Formula solvers starting with E — 80 solvers

Earthwork Volume by Average End Area

V = \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 = \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.

Eckert Number

\mathrm{Ec} = \frac{v^{2}}{c_p \, \Delta T}

Heat TransferKinetic energy against enthalpy: whether the energy carried in a flow's motion is large enough, compared with the heat the temperature difference is already moving, that slowing the flow down heats it noticeably. The group that says when viscous heating has to be in the energy equation.

Economic Order Quantity (Wilson EOQ)

Q = \sqrt{\frac{2\,D\,S}{H}}

Industrial EngineeringMoney & BusinessOrder size that minimises the sum of ordering cost and holding cost: the square root of twice the annual demand times the cost per order, divided by the annual cost of holding one unit.

EER to COP Conversion

\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.

Effect of Speed on Application Rate

V_2 = V_1 \frac{v_1}{v_2}

Crop ProductionHow the applied rate changes when ground speed changes and nothing else does: rate and speed are inversely proportional, so going 25% faster applies 20% less.

Effective Annual Rate from a Nominal Rate

\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

Money & BusinessWhat a quoted nominal annual rate really costs or earns once it compounds m times a year — 12% compounded monthly is 12.68% effective.

Effective Contact Time from Baffling Factor

T_{10} = \theta \times \mathrm{BF}

Water TreatmentThe effective contact time a basin actually delivers: its theoretical detention time multiplied by a baffling factor. T₁₀ is the time by which only 10% of a tracer has reached the outlet, so 90% of the water has had at least that much contact. This, and not the detention time, is the T in CT.

Effective Field Capacity

C = w \, S \, e

Farm OperationsHow fast a machine actually covers ground, from its working width, its field speed and its field efficiency. The number every other machinery calculation is built on.

Effective Population Size

N_e = \frac{4 N_m N_f}{N_m + N_f}

Plant Breeding & GeneticsThe size of an idealised population that would drift as fast as the real one, from the numbers of male and female parents. Unequal sex ratios shrink it sharply.

Effective R-Value with Framing (Parallel Path)

\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.

Effective Stack Height

H = h_s + \Delta h

Air Quality & DispersionThe height dispersion actually starts from: the physical stack plus the plume rise. Because ground-level concentration falls with the square of this height, plume rise is worth more than steel.

Effective Stress (Terzaghi, 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.

Effective Volume in Three-Point Bending

V_{\text{eff}} = \frac{V}{2\,(m+1)^2}

Ceramics & GlassOnly a sliver of a bend bar is meaningfully stressed: the stress falls to zero at the neutral axis and at both supports, so most of the bar is a spectator. The effective volume is the volume of uniformly stressed material that would carry the same risk of failure, and it is what belongs in every Weibull scaling calculation.

Effectiveness from NTU (Counterflow)

\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.

Ekman Number

\mathrm{Ek} = \frac{\nu}{f \, L^{2}}

Air Quality & DispersionViscosity against the Coriolis force. It is tiny almost everywhere in the atmosphere and ocean, which is why friction only matters in thin boundary layers at the top and bottom — and the thickness of those layers is exactly what this number sets.

Elastic Collision — Final Velocity of Body 1

v_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.

Elastic Modulus from Compressive Strength

E_c = k \sqrt{f'_c}

ConcreteThe square-root correlation every concrete code carries: stiffness estimated from compressive strength. The coefficient k is a USER INPUT here and always will be, because ACI, CSA and Eurocode fit different lines through different test populations and their coefficients are not interchangeable — which is exactly the thing a built-in constant would hide.

Elastic Potential Energy

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

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

Elastic Section Modulus (S = I/c)

S = \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.

Elastic Tunnel Convergence

u = \frac{(p_0 - p_i)\,R}{2G}

Rock MechanicsHow far the wall of a circular tunnel moves inwards while the ground around it is still elastic: the difference between the in-situ stress and whatever the support is pushing back with, times the radius, over twice the shear modulus. It is the straight elastic portion of the ground reaction curve, it assumes a hydrostatic stress field, and it is the one part of convergence-confinement that closes in a single line.

