Formula solvers starting with A — 84 solvers

Abrams' Water-Cement Ratio Law

f_c = \dfrac{A}{B^{\,w/c}}

ConcreteDuff Abrams' 1918 finding, and still the single most useful sentence in concrete technology: for a workable mix made with sound materials, compressive strength depends on the ratio of water to cement and on very little else. Strength falls exponentially as that ratio rises, which is why every extra litre of water added at the truck to make placing easier is paid for in strength.

Absolute Volume of a Mix Component

V_i = \dfrac{m_i}{SG_i \, \rho_w}

ConcreteThe volume one ingredient of a mix actually occupies as solid matter: its mass divided by its specific gravity times the density of water. This is the arithmetic the whole absolute-volume method of mix design rests on — every component's solid volume, plus the air, must add up to the volume of concrete you meant to make.

Absorbance and Transmittance

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.

Absorption Factor

A = \frac{L}{m V}

Chemical EngineeringThe single number that decides whether an absorber can do its job: the liquid's capacity to carry the solute away divided by the gas's capacity to deliver it. Below one, no number of stages will finish the separation.

Acceleration Down a Frictionless Incline

a = 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\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.

Acid Feed to Reduce Alkalinity

\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.

Activation Energy from an Arrhenius Plot

E_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.

Active Ingredient Rate

R_{ai} = c \, R_v

Crop ProductionThe active ingredient actually applied per unit area, from the formulation's concentration and the product rate — the quantity a label's legal maximum is written against, not the product itself.

Active Thrust on a Retaining Wall

P_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.

Addition Rule (Mutually Exclusive Events)

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.

Advancing Blade Tip Mach Number

M_{tip} = \frac{\Omega R + V}{a}

Aerospace & FlightHow fast the fastest part of the rotor is going, in Mach: tip speed plus forward speed, over the local speed of sound. It is the hard ceiling at one end of a helicopter's speed envelope, with retreating blade stall at the other.

Aeration Airflow Requirement

Q = q \, M

Farm OperationsTotal fan airflow for a stored quantity of grain, from the aeration rate per tonne. The first number in sizing an aeration fan — though not the last, because static pressure decides the rest.

Aggregate Moisture Correction (Free Water)

W_{free} = m_{st} \, \dfrac{MC - Ab}{1 + MC}

ConcreteHow much water a stockpiled aggregate gives to the mix — or takes from it. Moisture content and absorption are both quoted against the OVEN-DRY mass, while the aggregate is batched at its wet stock mass, so the correction is not simply the difference of two percentages. This is the arithmetic that keeps a mix design and the concrete in the truck describing the same material.

Air Base from Endlap

B = W_g \, (1 - p_{end})

Photogrammetry & Remote SensingHow far the aircraft travels between one exposure and the next: the along-track ground width of the frame, less the fraction of it that must overlap the previous frame. Endlap is the overlap between CONSECUTIVE frames on one line — not between lines, which is sidelap and a different number.

Air Changes per Hour (ACH)

\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.

Air Latent Heat (0.68 Rule)

\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 Sensible Heat (1.08 Rule)

\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 Total Heat (4.5 Rule)

\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.

Airflow to Dilute a Gas

Q = \frac{q_{gas}}{C_{limit} - C_{intake}}

Rock MechanicsHow much air a working place needs to keep a contaminant below its limit: the volume of gas being made, divided by the concentration headroom between the limit and whatever the intake air already carries. It is a mass balance and nothing more, and its one assumption — perfect mixing — is the one a real heading breaks.

Airway Resistance from Geometry

R = \frac{k \, O \, L}{A^{3}}

Rock MechanicsEverything an airway does to the air, collapsed into one number. Atkinson's equation splits neatly into a part that depends only on the opening — its lining, its rubbing surface, its length and its area — and a part that depends only on the airflow. This is the first part, and it is what a ventilation network model stores for every branch in the mine.

Albedo and Reflected Radiation

G_r = \alpha\,G

Heat TransferSolar & Wind PowerShortwave radiation reflected by a surface, as its albedo times the incident irradiance. Whatever is not reflected is absorbed, so the complement (1 − α)G is the part that actually warms the ground — and it is reported alongside every answer here.

