Formula solvers starting with G — 65 solvers

Galvanic Driving Voltage

\Delta E = E_c - E_a

Corrosion & ProtectionThe potential difference between two metals coupled in the same electrolyte: the noble one's potential less the active one's. The voltage that drives every galvanic cell, wanted or otherwise.

Gas Density from Molar Mass

\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).

Gas Volume at STP

V = n\,V_m

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

Gate Propagation Delay from an RC Load

t_p = R C \ln 2

Semiconductors & ChipsTime for a logic gate's output to cross the halfway point when it drives a capacitive load through its own on-resistance. The output is an RC step response, and the 50% crossing sits at exactly one time constant times the natural log of 2.

Gauge and Absolute Pressure

P_{abs} = P_{gauge} + P_{atm}

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

Gaussian Beam Divergence (θ = λ/πw₀)

\theta = \frac{\lambda}{\pi w_0}

OpticsThe far-field half-angle a diffraction-limited Gaussian beam spreads at, set entirely by its wavelength and the radius of its own waist. It is the hardest constraint in beam delivery: a small waist and a small divergence are the same thing asked for twice, and you cannot have both.

Gaussian Beam Radius at Distance z

w(z) = w_0 \sqrt{1 + \left(\frac{z}{z_R}\right)^{2}}

OpticsThe whole propagation of a Gaussian beam in one line: a hyperbola that is flat near the waist and straightens into the far-field cone once z passes the Rayleigh range. Use it to find how wide the beam will be on the target, or to work backwards from two measured widths to where the waist actually sits.

Gaussian Plume Ground-Level Concentration

C = \frac{Q}{\pi \sigma_y \sigma_z u} \, e^{-H^{2}/(2\sigma_z^{2})}

Air Quality & DispersionGround-level concentration directly downwind of an elevated point source, on the plume centreline, with full reflection from the ground. The core equation of regulatory air-quality modelling.

Gay-Lussac's Law

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

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

GCD–LCM Identity

a \cdot b = \gcd(a,b) \cdot \operatorname{lcm}(a,b)

AlgebraThe product of two whole numbers equals the product of their greatest common divisor and least common multiple — so any one of the four follows from the other three.

Gear Centre Distance

C = \frac{d_1 + d_2}{2}

Machine DesignHow far apart two meshing gear shafts sit: the mean of the two pitch diameters, which is the sum of the two pitch radii. The number that gets drilled into a housing and cannot be changed afterwards.

Gear Pitch Diameter

d = m \, N

Machine DesignThe pitch diameter of a spur gear from its module and tooth count — the diameter of the imaginary rolling cylinder the teeth are cut around. The metric module form, d = m N, with the North American diametral pitch convention P = N/d explained and bridged rather than silently substituted.

Gear Ratio

GR = \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.

Gear Tooth Tangential Force

W_t = \frac{2T}{d}

Machine DesignThe tangential load a gear tooth carries: torque divided by the pitch radius, which is twice the torque over the pitch diameter. The starting number for every bending and contact stress calculation, and for sizing the shaft and bearings behind the gear.

General Addition Rule

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.

General Multiplication Rule

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.

Generalized Hoek–Brown Failure Criterion (2002)

\sigma_1 = \sigma_3 + \sigma_{ci} \left( m_b \frac{\sigma_3}{\sigma_{ci}} + s \right)^{a}

Rock MechanicsThe strength of a jointed rock mass at a given confinement, in the 2002 edition of Hoek, Carranza-Torres and Corkum. Unlike Mohr–Coulomb, the envelope is CURVED: rock gains strength with confinement quickly at first and then far more slowly, which is why a straight line fitted at one stress level misleads badly at another.

Generator Sizing from Connected Load

P_{g} = P_{c} \, D_{f} \left( 1 + m \right)

Electrical TradeContinuous kilowatts a generator must deliver: connected load reduced by a demand factor, then raised by a margin for spare capacity.