Electric Charge (Q = It)

Q = I t

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

Electrical Energy (E = Pt)

E = P t

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

Electrical Power (P = I²R)

P = I^{2} R

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

Electrical Power (P = V²/R)

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

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

Electrical Power (P = VI)

P = V I

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

Elevation from Grade and Distance

E_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.

Elevation on a Parabolic Vertical Curve

E = 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.

Ellipse Area

A = \pi a b

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

Ellipse Perimeter (Ramanujan Approximation)

P \approx \pi \left[ 3(a+b) - \sqrt{(3a+b)(a+3b)} \right]

GeometryPerimeter of an ellipse by Ramanujan's second approximation — accurate to better than one part in a billion for ordinary shapes.

Ellipsoid Volume

V = \frac{4}{3}\pi a b c

GeometryVolume of an ellipsoid from its three semi-axes; setting a = b = c returns the sphere.

Elliptic Curve Security Strength

s = \frac{n}{2}

Cryptography & Key SizingClassical security strength of an elliptic curve group whose order is about n bits: half the key size, because the best known generic attack is Pollard's rho, which finishes in about the square root of the group order. P-256 gives 128 bits, P-384 gives 192, P-521 gives about 260.

Elo Expected Score

E_A = \frac{1}{1 + 10^{\,(R_B - R_A)/400}}

Games & RatingsStatisticsThe share of a point a player is expected to take against a given opponent, worked out from the difference between their two ratings alone. Half a point means an even match; the curve rises toward one as the gap grows and never quite reaches it. A 200-point advantage is worth about 0.76, which is the one number most players already carry in their heads.

Elo Rating Change After a Game

R' = R + K \, (S - E)

Games & RatingsStatisticsThe whole of an Elo update in one line: take the difference between what actually happened and what was expected, multiply by the K-factor, and add it to the old rating. Every point one player gains the other loses, so a closed pool's total rating never changes.

Emission Correction to Reference Oxygen

C_{corr} = C_{meas} \, \frac{20.9 - O_{2,ref}}{20.9 - O_{2,meas}}

Air Quality & DispersionRestates a measured stack concentration at the reference oxygen a limit is written at, so that adding dilution air can no longer make an emission look cleaner than it is. The US EPA convention, built on 20.9 % oxygen in ambient air.

Emission Rate from Stack Concentration

E = C \, Q_v

Air Quality & DispersionThe mass of pollutant leaving a stack per unit time, from the measured concentration and the volumetric flow. This is the number a permit limit is written against and the source strength every dispersion model asks for.

Energy Cost from a Utility Rate

C_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.

Energy Efficiency Ratio (EER)

\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.

Energy Loss in a Hydraulic Jump

\Delta E = \frac{(y_2 - y_1)^{3}}{4 \, y_1 \, y_2}

Fluid MechanicsCivil & SurveyingWater & WastewaterSpecific energy destroyed by a hydraulic jump in a rectangular channel, obtained by subtracting the downstream specific energy from the upstream one and simplifying with the momentum equation.

Energy per Switching Event

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

Semiconductors & ChipsEnergy delivered to a capacitive node each time it is charged to the supply rail. Answers come back in joules because the units engine has no femtojoule — a 1 pF node at 1 V stores 5e-13 J, which is 0.5 pJ.

Energy Ratio Between Two Magnitudes

\dfrac{E_2}{E_1} = 10^{\,1.5\,\Delta M}

SeismologyWhat a difference in magnitude is actually worth in energy: about 32 times per whole unit, and exactly 1000 times per two units. The fact that turns "only one point higher" into the most misleading phrase in earthquake reporting.

Energy Stored in a Capacitor

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

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

Energy Stored in an Inductor

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

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

Environmental Lapse Rate

\Gamma = \frac{T_1 - T_2}{z_2 - z_1}

Air Quality & DispersionHow fast the air cools with height, from two temperatures at two altitudes. Comparing it against the dry adiabatic rate of 9.8 °C/km is what decides whether the atmosphere is stable or unstable.

Epicentral Distance from the S–P Interval

d = \dfrac{\Delta t}{\frac{1}{V_s} - \frac{1}{V_p}}

SeismologyHow far away the earthquake was, from one seismogram and no synchronised clock: the gap between the P and S arrivals divided by the difference of the two slownesses. The oldest trick in observational seismology and still the first one taught.