Alcohol by Volume from Gravity

ABV = (OG - FG) \times 131.25

Brewing & FermentationThe standard homebrew estimate of alcohol by volume from the original and final gravity readings — the single most-used calculation in brewing, and one of the least exact.

Alkalinity Remaining After Alum

A_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.

Allowable Noise Exposure Time

T = \frac{8}{2^{(L - L_c)/q}}

Acoustics & NoiseHow long a person may be exposed to a steady noise level before reaching a full daily dose. The answer depends entirely on which exchange rate the jurisdiction uses, and the two in common use disagree by hours.

Alpert Ceiling Jet Temperature (far field)

\Delta T = \dfrac{5.38 \, (\dot{Q}/r)^{2/3}}{H}

Fire Protection & Fire DynamicsRonald Alpert's 1972 correlation for the gas temperature in the thin, fast layer that spreads radially under a ceiling when a plume strikes it. This is the far-field form, for a radius greater than about 0.18 of the ceiling height — the region where sprinklers and heat detectors actually sit.

Alpert Ceiling Jet Velocity (far field)

u = \dfrac{0.195 \, \dot{Q}^{1/3} H^{1/2}}{r^{5/6}}

Fire Protection & Fire DynamicsThe companion to Alpert's temperature correlation: how fast the ceiling jet is moving at a given radius. Velocity is what carries heat into a detector element, so it is half of any response-time calculation — the other half being the temperature the gas is at.

Amdahl's Law (Speedup from Parallelisation)

S = \frac{1}{(1 - p) + \dfrac{p}{N}}

Computer ScienceSpeedup of a fixed workload on N processors when a fraction p of it can be parallelised. The serial remainder sets a hard ceiling no amount of hardware can pass.

Angle Between Two 2D Vectors (Components)

\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.

Angle of Incidence on a Tilted Surface

\theta = \lvert \phi - \delta - \beta \rvert

Solar & Wind PowerHow far off square the noon sun strikes an equator-facing collector tilted at angle β, at latitude φ, on a day of declination δ. Set the tilt to φ − δ and the answer is zero: the rays arrive dead on.

Angle of Repose (μ = 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.

Angle of Twist (φ = TL/JG)

\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.

Angular Acceleration

\alpha = \frac{\omega - \omega_0}{t}

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

Angular Diameter (Small-Angle)

\theta = \frac{2R}{d}

Astronomy & GravitationPhysicsHow large an object of radius R looks from distance d, in radians — the small-angle rule behind eclipses and telescope limits.

Angular Displacement (θ = ω₀t + ½αt²)

\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

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

Angular Momentum (L = Iω)

L = I \omega

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

Angular Velocity (ω = θ/t)

\omega = \frac{\theta}{t}

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

Angular Velocity from Period

\omega = \frac{2\pi}{T}

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

Annual ESALs from a Truck Count (Truck Factor)

W_{18} = 365 \, \mathrm{AADTT} \cdot \mathrm{TF}

Axle Loads & PavementYearly equivalent single axle loads from a daily truck count and the ESALs an average truck delivers per pass. The truck factor is the sum of the equivalency factors of all the truck's axles, and it is a property of the LOAD that trucks carry on that road, not of the trucks themselves.

Annulus Area (Ring)

A = \pi (R^2 - r^2)

GeometryArea of the ring between two concentric circles.

Anode Current Output

I = \frac{\Delta E}{R}

Corrosion & ProtectionWhat a galvanic anode actually delivers: the driving voltage between the anode and the polarized structure, divided by the total resistance of the circuit. Ohm's law doing the whole of cathodic protection design.

Antoine Equation (Vapour Pressure)

\log_{10} P = A - \frac{B}{C + T}

ChemistryThermodynamicsThe three-constant fit that every handbook uses for pure-component vapour pressure. The published constants A, B and C are for pressure in mmHg and temperature in degrees Celsius; enter pressure and temperature in whatever units you like and the conversion is handled inside.

API RP 14E Erosional Velocity

V_e = \frac{C}{\sqrt{\rho_m}}

Erosion & SedimentThe velocity limit every offshore and onshore piping specification quotes, from API Recommended Practice 14E: the mixture density under a square root, divided into an empirical constant C. It is the first number written on a flowline sizing sheet — and it is a rule of thumb with no published experimental basis, which is why C is yours to choose here rather than ours to assume.

