Formula solvers starting with C — 181 solvers

Cable Arc Length (Parabolic Series)

s = L \left( 1 + \frac{8 d^{2}}{3 L^{2}} - \frac{32 d^{4}}{5 L^{4}} \right)

Strength of MaterialsHow much cable is actually strung between two supports, which is always more than the span. The series says the excess grows with the square of the sag ratio, so a shallow cable is barely longer than its span and a deep one is noticeably longer. It is what you order cable by, what a stringing chart is built on, and what tells you how much slack a temperature change has to absorb.

Cable Horizontal Tension (Parabolic)

H = \frac{w L^{2}}{8 d}

Strength of MaterialsThe one relation every suspended cable problem starts from. A flexible cable carrying a load spread evenly along the horizontal hangs in a parabola, and the horizontal pull it needs is the load times the span squared over eight times the sag. Read it backwards and it is the most useful statement in the subject: sag and tension trade against each other, and you cannot have both small.

Cable Sag Ratio

\frac{d}{L}

Strength of MaterialsSag divided by span. It is the single number that decides everything else about a suspended cable: how far above H the support tension runs, how much longer the cable is than the span, and — the reason this page exists — whether the parabola may be used at all. Below about 1 in 10 the parabola and the catenary agree within a percent; above it they part company.

Calories Burned from MET, Mass and Time

E = \mathrm{MET} \cdot m \cdot t

Everyday & HealthEnergy expended during an activity: one MET is defined as one kilocalorie per kilogram of body mass per hour.

Camera Trigger Interval

t = \frac{B}{V_g}

Photogrammetry & Remote SensingHow often the shutter must fire: the air base divided by the ground speed. The bridge between a geometric flight plan and the two settings the aircraft actually flies with — an interval and a speed. If the camera cannot cycle this fast, the plan is not flyable and the speed has to come down.

Canopy Light Extinction (Beer's Law for a Canopy)

I = I_0 \, e^{-k L}

Forestry & Forest EcologyLight falls off exponentially as it passes down through leaf layers, exactly as it falls off passing through a coloured solution or a slab of lead. Monsi and Saeki brought the physicists' attenuation law into plant ecology in 1953, and the extinction coefficient k that came with it encodes how the leaves are tilted.

Cantilever Deflection — End Load

\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.

Cantilever Deflection — Uniform Load

\delta = \frac{w L^{4}}{8 E I}

Strength of MaterialsCivil & SurveyingMechanicsTip deflection of a cantilever carrying a uniformly distributed load along its whole length, δ = wL⁴/8EI — three-eighths of the sag the same total load would cause at the tip.

Capacitance (C = Q/V)

C = \frac{Q}{V}

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

Capacitive Reactance (X_C = 1/2πfC)

X_C = \frac{1}{2\pi f C}

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

Capacity Factor

CF = \frac{E}{P_r \, t}

Solar & Wind PowerEnergy actually generated over a period, divided by the energy the plant would have made running flat out at its rated power for the whole of it. The number that separates a nameplate from a power station.

Capacity of a Signalised Approach

c = s\,\frac{g}{C}

Traffic Flow & SignalsAn approach discharges at its saturation flow while it is green and at nothing while it is red, so its capacity is the saturation flow scaled by the fraction of the cycle it owns. One line, and it is the bridge between a signal's timings and the vehicles per hour it actually delivers.

Capacity Rate Ratio (Cr)

C_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.

Capillary Number (Viscous Stress against Surface Tension)

Ca = \frac{\mu v}{\sigma}

Fluid MechanicsViscous stress dragging on an interface, divided by the surface tension pulling it back into shape. It is the number behind dip coating, wetting and dewetting, the thickness of the film a withdrawn plate carries, the trapping of residual oil in a rock pore, and whether a bubble in a tube leaves a liquid layer behind it. There is no length in it, which is what makes it different from everything else on this page.

Capital Recovery Factor

\mathit{CRF} = \frac{i\,(1+i)^{n}}{(1+i)^{n} - 1}

Industrial EngineeringMoney & BusinessFraction of a capital sum that must be recovered each year to repay it with interest over n years. Multiply a first cost by CRF and you get its annual equivalent.

Capstan Equation (Belt Tension Ratio)

\frac{F_1}{F_2} = e^{\mu \theta}

Machine DesignThe limit on how hard a belt or rope can pull before it slips: the tension ratio a wrapped flexible member can hold grows exponentially with the coefficient of friction and the wrap angle. Why three turns of line round a bollard hold a ship, and why the wrap angle must be in radians.

Carbon Equivalent (IIW)

CE = C + \frac{Mn}{6} + \frac{Cr + Mo + V}{5} + \frac{Ni + Cu}{15}

Welding & JoiningThe International Institute of Welding's screening number for weldability: every alloying element on the mill certificate converted into the amount of carbon that would harden the steel as much, then added up. It feeds a code's preheat table. It is not itself a verdict on anything.