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

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.

Equivalent Airspeed from True Airspeed

V_E = V \sqrt{\frac{\rho}{\rho_0}}

Aerospace & FlightThe speed the wing thinks it is doing: true airspeed scaled by the square root of the density ratio, so that ½ρV² and ½ρ₀V_E² come out the same. The bridge between the number on the panel and the number over the ground.

Equivalent Annual Cost

\mathit{EAC} = P\,\frac{i\,(1+i)^{n}}{(1+i)^{n} - 1} + M

Industrial EngineeringMoney & BusinessTotal yearly cost of owning and running an asset: the capital cost spread over its life at interest, plus the annual operating cost. The right way to compare a cheap machine that lasts five years with an expensive one that lasts fifteen.

Equivalent Dimension (De = span / ESR)

D_e = \frac{B}{ESR}

Rock MechanicsThe horizontal axis of Barton's support chart: the span, diameter or wall height of the opening divided by an excavation support ratio that says how much risk the opening's PURPOSE allows. Two tunnels of identical size in identical rock get different equivalent dimensions, and therefore different support, because one is a temporary mine drift and the other carries passengers.

Equivalent Horizontal Permeability of Layered Soil

k_{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.

Equivalent Length of a Fitting

L_{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.

Equivalent Round Duct Diameter

D_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.

Equivalent Tyre Contact Radius

a = \sqrt{\frac{P}{\pi \, p}}

Axle Loads & PavementThe radius of the circle that carries the same wheel load at the same contact pressure as the real footprint. Layered-elastic pavement analysis is built on circular loaded areas, so every real, roughly rectangular tyre print gets turned into this before anything can be calculated.

Equivalent Weight from Molar Mass and Valence

\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.

Erlang B Blocking Probability

B = \frac{A^{N}/N!}{\displaystyle\sum_{k=0}^{N} A^{k}/k!}

Computer ScienceProbabilityFraction of calls turned away by N circuits carrying A erlangs of offered traffic, on the assumption that a blocked call simply goes away. The standard sizing tool for trunks, agents and connection pools.

Escape Velocity

v = \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.

Estimated GFR (MDRD 4-Variable, IDMS-Traceable)

\mathrm{eGFR} = 175 \cdot S_{cr}^{-1.154} \cdot a^{-0.203} \cdot F

Biomedical & ClinicalEstimated glomerular filtration rate in mL/min per 1.73 m² by the IDMS-traceable four-variable MDRD equation.

Estimated Time En Route

t = \frac{d}{V_g}

Navigation & PositionDistance divided by ground speed. Trivial arithmetic, and the single most common place a navigation plan goes wrong, because the speed put into it is so often the airspeed or the log speed rather than the speed actually made good.

Euclid's Pythagorean Triple

c = m^2 + n^2

GeometryEuclid's generator turns any two whole numbers into a right triangle: legs m² − n² and 2mn, hypotenuse m² + n². Every primitive triple arises exactly once this way.

Euler Critical Buckling Load

P_{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.

Euler Number (Pressure against Inertia)

Eu = \frac{\Delta p}{\rho v^{2}}

Fluid MechanicsA pressure difference divided by the dynamic pressure scale ρv². It is what makes a pressure drop portable between a model and the real thing: two geometrically similar flows at the same Euler number lose the same fraction of their velocity head, whatever their size. The friction factor, the loss coefficient K and the pressure coefficient Cp are all this same group wearing different hats.

Exceedance Risk over a Design Life

R = 1 - \left(1 - \frac{1}{T}\right)^{n}

Water & WastewaterCivil & SurveyingProbabilityThe probability that a storm of return period T is equalled or exceeded at least once during n years of exposure. This is the calculation that turns a reassuring-sounding return period into the number that actually matters: a 100-year storm has a 26 percent chance of turning up during a 30-year mortgage.

