcting for the acid's equivalent weight and its commercial strength.

Newton's Law of Cooling (Q = hAΔT)

Q˙=hA ΔT\dot{Q} = h A \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsConvective heat rate from a surface, set by the film coefficient, the wetted area and the surface-to-fluid temperature difference.

Overall Heat Transfer Coefficient (U)

1U=1hi+Lk+1ho\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}

Heat TransferThermodynamicsHVAC & HydronicsOverall coefficient U for a plane wall with fluid on both sides, adding the inside film, the wall and the outside film as resistances in series.

Thermal Resistance of a Plane Wall

R=LkAR = \frac{L}{k A}

Heat TransferThermodynamicsConduction resistance of a flat slab in kelvin per watt, from its thickness, thermal conductivity and the area heat crosses.

Convection Film Resistance

R=1hAR = \frac{1}{h A}

Heat TransferThermodynamicsHVAC & HydronicsThermal resistance of a boundary-layer film in kelvin per watt, the reciprocal of the film coefficient times the wetted surface area.

Thermal Resistances in Series

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferThermodynamicsTotal resistance of a composite wall, where the same heat crosses each layer in turn so the layer resistances simply add.

Heat Flow from Thermal Resistance

Q˙=ΔTR\dot{Q} = \frac{\Delta T}{R}

Heat TransferThermodynamicsHVAC & HydronicsOhm's law for heat: the flow through an assembly equals the temperature difference across it divided by its total thermal resistance.

Overall U from Total Resistance

U=1RtotAU = \frac{1}{R_{tot} A}

Heat TransferThermodynamicsHVAC & HydronicsConverts an assembly's total resistance in kelvin per watt into the overall coefficient U quoted on exchanger and envelope datasheets.

Conduction Through a Pipe Wall

Q˙=2πkL ΔTln⁡(r2/r1)\dot{Q} = \frac{2 \pi k L \, \Delta T}{\ln(r_2 / r_1)}

Heat TransferThermodynamicsHVAC & HydronicsRadial conduction through a cylindrical pipe or insulation layer, where the area grows outward so the resistance follows a logarithm.

Critical Radius of Insulation

rcr=khr_{cr} = \frac{k}{h}

Heat TransferThermodynamicsThe outer radius below which adding insulation to a small cylinder increases heat loss, because added surface beats added resistance.

Log Mean Temperature Difference (Counterflow)

ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}

Heat TransferThermodynamicsHVAC & HydronicsEffective driving temperature difference in a counterflow exchanger, from the terminal differences at the hot and cold ends of the shell.

Log Mean Temperature Difference (Parallel Flow)

ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}

Heat TransferThermodynamicsHVAC & HydronicsEffective driving temperature difference when both streams enter at the same end, pairing the two inlets and the two outlets.

Heat Exchanger Duty (Q = U·A·F·LMTD)

Q˙=UAF ΔTlm\dot{Q} = U A F \, \Delta T_{lm}

Heat TransferThermodynamicsHVAC & HydronicsThe LMTD design equation with the correction factor F, which derates the counterflow driving force for shell-and-tube or crossflow arrangements.

Stream Duty from Mass Flow (Q = ṁcΔT)

Q˙=m˙ cp ΔT\dot{Q} = \dot{m} \, c_p \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsHeat picked up or given off by one exchanger stream, from its mass flow, specific heat and the temperature change across the unit.

Number of Transfer Units (NTU)

NTU=UAm˙ cp\mathrm{NTU} = \frac{U A}{\dot{m} \, c_p}

Heat TransferThermodynamicsDimensionless size of an exchanger: its conductance UA divided by the heat capacity rate of the minimum stream, ṁ times its specific heat.

Capacity Rate Ratio (Cr)

Cr=m˙mincminm˙maxcmaxC_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}

Heat TransferThermodynamicsRatio of the smaller to the larger stream heat capacity rate ṁcₚ, the second dimensionless group the effectiveness-NTU method needs.

Effectiveness from NTU (Counterflow)

ε=1−e−NTU(1−Cr)1−Cr e−NTU(1−Cr)\varepsilon = \frac{1 - e^{-\mathrm{NTU}(1 - C_r)}}{1 - C_r \, e^{-\mathrm{NTU}(1 - C_r)}}

Heat TransferThermodynamicsCounterflow effectiveness from the two dimensionless groups NTU and Cr, valid for any Cr from 0 to 1 with the balanced case handled as a limit.