Carbon from Dry Biomass (and CO₂ Equivalent)

C = f_C \, B

Forestry & Forest EcologyRoughly half of oven-dry plant matter is carbon, so a carbon stock is a biomass multiplied by a fraction — 0.47 is the IPCC default. Multiply the carbon by 44/12 to reach the CO₂ that would be released if it all burned, which is the figure carbon markets and inventories trade in.

Cardiac Output (Heart Rate × Stroke Volume)

\mathrm{CO} = \mathrm{HR} \cdot \mathrm{SV}

Biomedical & ClinicalCardiac output: the volume the left ventricle ejects each beat, multiplied by the number of beats each minute.

Carnot Efficiency

\eta = 1 - \frac{T_c}{T_h}

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

Carr Compressibility Index

CI = 1 - \frac{\rho_b}{\rho_t}

Additive ManufacturingThe same tapped-and-loose experiment as the Hausner ratio, reported as the percentage of its loose volume the powder gives up when tapped. Ralph Carr published it in Chemical Engineering in 1965. It is not a second measurement and it carries no information the Hausner ratio does not — CI = 100(1 − 1/HR) exactly — but it is the form most pharmaceutical and powder-handling specifications are written in, so both belong on the site.

Case Fatality Ratio (CFR)

\mathrm{CFR} = \frac{D}{C}

Epidemiology & Diagnostic AccuracyThe share of diagnosed cases that ended in death. Long called the case fatality RATE, and the CDC flags that name as a misnomer for the same reason as the attack rate: no time appears in the denominator, so it is a proportion.

Catenary Arc Length

s = 2 a \sinh \frac{L}{2a}

Strength of MaterialsThe exact length of cable hanging between two level supports under its own weight. No series, no truncation — the arc length of a catenary is a hyperbolic sine, and that is the closed form. Compare it against the parabolic series on the same span to see how little the difference is when the cable is taut and how quickly it grows when it is not.

Catenary Parameter

a = \frac{H}{w}

Strength of MaterialsOne length that fixes the entire shape of a hanging chain: the horizontal tension divided by the weight per unit length of cable. Every catenary is the same curve, y = a cosh(x/a), scaled by this number — a small a gives a sharply drooping rope, a large a gives a nearly straight one. It is also a real distance on the drawing, the height of the low point above the directrix.

Catenary Sag

d = a \left( \cosh \frac{L}{2a} - 1 \right)

Strength of MaterialsThe exact sag of a cable that carries only its own weight — a bare conductor, a chain, a slack guy wire, an unloaded haul rope. The parabola is not an approximation to this; it is the answer to a different loading. This is the answer when every metre of CABLE weighs the same, rather than every metre of ground beneath it.

Catenary Tension at a Point

T = H \cosh \frac{x}{a}

Strength of MaterialsThe tension anywhere along a free-hanging cable, from the horizontal distance to its lowest point. Written the other way it is the prettiest result in the subject: T = w·y, the tension at any point equals the weight per metre times the height of that point above the directrix. The cable holds itself up by hanging from a line that is not there.

Cathodic Protection Current Demand

I = A \, i \, f

Corrosion & ProtectionThe total current a cathodic protection system has to supply: the structure's surface area times the current density its environment demands, times the fraction of that surface actually exposed through the coating.

Cavitation Number (Margin above Vapour Pressure)

\sigma_c = \frac{p - p_v}{\tfrac{1}{2} \rho v^{2}}

Fluid MechanicsHow much pressure margin a flow has above the vapour pressure of its own liquid, measured in velocity heads. Low cavitation number means the pressure dips in the flow can reach vapour pressure, bubbles form, and they collapse somewhere downstream with enough violence to pit steel. Dimensionally it is the Euler number; in intent it is a completely different question.

Centripetal Acceleration (a = v²/r)

a_c = \frac{v^2}{r}

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

Centripetal Acceleration (a = ω²r)

a_c = \omega^{2} r

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

Centripetal Force (F = mv²/r)

F_c = \frac{m v^2}{r}

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

Chain Length in Pitches

L = \frac{2C}{p} + \frac{N_1 + N_2}{2} + \left( \frac{N_2 - N_1}{2\pi} \right)^{2} \frac{p}{C}

Machine DesignHow many pitches of roller chain a drive needs, from the two sprocket tooth counts, the chain pitch and the centre distance. The answer must be rounded up to a whole number — and preferably an even one, so the chain closes without an offset link.

Chance of Meeting a Target Number on One Die

p = \frac{d - t + 1}{d}

Games & RatingsStatisticsThe chance that a single fair die of d faces shows at least the target number t. Counting the faces that succeed is the whole derivation: there are d − t + 1 of them, and the plus one is the step everybody drops the first time.

