Formula solvers starting with L — 63 solvers

Ladder Setback and Angle (4-to-1 Rule)

\theta = \arctan\!\left(\frac{H}{B}\right)

Trades & ConstructionGeometryRelates the height a ladder reaches, the distance its feet sit out from the wall, and the resulting angle. The 4-to-1 rule sets the setback at a quarter of the height, which lands at about 75.5°.

Lambda Ratio (Specific Film Thickness)

\Lambda = \frac{h_{\min}}{\sqrt{R_{q1}^{2} + R_{q2}^{2}}}

Machine DesignThe minimum oil film thickness measured in units of the combined surface roughness. It is the single number that says which lubrication regime a contact is running in — boundary, mixed or full film — and that regime decides almost everything else about friction and wear, which is why a friction or wear figure quoted without it is not much use.

Laminar Friction Factor (f = 64/Re)

f = \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.

Langelier Saturation Index (LSI)

\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.

Larson–Miller Parameter

P = T \left( C + \log_{10} t \right)

Metallurgy & Heat TreatmentThe time–temperature trade that lets a 1000-hour rupture test at 700 °C stand in for a 100,000-hour life at 600 °C. Larson and Miller published it in 1952, and every creep-rupture master curve since has been plotted against it. Two warnings live on this page and both matter: the temperature is ABSOLUTE, and the logarithm is of a time in HOURS.

Larson–Skold Index

\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.

Laser Fluence (Energy per Unit Area)

F = \frac{E_p}{A} = \frac{4 E_p}{\pi d^{2}}

OpticsPulse energy divided by the area it lands on. Fluence in J/cm² is the quantity every ablation threshold, every damage rating and every laser process window is written in, and it is the number that decides whether a pulse marks a surface, cuts it, or does nothing at all.

Latent Heat

Q = m L

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

Latitude of a Traverse Leg

\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.

Law of Cosines

c^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

\frac{a}{\sin A} = \frac{b}{\sin B}

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

Layer Count from Build Height

n = \frac{H}{t}

Additive ManufacturingBuild height divided by layer thickness. The simplest relation in additive manufacturing and the one every schedule rests on, because the layer count multiplies every per-layer cost the machine has — the scan, the recoat, the dwell, the inert-gas sweep.

LC Resonant Frequency

f = \frac{1}{2\pi\sqrt{LC}}

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

Leaching Fraction

LF = 100 \, \frac{V_r}{V_a}

Indoor & GreenhouseCrop ProductionThe share of applied solution that drains away as runoff. Deliberate over-watering is how salts are flushed from a container before they accumulate to the point of damage.

Leaf Area Index (LAI)

L = \frac{A_L}{A_G}

Forestry & Forest EcologyThe total one-sided leaf area a canopy carries, divided by the ground area beneath it. A pure number, and the foundational canopy quantity: light interception, transpiration, rainfall interception and photosynthetic capacity are all read off it before anything else.

Leaf Vapour Pressure Deficit

VPD_L = e_s(T_L) - e_s(T_a)\frac{RH}{100}

Indoor & GreenhouseThe deficit the plant actually experiences, measured between the saturated air inside the leaf and the room. Leaves under lights run cooler than the air by transpiration, and this is the form that accounts for it.

LED Series Resistor

R = \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.

Length Contraction

L = 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.

Length Plus Girth

S = L + 2\,(W + H)

Freight & ShippingGirth is the distance a tape travels around the short way — twice the width plus twice the height. Added to the longest side it gives the single number carriers use as a size limit, and it is why a long thin parcel and a fat short one can be charged alike.

Lens Magnification (m = −d_i/d_o)

m = -\frac{d_i}{d_o}

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

Lens Power in Diopters

P = \frac{1}{f}

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

Lerner Index of Market Power

L = \frac{P - \mathit{MC}}{P}

Prices & MarketsAbba Lerner's 1934 measure of market power: the gap between price and MARGINAL cost, as a fraction of price. Zero under perfect competition, and approaching one for a seller facing almost no competitive constraint.

Lever Rule

W_\alpha = \dfrac{C_0 - C_\beta}{C_\alpha - C_\beta}

Metallurgy & Heat TreatmentHow much of each phase is present in a two-phase field, read straight off a tie line. It is nothing more than a mass balance — the solute has to be somewhere — and that is exactly why it is exact, and exactly why it knows nothing whatever about how long anything takes.

Lewis Number

\mathrm{Le} = \frac{\alpha}{D}

Heat TransferThermal diffusivity divided by mass diffusivity: whether heat or molecules spread faster through the same fluid. Equal to Schmidt divided by Prandtl, and the group that lets a wet-bulb thermometer, a cooling tower and a drying oven be analysed by one set of equations.