Maximum Possible Heat Transfer (Qmax)

Q˙max=m˙mincmin(Th,in−Tc,in)\dot{Q}_{max} = \dot{m}_{min} c_{min} (T_{h,in} - T_{c,in})

Heat TransferThermodynamicsThe thermodynamic ceiling on exchanger duty: the minimum capacity rate multiplied by the full inlet-to-inlet temperature difference.

Heat Exchanger Effectiveness (ε = Q/Qmax)

ε=Q˙Q˙max\varepsilon = \frac{\dot{Q}}{\dot{Q}_{max}}

Heat TransferThermodynamicsHVAC & HydronicsEffectiveness as the ratio of actual duty to the thermodynamic maximum, the performance figure that needs no outlet temperatures to interpret.

Fouled Overall Coefficient

1Uf=1Uc+1hf\frac{1}{U_f} = \frac{1}{U_c} + \frac{1}{h_f}

Heat TransferThermodynamicsHVAC & HydronicsAdds a fouling deposit as one more resistance in series, reducing the clean overall coefficient to the fouled value used for design margin.

Fin Parameter mL (Straight Fin)

mL=L2hktmL = L \sqrt{\frac{2h}{k t}}

Heat TransferThermodynamicsThe dimensionless group governing straight-fin performance, combining fin length, thickness, material conductivity and the surface film coefficient.

Fin Efficiency (Straight Fin)

ηf=tanh⁡(mL)mL\eta_f = \frac{\tanh(mL)}{mL}

Heat TransferThermodynamicsEfficiency of a straight fin with an adiabatic tip, comparing its real duty with the duty it would give if it were all at base temperature.

Fin Heat Transfer Rate

Q˙f=ηf hAf ΔTb\dot{Q}_f = \eta_f \, h A_f \, \Delta T_b

Heat TransferThermodynamicsHVAC & HydronicsDuty of a fin or finned surface: the ideal convective rate over the whole fin area, derated by the fin efficiency.

Biot Number

Bi=hLck\mathrm{Bi} = \frac{h L_c}{k}

Heat TransferThermodynamicsRatio of internal conduction resistance to surface convection resistance; below 0.1 a body may be treated as having one uniform temperature.

Fourier Number

Fo=k tρ c L2\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}

Heat TransferThermodynamicsDimensionless time for transient conduction, written from conductivity, density and specific heat so no diffusivity input is needed.

Lumped Capacitance Time Constant

τ=ρVchA\tau = \frac{\rho V c}{h A}

Heat TransferThermodynamicsThermal time constant of a body cooling at uniform temperature, its stored heat per kelvin divided by the surface conductance hA.

Lumped Capacitance Cooling Curve

T=T∞+(T0−T∞)e−t/τT = T_\infty + (T_0 - T_\infty) e^{-t/\tau}

Heat TransferThermodynamicsExponential temperature history of a body at uniform temperature, and the time it needs to reach any temperature between start and ambient.

Prandtl Number

Pr=μcpk\mathrm{Pr} = \frac{\mu c_p}{k}

Heat TransferFluid MechanicsThermodynamicsFluid property group comparing how fast momentum diffuses with how fast heat does, setting the relative thickness of the two boundary layers.

Nusselt Number

Nu=hLk\mathrm{Nu} = \frac{h L}{k}

Heat TransferFluid MechanicsThermodynamicsDimensionless convection coefficient: the ratio of convective transfer at a surface to pure conduction through the same fluid layer.

Dittus-Boelter Correlation

Nu=0.023 Re0.8 Prn\mathrm{Nu} = 0.023 \, \mathrm{Re}^{0.8} \, \mathrm{Pr}^{n}

Heat TransferFluid MechanicsTurbulent tube-flow Nusselt number, valid for Re above 10,000, Pr from 0.6 to 160 and L/D over 10, with n = 0.4 heating and 0.3 cooling.

Net Radiation Exchange Between Surfaces

Q˙=εσA(T14−T24)\dot{Q} = \varepsilon \sigma A (T_1^4 - T_2^4)

Heat TransferThermodynamicsNet radiant heat from a grey surface to large surroundings, using the Stefan-Boltzmann constant and the difference of fourth-power temperatures.

View Factor Reciprocity

A1F1→2=A2F2→1A_1 F_{1 \to 2} = A_2 F_{2 \to 1}

Heat TransferThermodynamicsReciprocity relation for radiation view factors, which lets you recover the unknown factor between two surfaces from the known one and their areas.

Combined Convection and Radiation Coefficient

ht=hc+εσ(Ts+Tsur)(Ts2+Tsur2)h_t = h_c + \varepsilon \sigma (T_s + T_{sur})(T_s^2 + T_{sur}^2)

Heat TransferThermodynamicsHVAC & HydronicsTotal surface coefficient adding a linearised radiation term to the convective film, so one h covers both mechanisms over a modest ΔT.