Genetic Gain per Year

\Delta G = \frac{R}{L}

Plant Breeding & GeneticsResponse to selection divided by the years one breeding cycle takes. The figure that decides whether a programme is actually improving anything at a useful rate.

Geometric Distribution (First Success)

P(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.

Geometric Mean of Two Numbers

G = \sqrt{ab}

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

Geometric Sequence nth Term

a_n = a_1 \, r^{\,n-1}

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

Geometric Series Sum

S_n = \frac{a_1 (1 - r^{n})}{1 - r}

AlgebraSum of the first n terms of a geometric sequence.

Ghyben-Herzberg Freshwater Lens

z = \frac{\rho_f}{\rho_s - \rho_f} \, h \;\approx\; 40 \, h

Water & WastewaterDepth to the freshwater-saltwater interface below sea level in a coastal or island aquifer, from the hydrostatic balance Ghyben (1888) and Herzberg (1901) found independently. With fresh water at 1000 kg/m³ and seawater at 1025, the ratio is exactly 40, which is where the familiar 40:1 rule comes from.

Gibbs Free Energy and the Equilibrium Constant

\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.

Gibbs Free Energy Change (ΔG = ΔH − TΔS)

\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.

Gilliland Correlation (Actual Stages)

\frac{N - N_{min}}{N + 1} = 1 - \exp\!\left[\left(\frac{1 + 54.4X}{11 + 117.2X}\right)\!\left(\frac{X - 1}{\sqrt{X}}\right)\right],\quad X = \frac{R - R_{min}}{R + 1}

Chemical EngineeringThe empirical bridge across the middle of column design: given the two unbuildable limits — Fenske's fewest stages and Underwood's least reflux — it returns the stage count a column actually needs at the reflux you intend to run.

Glicko Expected Score Against One Opponent

E = \frac{1}{1 + 10^{\,-g \, (r - r_j)/400}}

Games & RatingsStatisticsThe Elo curve with the rating difference first shrunk by the opponent's uncertainty. When the opponent's rating is exactly known the two agree; when it is not, this prediction sits closer to an even result, because a rating you cannot trust cannot support a confident forecast.

Glicko g Attenuation Factor

g(RD) = \frac{1}{\sqrt{1 + \dfrac{3 q^{2} RD^{2}}{\pi^{2}}}}, \qquad q = \frac{\ln 10}{400}

Games & RatingsStatisticsThe factor by which a rating difference is discounted when the opponent's own rating is uncertain. It is one when the opponent is perfectly known and falls toward zero as their rating deviation grows, which is Glicko's way of saying that a prediction can only be as confident as the weaker of the two ratings involved.

Glicko Rating Deviation Growth While Idle

RD = \sqrt{RD_0^{\,2} + c^{2} t}

Games & RatingsStatisticsHow the uncertainty attached to a rating grows during a layoff. Glickman's insight was that a rating is a claim with an error bar, and that the error bar widens whenever a player stops producing evidence — so a 1900 who last played eight years ago is a far weaker claim than a 1900 who played last week.

Glicko Rating Update (One Game)

r' = r + \frac{q \, g \, (s - E)}{\dfrac{1}{RD^{2}} + q^{2} g^{2} E (1 - E)}

Games & RatingsStatisticsGlicko's replacement for Elo's fixed K-factor: the step size is worked out from how well the rating was already known. A player with a wide rating deviation moves a long way on one result; a player whose rating rests on hundreds of games barely moves at all. The two terms in the denominator are exactly that trade — prior knowledge against the information one game carries.

Glide Distance from Altitude

x = h \cdot \frac{L}{D}

Aerospace & FlightHow far an aircraft travels over the ground while descending a given height, in still air: altitude multiplied by the lift-to-drag ratio. The calculation every pilot does silently when the engine goes quiet.

Glycol Dilution

C_1 V_1 = C_2 V_2

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

Glycol Loop Heat Transfer (Capacity Derate)

\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.