Lift Equation

L = q \, S \, C_L

Aerospace & FlightThe lift a wing produces: dynamic pressure times reference area times the lift coefficient. Three of the four terms are things you can measure with a tape and a gauge, and the fourth is where the whole of aerodynamics lives.

Lift-to-Drag Ratio

L/D = \frac{L}{D}

Aerospace & FlightLift divided by drag: how many newtons of weight an aircraft carries for every newton it has to push against. The single number that says how good a flying machine is, and the one a sailplane pilot lives by.

Light Intensity at a New Distance

E_2 = E_1 \left(\frac{d_1}{d_2}\right)^{2}

Indoor & GreenhouseHow photon flux density falls as a fixture is raised above the canopy: intensity drops with the square of distance, so doubling the height quarters the light.

Lighting Power Density

P_d = \frac{P}{A}

Indoor & GreenhouseElectrical lighting load per unit of growing area — the figure a grow space is planned and its cooling sized against.

Lightning Channel Length from the Duration of the Rumble

L = \left(331.3 + 0.606\,T_C\right) \Delta t

Storm & SkyThunder rumbles because the whole channel goes off at once and you hear it a piece at a time: sound from the near end reaches you first, sound from the far end last. The duration of the roll is therefore the difference in range across the channel divided by the speed of sound — three seconds of rumble is about a kilometre of lightning.

Lightning Collection Area of a Rectangular Structure

A_D = L W + 2\,(3H)(L + W) + \pi (3H)^2

Storm & SkyIEC 62305's equivalent collection area: the patch of ground that, if lightning fell on it, would instead have struck the building. It is the footprint, plus a strip three building-heights wide down every side, plus a quarter-circle of the same radius at each corner — one rectangle, two side strips and one full circle when you add the four corners up.

Lightning Striking Distance (Rolling Sphere Radius)

r_s = 10\, I^{0.65}

Storm & SkyThe distance at which a descending lightning leader commits to whatever it is going to hit, as a function of the peak current the flash will carry. This is the electrogeometric relation of IEC 62305, and it is the geometry behind the rolling sphere: roll a sphere of this radius over a structure, and every surface it touches is a surface lightning can reach.

Limiting Drawing Ratio

LDR = \frac{D_{max}}{d}

Machining & TurningThe largest blank a material will draw into a cup in one operation, divided by the cup diameter. The standard measure of drawability, determined by experiment rather than by calculation, and governed by the sheet's plastic strain ratio r — the quantity Lankford and his colleagues introduced in 1950.

Lincoln-Petersen Mark-Recapture Estimate

N = \frac{M \, C}{R}

Ecology & BiodiversityStatisticsPopulation size estimated from two samples: mark M individuals, later catch C, and count how many of those carry a mark. Assumes a closed population, equal catchability and no lost marks — every one of those failing biases the answer, and the Chapman correction is what small samples actually use.

Line Balancing Efficiency

E = \frac{\sum t_i}{N\,C}

Industrial EngineeringFraction of paid station time that is actually working: total task time divided by the stations you staffed times the cycle time. What is left over is balance delay.

Linear Momentum (p = mv)

p = m v

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

Linear Speed from Rotation (v = ωr)

v = \omega r

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

Linear Wave Dispersion Relation

\omega^{2} = g\,k\,\tanh(k d), \qquad \omega = \frac{2\pi}{T},\quad k = \frac{2\pi}{L}

Coastal & Ocean WavesThe one relation the whole of linear wave theory rests on, valid at every depth. The deep-water and shallow-water formulas are both limits of this, and between those limits — which is where most coastal engineering happens — there is nothing else to use. It is transcendental in the wavenumber, so that direction is solved numerically here rather than pretended into a closed form.

Lineweaver–Burk (Double-Reciprocal) Plot

\frac{1}{v} = \frac{K_m}{V_{max}} \cdot \frac{1}{[S]} + \frac{1}{V_{max}}

Chemical EngineeringTake reciprocals of both sides of Michaelis–Menten and the hyperbola straightens into a line with slope K_m/V_max, an intercept of 1/V_max on the vertical axis and −1/K_m on the horizontal. Lineweaver and Burk published it in 1934, before anyone could fit a curve without a slide rule. It is still the clearest picture of enzyme kinetics ever drawn and it is no longer the right way to get numbers.

Liquid Product per Tank

V_p = R_v \, A

Crop ProductionHow much liquid formulation to measure into one tank: the label's volume rate per unit area times the area that tank will cover.