R-Value of an Insulation Layer (R = L/k)

R=LkR = \frac{L}{k}

Heat TransferHVAC & HydronicsThermodynamicsArea-specific thermal resistance of one layer, from its thickness and thermal conductivity — the RSI or R-value quoted on every insulation label.

Total R-Value of an Assembly

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferHVAC & HydronicsThermodynamicsTotal R-value of a wall, roof or floor built up from three layers in series, where the same heat crosses each layer so the R-values simply add.

U-Factor from Total R-Value (U = 1/R)

U=1RtotU = \frac{1}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsConverts an assembly's total R-value into the U-factor used by energy codes and window labels, and back — the two are simple reciprocals.

Heat Loss Through an Assembly (Q = A·ΔT/R)

Q˙=A ΔTRtot\dot{Q} = \frac{A \, \Delta T}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsSteady heat loss through a wall, roof or floor from its area, the inside-to-outside temperature difference and the assembly's total R-value.

Heat Flux Through Insulation (q = ΔT/R)

q′′=ΔTRq'' = \frac{\Delta T}{R}

Heat TransferHVAC & HydronicsThermodynamicsHeat flow per unit area through an insulated assembly, straight from the temperature difference and the R-value, with no area needed.

Effective R-Value with Framing (Parallel Path)

1Reff=ffrRfr+1−ffrRcav\frac{1}{R_{eff}} = \frac{f_{fr}}{R_{fr}} + \frac{1 - f_{fr}}{R_{cav}}

Heat TransferHVAC & HydronicsThermodynamicsWhole-wall R-value once the studs are counted, area-weighting the framing and cavity paths as parallel conductances rather than averaging their R-values.

Fouling Factor on an Overall Coefficient

1Uf=1Uc+Rf\frac{1}{U_f} = \frac{1}{U_c} + R_f

Heat TransferThermodynamicsHVAC & HydronicsService-condition U of a heat exchanger, adding the TEMA fouling factor as an extra area-specific resistance on top of the clean coefficient.

Total Vertical Stress (σ = γz)

σv=γz\sigma_v = \gamma z

Soil MechanicsMechanicsTotal vertical stress at depth in a uniform soil layer, the weight of the overburden column standing on one unit of area.

Pore Water Pressure (u = γw zw)

u=γwzwu = \gamma_w z_w

Soil MechanicsFluid MechanicsHydrostatic pore water pressure at a point below a static water table, from the depth of water standing above it.

Effective Stress (Terzaghi, σ′ = σ − u)

σ′=σ−u\sigma' = \sigma - u

Soil MechanicsMechanicsTerzaghi's effective stress principle: the grain-to-grain stress that controls soil strength equals total stress minus pore water pressure.

Void Ratio and Porosity (e = n/(1 − n))

e=n1−ne = \frac{n}{1 - n}

Soil MechanicsStrength of MaterialsConverts between void ratio, the void volume per unit of solid, and porosity, the void volume per unit of total soil volume.

Degree of Saturation (Se = wGs)

S=w GseS = \frac{w\,G_s}{e}

Soil MechanicsStrength of MaterialsDegree of saturation from water content, specific gravity of solids and void ratio, using the phase identity Se = wGs.

Water (Moisture) Content

w=MwMs×100w = \frac{M_w}{M_s}\times 100

Soil MechanicsStrength of MaterialsGravimetric water content of a soil as the mass of pore water divided by the mass of oven-dry solids, expressed as a percentage.

Dry Unit Weight from Moist Unit Weight

γd=γ1+w100\gamma_d = \frac{\gamma}{1 + \dfrac{w}{100}}

Soil MechanicsStrength of MaterialsStrips the pore water out of a measured bulk unit weight to give the dry unit weight used for compaction control and phase work.

Saturated Unit Weight

γsat=(Gs+e) γw1+e\gamma_{sat} = \frac{(G_s + e)\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsUnit weight of a soil whose voids are completely full of water, from the specific gravity of the solids and the void ratio.

Submerged (Buoyant) Unit Weight

γ′=γsat−γw\gamma' = \gamma_{sat} - \gamma_w

Soil MechanicsFluid MechanicsEffective or buoyant unit weight of soil below the water table, the saturated unit weight less the uplift of the water it displaces.