Good Engineering Practice Stack Height

H_{GEP} = h_b + 1.5 L

Air Quality & DispersionThe stack height that just clears the turbulent wake of a nearby building, by the US EPA formula: the building height plus one and a half times its lesser dimension. It is a ceiling on credit, not a permit to build tall.

Goodman Fatigue Criterion

\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}

Strength of MaterialsMechanicsThe modified Goodman line for infinite life under a fluctuating stress: alternating stress over endurance limit plus mean stress over ultimate strength equals one over the factor of safety.

Grade to Slope Angle

\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.

Graetz Number

\mathrm{Gz} = \frac{D}{L} \, \mathrm{Re} \, \mathrm{Pr}

Heat TransferHow far into a duct the temperature profile is still developing. It compares the time a fluid spends in the tube with the time heat needs to diffuse from the wall to the centreline, and it decides whether an entry-length correlation or a fully developed one is the right tool.

Graham's Law of Effusion

\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.

Grain Cone Volume

V = \frac{\pi}{3} \left(\frac{d}{2}\right)^{2} h

Farm OperationsThe volume of a cone of grain: the peak above a level fill in a bin, or a whole pile dumped on the ground. One third of the cylinder that would enclose it, which is more than most people guess.

Grain Drying Energy

E = \frac{m_w \, \lambda}{\eta}

Farm OperationsThe fuel energy a dryer needs to evaporate a given mass of water, once its efficiency is accounted for. The calculation behind a propane bill in a wet harvest.

Grain Mass from Volume

m = \rho \, V

Farm OperationsThe mass of grain filling a known volume, from its bulk density. The step between a bin measurement and a quantity that can be sold, hauled or insured.

Grain Weight at a New Moisture

W_2 = W_1 \, \frac{100 - M_1}{100 - M_2}

Crop ProductionWhat a lot of grain weighs once its moisture changes, holding the dry matter constant. This is the core of every grain contract, every drying invoice and every elevator settlement.

Grains per Gallon ↔ ppm Hardness

H = 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.

Grashof Number

\mathrm{Gr} = \frac{g \, \beta \, \Delta T \, L^{3}}{\nu^{2}}

Heat TransferBuoyancy against viscosity: how hard the density difference a temperature difference creates pushes a fluid, compared with how hard the fluid's own stickiness resists. The natural-convection counterpart of the Reynolds number, and the starting point for every hot-wall-in-still-air calculation.

Gravimetric Air Content

A = \dfrac{T - D}{T}

ConcreteAir content worked out from two densities: what the concrete WOULD weigh per unit volume with no air in it at all, and what it actually weighs. The shortfall is air. It needs no pressure meter and no special apparatus beyond a scale and a measure of known volume, and it is the method that catches a pressure meter that has drifted.

Gravitational Field Strength

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

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

Gravitational Potential Energy (Orbital)

U = -\frac{GMm}{r}

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

Gravitational Potential Energy (U = mgh)

U = m g h

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

Gravity Points from a Grain Bill

P = \frac{PPG \times W \times E}{V}

Brewing & FermentationThe gravity a grain bill should produce, from each malt's potential, the weight used, the brewhouse efficiency and the batch volume. The forecast every recipe is built on.

Grazing Days from Available Forage

D = \frac{F \, u}{I}

Livestock & FeedHow many days a herd can graze a paddock, from the forage standing on it, the share of that forage the animals will actually harvest, and what the herd eats each day.

Great Circle Distance (Haversine)

d = 2R\arcsin\sqrt{\sin^{2}\frac{\varphi_2-\varphi_1}{2} + \cos\varphi_1\cos\varphi_2\sin^{2}\frac{\lambda_2-\lambda_1}{2}}

Navigation & PositionThe shortest distance between two points on a sphere, from their latitudes and longitudes. Sinnott's haversine form, which keeps its precision on short legs where the older law-of-cosines version loses it.

Green Split by Critical Flow Ratio

g_i = \frac{y_i}{Y}\,(C - L)

Traffic Flow & SignalsOnce a cycle length is chosen, the effective green left over after lost time is shared among the phases in proportion to their critical flow ratios. Splitting it this way gives every phase the same degree of saturation, which is the arrangement that minimises the worst delay at the intersection.