Apparent Attenuation

AA = \frac{OG - FG}{OG - 1}

Brewing & FermentationThe share of the wort's original extract that the yeast has fermented away, read off the hydrometer. The number that says whether a fermentation has finished or has stalled.

Apparent Brightness (Inverse-Square Law)

b = \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.

Apparent Depth

d' = \frac{d}{n}

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

Apparent Wall Shear Rate

\dot{\gamma}_{app} = \frac{4Q}{\pi R^{3}}

Polymers & Plastics ProcessingThe shear rate at the wall of a round channel, worked as if the fluid were Newtonian. It is the x-axis of every capillary-rheometer flow curve and the first number to reach for when asking whether a gate is shearing a melt too hard — and it is called APPARENT because for a real polymer melt it is not the true wall shear rate until Rabinowitsch has corrected it.

Arc Length

s = r \theta

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

Arc-Flash Boundary

D_{B} = D \sqrt{\frac{E}{E_{B}}}

Electrical TradeDistance at which incident energy has fallen to the onset-of-burn threshold, scaled from a known energy at a known distance by the inverse-square law.

Arc-Flash Incident Energy

E = \frac{P_{arc} \, t}{4\pi \left(4.184 \times 10^{4}\right) D^{2}}

Electrical TradeHeat energy landing on a worker at a given distance from an arc, from the arc's power, how long it burns, and inverse-square spreading.

Archard Wear Depth

h = \frac{K \, W \, s}{H \, A}

Machine DesignArchard's wear volume spread over the apparent contact area, which turns it into the number a clearance limit is actually written in: how far the surface recedes. Note that the area is the APPARENT one — the nominal projected area of the contact — and not the real area at the asperity tips, which is hundreds of times smaller.

Archard Wear Equation (Volume Lost)

V = \frac{K \, W \, s}{H}

Machine DesignThe volume of material a sliding contact loses: load times distance, divided by the hardness of the softer surface, times a dimensionless wear coefficient. Archard published it in 1953 and it is still the equation wear is estimated with — but the coefficient K spans eight orders of magnitude, so what comes out is an order of magnitude, not a life.

Archimedes Number (Buoyancy against Viscosity)

Ar = \frac{g L^{3} \rho \, (\rho_s - \rho)}{\mu^{2}}

Fluid MechanicsThe buoyant driving force on a particle, non-dimensionalised by viscosity — and, like Ohnesorge, deliberately free of velocity. That is what makes it useful: it is built entirely from the properties of the particle and the fluid, so it can be computed before anything is moving, and it is the standard entry point to fluidisation, settling and pneumatic conveying correlations.

Area Covered per Tank

A = \frac{T}{V}

Crop ProductionHow much ground one tankful covers: the tank volume divided by the application rate. The number that decides where in the field the sprayer runs dry.

Area Moment of Inertia — Rectangle

I = \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 = \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.

Area of a Circle

A = \pi r^{2}

GeometryArea enclosed by a circle of radius r.

Area of a Three-Sided Parcel by Coordinates

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.

Area of a Triangle

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

GeometryArea from base and perpendicular height.

Areal Energy Density (Laser Powder Bed Fusion)

E_a = \frac{P}{v \, h}

Additive ManufacturingThe same beam accounted per unit of scanned AREA rather than per unit volume: power divided by scan speed and hatch spacing. It drops the layer thickness, which is the one term in volumetric energy density that describes the powder rather than the beam — so it is a slightly more honest description of what a single hatched layer receives, and it is still an average over things that are not uniform.

Arithmetic Mean of Two Numbers

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

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

Arithmetic Sequence nth Term

a_n = a_1 + (n - 1) d

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

Arithmetic Series Sum

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

AlgebraSum of the first n terms of an arithmetic sequence.

Arrangements with Repetition

N = n^{r}

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

Arrhenius Equation

k = 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 Equivalent Age (Freiesleben Hansen and Pedersen)

t_e = t \, \exp\!\left[ -\dfrac{E}{R} \left( \dfrac{1}{T} - \dfrac{1}{T_r} \right) \right]

ConcreteThe maturity function Freiesleben Hansen and Pedersen put forward in 1977, and the one most maturity systems now use: hydration is treated as a thermally activated reaction, so the rate follows an Arrhenius exponential in absolute temperature rather than a straight line above a datum. It holds over a far wider temperature range than the linear form, at the price of a second parameter that has to be measured.