Dry Unit Weight from Gs and Void Ratio

γd=Gs γw1+e\gamma_d = \frac{G_s\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsDry unit weight of a soil from the specific gravity of its solids and its void ratio, the phase-diagram route used to back out e in the lab.

Relative Density of a Granular Soil

Dr=emax−eemax−emin×100D_r = \frac{e_{max} - e}{e_{max} - e_{min}}\times 100

Soil MechanicsStrength of MaterialsRelative density of a sand or gravel, placing its in-situ void ratio on the scale between its loosest and densest laboratory states.

Relative Compaction (Percent Proctor)

R=γd,fieldγd,max×100R = \frac{\gamma_{d,field}}{\gamma_{d,max}}\times 100

Soil MechanicsStrength of MaterialsRelative compaction of placed fill as the field dry unit weight divided by the Proctor maximum dry unit weight, in percent.

Plasticity Index (PI = LL − PL)

PI=LL−PLPI = LL - PL

Soil MechanicsStrength of MaterialsPlasticity index of a fine-grained soil as the liquid limit minus the plastic limit, the width of the moisture range where clay behaves plastically.

Liquidity Index

LI=w−PLPILI = \frac{w - PL}{PI}

Soil MechanicsStrength of MaterialsLiquidity index locating the natural water content of a clay between its plastic limit and liquid limit, a direct index of consistency.

Hydraulic Gradient

i=ΔhLi = \frac{\Delta h}{L}

Soil MechanicsFluid MechanicsHydraulic gradient as the loss of total head divided by the length of the flow path, the dimensionless driving force behind all seepage.

Darcy's Law for Groundwater Flow

Q=k i AQ = k\,i\,A

Soil MechanicsFluid MechanicsDarcy's law for laminar flow through soil: discharge equals hydraulic conductivity times hydraulic gradient times gross cross-sectional area.

Seepage Velocity from Discharge Velocity

vs=vnv_s = \frac{v}{n}

Soil MechanicsFluid MechanicsActual seepage velocity through the pores, obtained by dividing Darcy's fictitious discharge velocity by the porosity of the soil.

Equivalent Horizontal Permeability of Layered Soil

keq=k1H1+k2H2H1+H2k_{eq} = \frac{k_1 H_1 + k_2 H_2}{H_1 + H_2}

Soil MechanicsFluid MechanicsThickness-weighted equivalent permeability for flow parallel to the bedding of two soil layers, the parallel-resistance case of stratified seepage.

Mohr–Coulomb Shear Strength

τf=c′+σ′tan⁡ϕ′\tau_f = c' + \sigma'\tan\phi'

Soil MechanicsMechanicsMohr–Coulomb failure criterion giving the shear strength of soil from effective cohesion and the friction mobilised by effective normal stress.

Rankine Active Earth Pressure Coefficient

Ka=tan⁡2 ⁣(45∘−ϕ2)K_a = \tan^{2}\!\left(45^\circ - \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of active earth pressure for a smooth vertical wall retaining level cohesionless backfill that has yielded away from the soil.

Rankine Passive Earth Pressure Coefficient

Kp=tan⁡2 ⁣(45∘+ϕ2)K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of passive earth pressure, the resistance mobilised when a wall or footing is pushed into level cohesionless soil.

At-Rest Earth Pressure Coefficient (Jaky)

K0=1−sin⁡ϕK_0 = 1 - \sin\phi

Soil MechanicsMechanicsJaky's 1944 empirical coefficient of earth pressure at rest for a normally consolidated soil that is not permitted to strain laterally.

Active Thrust on a Retaining Wall

Pa=12 Ka γ H2P_a = \tfrac{1}{2}\,K_a\,\gamma\,H^{2}

Soil MechanicsMechanicsTotal Rankine active thrust per unit length of wall from a dry cohesionless backfill, acting at one third of the wall height above the base.

Bearing Capacity Factor Nq

Nq=eπtan⁡ϕ tan⁡2 ⁣(45∘+ϕ2)N_q = e^{\pi\tan\phi}\,\tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsPrandtl–Reissner surcharge bearing capacity factor Nq from the friction angle, the value tabulated by Meyerhof, Hansen and Vesic.

Bearing Capacity Factor Nc

Nc=(Nq−1)cot⁡ϕN_c = (N_q - 1)\cot\phi

Soil MechanicsMechanicsPrandtl's cohesion bearing capacity factor Nc derived from Nq and the friction angle, tending to 5.14 as the friction angle goes to zero.