Excess Air from Flue Gas Oxygen

EA = \frac{O_2}{20.9 - O_2}

Air Quality & DispersionHVAC & HydronicsHow much air beyond stoichiometric is passing through a burner, read straight off the oxygen in the flue gas. The single most useful number a combustion analyser gives you, because everything about efficiency follows from it.

Expansion Loop Leg Length (Guided Cantilever)

L = \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.

Expansion Tank Acceptance Volume

V_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.

Expected Annual Lightning Strikes to a Structure

N_D = N_G \, A_D \, C_D \times 10^{-6}

Storm & SkyIEC 62305's first arithmetic step: multiply the local ground flash density by the structure's collection area and a factor for how exposed its surroundings are, and you have the expected number of direct strikes per year. It is a long-run average over a very lumpy process, and the number it produces is usually far smaller than one.

Expected Successes in a Dice Pool

E = \frac{n \, (d - t + 1)}{d}

Games & RatingsStatisticsThe average number of dice in a pool of n that meet or beat a target number t. It is the per-die chance multiplied by the number of dice — the binomial mean, arrived at without ever needing the binomial distribution, because expectation adds whether or not the dice are independent.

Expected Sum of Several Dice

E = \frac{n \, (d + 1)}{2}

Games & RatingsStatisticsThe long-run average total from rolling n identical dice of d faces each. One die averages the midpoint of its faces, and expectation adds, so n of them average n times that — which is why the answer lands on a half whenever the number of dice is odd.

Expected Trials Until First Success

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

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

Expected Value of a Bet

E = 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.

Expected Value of an Exploding Die

E = \frac{d \, (d + 1)}{2 \, (d - 1)}

Games & RatingsStatisticsThe long-run average of a die that is rolled again and added whenever it lands on its highest face, with the rerolls themselves able to explode without limit. An ordinary six-sided die averages 3.5; the same die exploding averages 4.2, and the whole of that extra 0.7 comes from a geometric series that converges because each further explosion is six times rarer than the last.

Expected Value of the Higher of Two Dice

E_{\max} = \frac{(d + 1)(4d - 1)}{6d}

Games & RatingsStatisticsThe long-run average when two identical dice are rolled and only the larger is kept. On six-sided dice it is 161/36, about 4.47, against 3.5 for a single die — the extra is what an advantage on a roll is actually worth, and it is smaller than most people guess.

Expected Value of the Lower of Two Dice

E_{\min} = \frac{(d + 1)(2d + 1)}{6d}

Games & RatingsStatisticsThe long-run average when two identical dice are rolled and only the smaller is kept. On six-sided dice it is 91/36, about 2.53, against 3.5 for a single die. Added to the average of the higher die it gives exactly d + 1, and that identity is a complete proof that both formulas are correct.

Exponential Decay

A = A_0 (1 - r)^{t}

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

Exponential Growth

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

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

Exponential Growth by Doubling Time

N = N_0 \cdot 2^{t/T}

AlgebraGrowth of a quantity that doubles every fixed interval T.

Exponential Growth of Biomass

X = X_0 \, e^{\mu t}

Chemical EngineeringUnrestricted growth: every cell divides at the same average rate, so the population's rate of increase is proportional to how much of it there already is. Plot the natural log of biomass against time and the exponential phase is the straight stretch, whose slope is µ.

External Load That Separates a Preloaded Joint

P_{0} = \frac{F_{i}}{1 - C}

Machine DesignThe external tensile load at which the clamped members lose all their compression and the joint opens. Past this point the bolt carries the whole load on its own, and every advantage of preload disappears at once.

Extruder Net Output

Q = Q_d - Q_p

Polymers & Plastics ProcessingWhat actually leaves the die: the drag the barrel puts into the melt, less what the die pressure pushes back. Plotted against head pressure it is the SCREW CHARACTERISTIC — a falling straight line — and where it crosses the die's own rising line is the operating point the machine settles at.

Eyring Reverberation Time

T_{60} = \frac{0.161\,V}{-S\,\ln(1-\bar{\alpha})}

Acoustics & NoiseSabine's equation, corrected for rooms that actually absorb. Once a room is treated, sound is lost on every reflection rather than continuously, and the logarithm in the denominator is what accounts for it — Sabine's form runs long in exactly the rooms people pay to have treated.