Green-Ampt Infiltration Rate

f = K\left(1 + \frac{\psi \, \Delta\theta}{F}\right)

Water & WastewaterCivil & SurveyingSoil MechanicsThe Green-Ampt model: infiltration rate under ponded conditions, derived by applying Darcy's law across a sharp wetting front that advances into the soil like a piston. Unlike Horton's curve it is not a fit — every parameter is a measurable soil property.

Greenshields Capacity

q_{max} = \frac{v_f\,k_j}{4}

Traffic Flow & SignalsThe apex of the Greenshields parabola, found by setting its derivative to zero: capacity is a quarter of the free-flow speed times the jam density, and it happens at half the jam density and half the free-flow speed. Three of the tidiest halves in engineering, and all three are consequences of the straight line rather than facts about roads.

Greenshields Flow–Density Parabola

q = v_f\left(k - \frac{k^{2}}{k_j}\right)

Traffic Flow & SignalsSubstitute Greenshields' straight line into q = k v and the flow becomes a downward parabola in density: nothing on an empty road, nothing in a jam, and a maximum in between. It is the single most important picture in traffic theory, because it shows that the same flow is carried at two entirely different densities.

Greenshields Speed–Density Relation

v = v_f\left(1 - \frac{k}{k_j}\right)

Traffic Flow & SignalsGreenshields' 1935 proposal that speed falls in a straight line from the free-flow speed at an empty road to zero at jam density. It is the simplest model that gets the two endpoints right, and everything else in the classical theory of traffic streams is built on it.

Griffith Critical Stress

\sigma_c = \sqrt{\dfrac{2 E \gamma}{\pi a}}

Strength of MaterialsA. A. Griffith's 1921 answer to why glass breaks at a thousandth of the strength its atomic bonds promise: cracks. A crack only grows if the elastic energy released pays for the two new surfaces it creates, and that balance gives a fracture stress falling as the square root of the flaw size. The foundation the whole subject is built on.

Gross Dies per Wafer

N = \frac{\pi D^2}{4A} - \frac{\pi D}{\sqrt{2A}}

Semiconductors & ChipsStandard geometric estimate of how many square dies fit on a round wafer. The first term is the wafer area divided by the die area; the second subtracts the ring of partial dies that fall off the curved edge.

Gross Margin per Unit Area

G = R - C_v

Farm OperationsRevenue per hectare less the variable cost of producing it. The figure that ranks one crop against another on the same field — and the figure most often mistaken for a profit.

Gross Margin Percentage (on Price)

g = \frac{P - C}{P}

Money & BusinessMargin states the same profit as a fraction of the SELLING PRICE. The $80 cost sold at $100 is a 20% margin, not 25% — the classic small-business mix-up.

Ground Sample Distance (GSD)

\mathrm{GSD} = \frac{H \, p}{f}

Photogrammetry & Remote SensingHow much ground one pixel covers: flying height times pixel pitch, divided by focal length. It is the number every drone mapping flight is planned around, and it is the number most often misquoted as an accuracy. GSD is the spacing of pixel centres on the ground — not the smallest feature you can resolve, and not the accuracy of your survey.

Grow Space Air Exchange

Q = \frac{V}{t}

Indoor & GreenhouseExtraction airflow needed to replace the whole volume of a growing space in a given time — the basis on which a tent or room fan is sized.

Gustafson's Law (Scaled Speedup)

S = (1 - p) + p\,N

Computer ScienceSpeedup when the problem grows to fill the machine rather than staying fixed: linear in the processor count, with no ceiling. Here p is the parallel share of the run time on the large machine.

Gutenberg–Richter Frequency–Magnitude Relation

\log_{10} N = a - bM

SeismologyHow many earthquakes of magnitude M or larger a region produces: a straight line on a log-count axis. Fitted to a catalogue rather than derived from physics, and the a and b that come out belong to that catalogue and that region.