Terzaghi Ultimate Bearing Capacity (Strip Footing)

qu=c Nc+q Nq+12 γ B Nγq_u = c\,N_c + q\,N_q + \tfrac{1}{2}\,\gamma\,B\,N_\gamma

Soil MechanicsMechanicsTerzaghi's three-term ultimate bearing capacity of a shallow strip footing, summing the cohesion, surcharge and footing-width contributions.

Net Allowable Bearing Pressure

qall=qu−qFSq_{all} = \frac{q_u - q}{FS}

Soil MechanicsMechanicsNet allowable bearing pressure for a footing, the ultimate capacity less the existing overburden, divided by the chosen factor of safety.

SPT Overburden Correction (Liao–Whitman)

(N1)60=N60paσv′(N_1)_{60} = N_{60}\sqrt{\frac{p_a}{\sigma'_v}}

Soil MechanicsMechanicsCorrects a field SPT blow count to a reference overburden of one atmosphere using the Liao and Whitman square-root factor CN.

Consolidation Settlement of Normally Consolidated Clay

Sc=Cc H1+e0log⁡10 ⁣σf′σ0′S_c = \frac{C_c\,H}{1 + e_0}\log_{10}\!\frac{\sigma'_f}{\sigma'_0}

Soil MechanicsStrength of MaterialsPrimary consolidation settlement of a normally consolidated clay layer from its compression index, thickness and the stress increase applied.

Time Factor for Consolidation

Tv=cv tHdr2T_v = \frac{c_v\,t}{H_{dr}^{2}}

Soil MechanicsFluid MechanicsDimensionless time factor of Terzaghi consolidation theory, with the coefficient of consolidation entered in m²/s and the longest drainage path.

Time Factor from Degree of Consolidation (U ≤ 60%)

Tv=π4(U100)2T_v = \frac{\pi}{4}\left(\frac{U}{100}\right)^{2}

Soil MechanicsFluid MechanicsTerzaghi's parabolic approximation relating the time factor to the average degree of consolidation, valid for U of 60 percent or less.

Factor of Safety Against Sliding

FS=Wtan⁡δPhFS = \frac{W\tan\delta}{P_h}

Soil MechanicsMechanicsFactor of safety of a retaining structure against base sliding, comparing frictional resistance under its weight with the driving horizontal thrust.

Percent Grade from Rise and Run

G=100 ΔhLG = \frac{100\,\Delta h}{L}

Civil & SurveyingGeometryPercent grade is one hundred times the vertical rise divided by the horizontal run, measured level, never along the slope.

Grade to Slope Angle

θ=arctan⁡ ⁣(G100)\theta = \arctan\!\left(\frac{G}{100}\right)

Civil & SurveyingTrigonometryConverts a percent grade into the slope angle measured from horizontal, and back again with the tangent function.

Slope Ratio (H:V) to Percent Grade

G=100nG = \frac{100}{n}

Civil & SurveyingSoil MechanicsConverts an embankment slope quoted as n horizontal to one vertical into the equivalent percent grade, and back.

Elevation from Grade and Distance

E2=E1+G L100E_2 = E_1 + \frac{G\,L}{100}

Civil & SurveyingGeometryProjects an elevation along a uniform grade, adding the rise over a measured horizontal distance to the known starting elevation.

Earthwork Volume by Average End Area

V=L (A1+A2)2V = \frac{L\,(A_1 + A_2)}{2}

Civil & SurveyingGeometryVolume between two cross sections, averaging their end areas over the distance between them; the answer is reported in cubic yards.

Earthwork Volume by the Prismoidal Formula

V=L (A1+4Am+A2)6V = \frac{L\,(A_1 + 4A_m + A_2)}{6}

Civil & SurveyingGeometrySimpson's rule applied to earthwork, weighting the middle cross section four times the ends; the answer is reported in cubic yards.

Stockpile Volume (Truncated Pyramid)

V=h3(A1+A2+A1A2)V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)

Civil & SurveyingGeometryVolume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer is reported in cubic yards.

Swell: Loose Volume from Bank Volume

VL=VB(1+S100)V_L = V_B\left(1 + \frac{S}{100}\right)

Civil & SurveyingSoil MechanicsConverts undisturbed bank volume into the loose volume the same soil occupies once excavated, using its percent swell.

Shrinkage: Compacted Volume from Bank Volume

VC=VB(1−Sh100)V_C = V_B\left(1 - \frac{S_h}{100}\right)

Civil & SurveyingSoil MechanicsConverts bank volume into the compacted volume it fills in an engineered embankment, using the soil's percent shrinkage.