Arrhenius Two-Temperature Form

\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.

Arterial Oxygen Content (CaO₂)

C_a\mathrm{O_2} = 1.34\,\mathrm{Hb}\,S_a\mathrm{O_2} + 0.003\,P_a\mathrm{O_2}

Biomedical & ClinicalOxygen carried per decilitre of arterial blood: the large amount bound to haemoglobin, plus the small amount dissolved in plasma.

As-Fed to Dry Matter

M_{DM} = M_{AF} \times DM

Livestock & FeedConverts a weight of feed between the as-fed basis it is handled in and the dry matter basis every ration and feed test is written on. The conversion behind the most expensive mistake in feeding.

Asphalt Tonnage from Area and Thickness

M = A\,t\,\rho

Civil & SurveyingConverts a paving area, lift thickness and compacted mix density into the tonnage to order, reported in US short tons.

At-Rest Earth Pressure Coefficient (Jaky)

K_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.

Atkinson's Equation for Airway Pressure Drop

p = \frac{k \, O \, L \, Q^{2}}{A^{3}}

Rock MechanicsJohn Atkinson's 1854 equation for the pressure a mine airway takes out of the ventilating current: the friction factor of the lining, times the rubbing surface, times the square of the airflow, divided by the cube of the cross-section. It is Darcy-Weisbach rearranged for a passage described by its perimeter and area rather than by a diameter, and after 170 years it is still how mine ventilation is done.

Attack Rate

\mathrm{AR} = \frac{I}{N_{e}}

Epidemiology & Diagnostic AccuracyThe share of an exposed group who fell ill during an outbreak. The CDC's own Principles of Epidemiology notes that this is a misnomer: it is a proportion, not a rate, because there is no time in the denominator. The name has stuck anyway.

Atwood Machine Acceleration

a = \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.

Auger or Conveyor Transfer Time

t = \frac{Q}{R}

Farm OperationsThe hours an auger, belt or conveyor takes to move a quantity of grain at its rated capacity. The check that decides whether the handling system can keep up with the combine.

Availability from MTBF and MTTR

A = \frac{\mathit{MTBF}}{\mathit{MTBF} + \mathit{MTTR}}

Computer ScienceFraction of time a system is up: mean time between failures divided by the whole cycle of running plus repairing. This is where the marketing nines come from.

Available Short-Circuit Current from Percent Impedance

I_{SC} = \frac{100 \, I_{FL}}{\%Z}

Electricity & MagnetismElectrical TradeFault current a transformer can deliver at its secondary terminals, from its full-load current and nameplate percent impedance, assuming an infinite primary source.

Available Water Capacity

AWC = (\theta_{fc} - \theta_{pwp}) \, D

Crop ProductionThe depth of water a soil can hold that a crop can actually use — the gap between field capacity and the permanent wilting point, multiplied by the depth of rooting that reaches it.

Average Brute-Force Time

t = \frac{2^{\,H-1}}{R}

Cryptography & Key SizingExpected time for an exhaustive search over a space of 2^H values at R guesses per second. The average is half the space, hence the H − 1. It assumes an OFFLINE attack on a fast hash with no rate limiting — the worst realistic case, and the only one worth sizing against.

Average Daily Gain

ADG = \frac{W_f - W_i}{t}

Livestock & FeedLiveweight gained per day across a weighing period — the measurement that turns two scale readings and a calendar into a growth rate, and the basis of every projection to a target weight.

Average Memory Access Time (AMAT)

\mathit{AMAT} = t_h + m\,t_p

Computer ScienceWhat a memory reference costs on average once cache misses are counted: the hit time every access pays, plus the miss penalty paid on the fraction that miss. Stated in clock cycles.

Average Shear Stress (τ = V/A)

\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.

Axial Deformation (δ = PL/AE)

\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.

Axle Group Load Against the Legal Cap

U = \frac{P}{P_{\mathrm{lim}}}

Axle Loads & PavementHow close an axle, an axle group or a whole vehicle sits to the weight it is allowed, as a percentage of the cap. The caps are INPUTS, because they are jurisdiction-specific: 20,000 lb single, 34,000 lb tandem and 80,000 lb gross are the US interstate figures, and a spring load restriction can cut them to half overnight.