Liquidity Index

LI = \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.

Litter Turnover Time and the Steady-State Forest Floor

\tau = \frac{L_{ss}}{I}

Forestry & Forest EcologyAt steady state the forest floor stops changing because what falls onto it each year equals what rots off it, and the standing stock divided by the annual litterfall is the mean time a leaf spends lying there. It is the same k as Olson's decay model, measured with a scale and a frame instead of a litterbag: τ = 1/k.

Little's Law (Work in Process)

\mathit{WIP} = \mathit{TH} \times \mathit{CT}

Industrial EngineeringThe most general law in operations: the work sitting in a system equals the rate it flows through times how long each job stays. True for any stable system, whatever the arrival pattern.

Load Equivalency Factor (the Fourth Power Law)

\mathrm{LEF} = \left( \frac{P}{P_{18}} \right)^{\,n}

Axle Loads & PavementHow much pavement damage one axle does compared with the standard 18,000 lb (80 kN) single axle, as a power of the load ratio. The exponent is an INPUT here and defaults to 4, because the famous fourth power is a later curve fit and not a finding of the AASHO Road Test, whose own load exponents were 4.79 flexible and 4.62 rigid.

Loading Dose from Volume of Distribution

D_L = V_d \cdot C_p

Biomedical & ClinicalThe dose that fills the apparent volume of distribution to a target plasma concentration in one step, before any elimination.

Loan Payment (Amortized Loan or Mortgage)

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

Money & BusinessLevel payment that retires a loan of principal P in exactly n payments at periodic interest rate i — the mortgage, truck loan and equipment finance formula.

Local Scour Depth at a Bridge Pier (HEC-18)

y_s = 2.0 \, K_1 K_2 K_3 K_4 \, y_1 \left( \frac{a}{y_1} \right)^{0.65} Fr_1^{\,0.43}

Erosion & SedimentThe equation the US Federal Highway Administration recommends in HEC-18, "Evaluating Scour at Bridges": the depth of the hole a pier digs for itself in an erodible bed. Scour is the leading cause of bridge failure in North America, and this is the number that decides how deep the foundation goes.

Log Inactivation from Counts

\mathrm{LR} = \log_{10}\!\left(\frac{N_0}{N}\right)

Water TreatmentThe log reduction actually measured: the base-10 logarithm of the count before a process divided by the count after. This is the reduction observed, not the reduction credited — regulatory credit is granted for a demonstrated CT or an approved process, not for a favourable pair of samples.

Log Mean Temperature Difference (Counterflow)

\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)

\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.

Log Reduction to Percent Kill

P = 1 - 10^{-\mathrm{LR}}

Water TreatmentThe conversion between a log reduction and a percent kill. They are the same statement: 1-log is 90%, 2-log is 99%, 3-log is 99.9%, 4-log is 99.99%. Percent enters and leaves as a percentage when you pick the % unit; internally it is a fraction between 0 and 1.

Log5 Matchup Probability

p = \frac{p_A - p_A p_B}{p_A + p_B - 2 p_A p_B}

Games & RatingsStatisticsPredicts a head-to-head result from two win percentages against a common field. A .500 side beats a .400 side exactly 60% of the time, an even pair gives 0.500 whatever their shared strength, and the whole thing is the Bradley-Terry paired-comparison model wearing different clothes.

Logarithm Change of Base

\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 Power

\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.

Logarithm of a Product

\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\!\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.

Logarithmic Decrement

\delta = \frac{1}{n} \ln\!\left(\frac{x_1}{x_2}\right)

Machine DesignDamping measured the way it is actually measured on site: hit the machine, record the decay, and take the natural log of how much the amplitude shrank over n cycles. No dashpot rating required.

Logistic Population Growth

N(t) = \frac{K}{1 + \dfrac{K - N_0}{N_0}\,e^{-rt}}

Ecology & BiodiversityStatisticsPopulation at time t under density-dependent growth toward a carrying capacity K. Growth is nearly exponential while the population is small, slows through an inflection at K/2, and flattens as the resource runs out.

Longitudinal Stress in a Thin-Walled Cylinder

\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.

Loop Water Expansion Volume

\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.

Lorentz Factor

\gamma = \frac{1}{\sqrt{1 - \beta^{2}}}

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

LRFD Load Combination (1.2D + 1.6L)

U = 1.2 D + 1.6 L

Civil & SurveyingThe everyday strength-design load combination from ASCE 7 (combination 2) and the National Building Code family: dead load at 1.2 plus live load at 1.6.

Lumped Capacitance Cooling Curve

T = 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.

Lumped Capacitance Time Constant

\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.