Concrete Volume with Waste Allowance

V=L W T(1+w100)V = L\,W\,T\left(1 + \frac{w}{100}\right)

Civil & SurveyingGeometryOrders concrete for a rectangular slab, footing or column by adding a percent waste allowance to the neat volume, in cubic yards.

Asphalt Tonnage from Area and Thickness

M=A t ρM = A\,t\,\rho

Civil & SurveyingConverts a paving area, lift thickness and compacted mix density into the tonnage to order, reported in US short tons.

Radius from Degree of Curve (Arc Definition)

R=5729.578DR = \frac{5729.578}{D}

Civil & SurveyingGeometryConverts degree of curve to radius using the arc definition, where D is the central angle subtending one 100 ft station of arc.

Horizontal Curve Tangent Length

T=Rtan⁡Δ2T = R\tan\frac{\Delta}{2}

Civil & SurveyingTrigonometryDistance from the point of intersection back to the point of curvature, from the curve radius and its total deflection angle.

Horizontal Curve Length from Degree of Curve

L=100 ΔDL = \frac{100\,\Delta}{D}

Civil & SurveyingGeometryLength of a circular curve in 100 ft stations, from the total deflection angle and the degree of curve, on the arc definition.

Horizontal Curve External Distance

E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)

Civil & SurveyingTrigonometryClearance from the point of intersection to the midpoint of the curve, the distance a curve cuts back from the corner.

Horizontal Curve Middle Ordinate

M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)

Civil & SurveyingTrigonometryOffset from the middle of the long chord to the middle of the arc, the number that governs sight distance around obstructions.

Horizontal Curve Long Chord

C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}

Civil & SurveyingTrigonometryStraight-line distance from the point of curvature to the point of tangency, the chord that spans the entire circular curve.

Vertical Curve Length from K Value

L=K AL = K\,A

Civil & SurveyingLength of a parabolic vertical curve as the design K value times the algebraic grade change, K being length per percent of grade.

Elevation on a Parabolic Vertical Curve

E=EBVC+g1x100+A x2200 LE = E_{BVC} + \frac{g_1 x}{100} + \frac{A\,x^{2}}{200\,L}

Civil & SurveyingGeometryElevation at any station on an equal-tangent parabolic vertical curve, measured from the beginning of vertical curve.

High or Low Point on a Vertical Curve

xt=−g1LAx_t = -\frac{g_1 L}{A}

Civil & SurveyingDistance from the BVC to the crest or sag of a parabolic vertical curve, where the profile grade passes through zero.

Crest Vertical Curve Length for Sight Distance

L=A S2200(h1+h2)2L = \frac{A\,S^{2}}{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}

Civil & SurveyingGeometryCrest curve length needed to see an object over the hill, for the case where sight distance is shorter than the curve.

Stopping Sight Distance

d=v tr+v22ad = v\,t_r + \frac{v^{2}}{2a}

Civil & SurveyingDistance to stop from a design speed, adding the reaction distance travelled before braking to the braking distance itself.

Superelevation Rate for a Horizontal Curve

e100+f=v2g R\frac{e}{100} + f = \frac{v^{2}}{g\,R}

Civil & SurveyingBalances banking and side friction against centripetal demand on a horizontal curve, using standard gravity of 9.80665 m/s².

Back Azimuth

αb=α±180∘\alpha_b = \alpha \pm 180^{\circ}

Civil & SurveyingTrigonometryReverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside the zero to three sixty range.

Latitude of a Traverse Leg

Lat=Lcos⁡α\text{Lat} = L\cos\alpha

Civil & SurveyingTrigonometryNorth-south component of a traverse course, from the measured slope-corrected distance and the azimuth of the line.

Departure of a Traverse Leg

Dep=Lsin⁡α\text{Dep} = L\sin\alpha

Civil & SurveyingTrigonometryEast-west component of a traverse course, taken from the measured horizontal distance and the azimuth of the line.

Traverse Closure Error

Ec=(ΣLat)2+(ΣDep)2E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}

Civil & SurveyingGeometryLinear misclosure of a closed traverse, combining the residual sums of the latitudes and departures as a right triangle.

Traverse Precision Ratio

N=PEcN = \frac{P}{E_c}

Civil & SurveyingExpresses traverse accuracy as one part in N, dividing the total perimeter walked by the linear closure error.

Area of a Three-Sided Parcel by Coordinates

A=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]

Civil & SurveyingGeometryShoelace area of a parcel from the coordinates of its three corners, listed counter-clockwise so the result comes out positive.

Differential Levelling Elevation

E2=E1+BS−FSE_2 = E_1 + \text{BS} - \text{FS}

Civil & SurveyingCarries an elevation forward one instrument setup, adding the backsight to the known point and subtracting the foresight.

Stadia Distance from Rod Intercept

D=K s+CD = K\,s + C

Civil & SurveyingHorizontal distance from a stadia rod intercept, with the instrument's stadia interval factor and additive constant.

Hydraulic Detention Time

t=VQt = \frac{V}{Q}

Water & WastewaterWater TreatmentFluid MechanicsTheoretical detention time of a tank, clarifier or contact basin: the working volume divided by the flow passing through it.

Surface Overflow Rate

vo=QAv_o = \frac{Q}{A}

Water & WastewaterWater TreatmentFluid MechanicsSurface overflow (surface loading) rate of a settling basin — flow divided by plan area, reported here in metres per day.

Filtration Rate (Filter Loading Rate)

vf=QAv_f = \frac{Q}{A}

Water & WastewaterWater TreatmentFluid MechanicsFiltration rate through a granular media filter: flow divided by filter bed area, the approach velocity reported in metres per day.

Backwash Water Volume

Vbw=vb A tV_{bw} = v_b \, A \, t

Water & WastewaterWater TreatmentFluid MechanicsWater consumed by one filter backwash, from the backwash rise rate, the filter bed area and the duration of the wash.

Percent Backwash Water

%BW=VbwVf×100\%BW = \frac{V_{bw}}{V_f} \times 100

Water & WastewaterWater TreatmentShare of a filter's production consumed by its own backwash, the housekeeping ratio that flags short runs and poor coagulation.

CT Value for Disinfection Credit

CT=C t\text{CT} = C \, t

Water & WastewaterWater TreatmentChemistryDisinfection CT: residual concentration multiplied by contact time, returned in the regulatory unit of mg·min/L.

BOD Mass Loading

W=Q CW = Q \, C

Water & WastewaterWater TreatmentChemistryMass of BOD, COD or solids arriving per unit time from a flow and its concentration, on the sanitary basis of 1 mg/L = 1 g/m³.

BOD Removal Efficiency

E=Ci−CeCi×100E = \frac{C_i - C_e}{C_i} \times 100

Water & WastewaterWater TreatmentPercent removal across a treatment unit or a whole plant, from the influent and effluent concentrations of BOD, TSS or any pollutant.

Volumetric Organic Loading Rate

Lv=Q S0VL_v = \frac{Q \, S_0}{V}

Water & WastewaterWater TreatmentOrganic load applied per unit of reactor volume, returned in kg BOD per cubic metre per day — the sizing number for basins and digesters.

Food-to-Microorganism (F/M) Ratio

FM=Q S0V X\frac{F}{M} = \frac{Q \, S_0}{V \, X}

Water & WastewaterWater TreatmentFood-to-microorganism ratio: BOD applied per day divided by the mass of mixed liquor solids holding it, in kg BOD per kg MLVSS per day.

Mean Cell Residence Time (Sludge Age)

SRT=V XQw Xw\text{SRT} = \frac{V \, X}{Q_w \, X_w}

Water & WastewaterWater TreatmentSolids retention time of an activated sludge plant: the mass of solids under aeration divided by the mass wasted each day.

Sludge Volume Index (SVI)

SVI=SV30X\text{SVI} = \frac{SV_{30}}{X}

Water & WastewaterWater TreatmentSludge volume index in mL/g: the volume a gram of mixed liquor solids occupies after settling thirty minutes in a litre cylinder.

Return Activated Sludge Rate

Qr=Q XXr−XQ_r = \frac{Q \, X}{X_r - X}

Water & WastewaterWater TreatmentReturn sludge flow needed to hold a target mixed liquor concentration, from a solids balance around the aeration basin.

Clarifier Solids Loading Rate

SLR=(Q+Qr) XA\text{SLR} = \frac{(Q + Q_r) \, X}{A}

Water & WastewaterWater TreatmentSolids applied to a secondary clarifier per unit of surface area, in kg of MLSS per square metre per day, including the return flow.

Sludge Volume from Dry Solids

V=mS ρw (P/100)V = \frac{m}{S \, \rho_w \, (P/100)}

Water & WastewaterWater TreatmentVolume of wet sludge holding a given mass of dry solids, from the solids percentage and the sludge specific gravity, at 1000 kg/m³ water.

Specific Gravity of Sludge

1Ssl=P/100Ss+(1−P100)\frac{1}{S_{sl}} = \frac{P/100}{S_s} + \left(1 - \frac{P}{100}\right)

Water & WastewaterWater TreatmentSpecific gravity of a wet sludge from its percent solids and the specific gravity of the dry solids, by a reciprocal mass balance.

Solids Capture in Dewatering

R=Cc(Cf−Cfil)Cf(Cc−Cfil)×100R = \frac{C_c (C_f - C_{fil})}{C_f (C_c - C_{fil})} \times 100

Water & WastewaterWater TreatmentPercent solids capture of a belt press, centrifuge or filter from the feed, cake and filtrate solids concentrations by mass balance.

Van Kleeck Volatile Solids Reduction

R=Vin−VoutVin−VinVout×100R = \frac{V_{in} - V_{out}}{V_{in} - V_{in} V_{out}} \times 100

Water & WastewaterWater TreatmentPercent volatile solids destroyed in a digester by the Van Kleeck equation, using only the volatile fractions in and out.

Trickling Filter Hydraulic Loading

Lh=Q+QrAL_h = \frac{Q + Q_r}{A}

Water & WastewaterWater TreatmentFluid MechanicsHydraulic loading on a trickling filter including recirculation — total flow per unit of media surface area, in metres per day.

Trickling Filter Recirculation Factor

F=1+R(1+0.1R)2F = \frac{1 + R}{(1 + 0.1R)^2}

Water & WastewaterWater TreatmentNRC recirculation factor for trickling filter design, converting a recirculation ratio into the effective number of passes through the media.

Population Equivalent

PE=WwPE = \frac{W}{w}

Water & WastewaterWater TreatmentPopulation equivalent of a waste stream: its pollutant load divided by the load one person contributes, typically 60 g BOD per day.

Per-Capita Wastewater Flow

q=QPq = \frac{Q}{P}

Water & WastewaterWater TreatmentFluid MechanicsAverage wastewater contributed per person per day, from the plant flow and the population served, reported in gallons per capita per day.

Harmon Peaking Factor

PF=1+144+P/1000PF = 1 + \frac{14}{4 + \sqrt{P/1000}}

Water & WastewaterWater TreatmentFluid MechanicsHarmon peaking factor for sanitary sewer design: the ratio of peak hourly to average daily flow for a served population.

Manning's Equation for Velocity

v=1nR2/3S1/2v = \frac{1}{n} R^{2/3} S^{1/2}

Water & WastewaterFluid MechanicsOpen-channel velocity by Manning's equation in SI form, from the roughness coefficient, hydraulic radius and channel slope.

Manning's Equation for Flow

Q=1nAR2/3S1/2Q = \frac{1}{n} A R^{2/3} S^{1/2}

Water & WastewaterFluid MechanicsOpen-channel discharge by Manning's equation in SI form, from flow area, roughness, hydraulic radius and slope.

Francis Formula: Rectangular Weir

Q=3.33 L H3/2Q = 3.33 \, L \, H^{3/2}

Water & WastewaterFluid MechanicsFlow over a suppressed rectangular weir by the Francis formula, with the trade constant 3.33 for crest length and head in feet.

V-Notch (Triangular) Weir Flow

Q=815Cd2g tan⁡ ⁣θ2 H5/2Q = \frac{8}{15} C_d \sqrt{2g} \, \tan\!\frac{\theta}{2} \, H^{5/2}

Water & WastewaterFluid MechanicsDischarge over a sharp-crested triangular weir from the notch angle, head and discharge coefficient, in the standard theoretical form.

Parshall Flume Free Flow

Q=4 W H1.522 W0.026Q = 4 \, W \, H^{1.522 \, W^{0.026}}

Water & WastewaterFluid MechanicsFree-flow discharge through a Parshall flume of 1 to 8 ft throat width, with W and the head H in feet and Q in cubic feet per second.

Stokes Settling Velocity

vs=g(ρs−ρ)d218μv_s = \frac{g (\rho_s - \rho) d^2}{18 \mu}

Water & WastewaterFluid MechanicsTerminal settling velocity of a small sphere in laminar flow by Stokes' law — the grit chamber and clarifier design relation.

Jar Test Dose Scale-Up

D=Vst CstVsD = \frac{V_{st} \, C_{st}}{V_{s}}

Water & WastewaterWater TreatmentChemistryConverts millilitres of stock solution added to a jar test beaker into the equivalent plant dose in mg/L of raw water.

Alkalinity Remaining After Alum

Af=A0−0.45 DA_f = A_0 - 0.45 \, D

Water & WastewaterWater TreatmentChemistryAlkalinity left after coagulation, since each mg/L of alum destroys 0.45 mg/L of alkalinity as CaCO₃ in forming aluminium hydroxide floc.