Characteristic Velocity (c*)

c^{*} = \frac{p_c A_t}{\dot m}

Aerospace & FlightChamber pressure times throat area divided by mass flow. It measures the COMBUSTION alone — how much pressure the propellant makes per unit of flow — and is deliberately blind to whatever the nozzle does downstream of the throat.

Chargeable Weight

W_c = \max\!\left(W_a,\; W_{\mathrm{dim}}\right)

Freight & ShippingThe weight the invoice is built on: whichever is larger, the scale weight or the dimensional weight. One line of arithmetic, and the reason light bulky freight costs what it does.

Charles's Law

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

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

Chemical Feed Rate (lb/day = mg/L × MGD × 8.34)

\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.

Chemical Feed Rate from Dose

W = 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.

Chemical Mass Consumed Over a Period

m = 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.

Chemical Treatment Cost

C_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.

Chemical-Consuming Loss Rate

L = 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.

Chemostat Steady State — µ = D = F/V

\mu = D = \frac{F}{V}

Chemical EngineeringThe most counterintuitive result in bioprocess engineering, and the one worth learning first: in a steady-state chemostat the operator sets the culture's specific growth rate by turning a pump. Cells are washed out at the dilution rate D = F/V, so at steady state they must be replaced at exactly that rate, which means µ = D. The organism does not get a vote.

Chemostat Washout (Critical Dilution Rate)

D_{crit} = \frac{\mu_{max} \, S_0}{K_s + S_0}

Chemical EngineeringThe pump setting past which the chemostat stops working. Since µ = D at steady state, and µ can never exceed what the incoming substrate supports, there is a dilution rate above which cells are removed faster than they can possibly be made. Cross it and the culture washes out — not slowly, and not recoverably without reinoculating.

Chezy Equation

V = C \sqrt{R \, S}

Fluid MechanicsCivil & SurveyingWater & WastewaterAntoine Chezy's 1769 uniform-flow equation: mean velocity equals the Chezy coefficient times the square root of the hydraulic radius times the energy slope.

Chi-Square Contribution of One Cell

\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.

Chick–Watson Inactivation

\log_{10}\!\left(\frac{N_0}{N}\right) = k \, C^{\,n} \, t

Water TreatmentChick's 1908 first-order disinfection law with Watson's concentration exponent: the log reduction achieved rises with the disinfectant concentration raised to a dilution coefficient n, and linearly with contact time. When n = 1 it reduces to the familiar CT dose law. It is an idealisation — real curves show a shoulder at the start and a tail at the end.

Chiller Efficiency (kW per Ton)

\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.

Chiller Heat Rejection

Q_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.

Chilton–Colburn Analogy for Mass Transfer

\mathrm{Sh} = \frac{f}{2} \, \mathrm{Re} \, \mathrm{Sc}^{1/3}

Chemical EngineeringMass transfer inferred from pressure drop. Chilton and Colburn found in 1934 that the j-factor for mass transfer equals half the Fanning friction factor, which lets a Sherwood number be predicted from nothing but a Reynolds number and a Schmidt number.

Chlorine Dose from a Weight of Product

C = \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.

Chlorine Dose, Demand and Residual

D = 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.

Chvorinov's Rule

t_s = B \left( \dfrac{V}{A} \right)^{2}

Metallurgy & Heat TreatmentA casting freezes in a time proportional to the square of its volume-to-surface-area ratio. Chvorinov published it in 1940, and it is the whole basis of riser design: make the riser's modulus larger than the casting's and the riser will still be liquid when the casting needs feeding.

Circle Radius from Centre and a Point

r = \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.

Circular Sector Area

A = \frac{1}{2} r^{2} \theta

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

Circular Segment Area

A = \frac{r^2}{2}(\theta - \sin\theta)

GeometryArea of the segment cut off by a chord — the sector minus the triangle, from radius and central angle.

Circumference of a Circle

C = 2 \pi r

GeometryDistance around a circle of radius r.

Clarifier Solids Loading Rate

\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.

Classical Probability

P = \frac{f}{n}

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

Clausius–Clapeyron Equation (Two-Point Form)

\ln\!\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

ChemistryThermodynamicsRelates two points on a liquid's vapour-pressure curve to its molar enthalpy of vaporisation, assuming ΔH is constant over the interval and the vapour behaves ideally.

Clear-Sky Temperature (Berdahl-Martin)

T_{sky} = T_{air}\left[\,0.711 + 0.56\left(\tfrac{t_{dp}}{100}\right) + 0.73\left(\tfrac{t_{dp}}{100}\right)^{2}\right]^{1/4}

Air Quality & DispersionHeat TransferEffective radiating temperature of a cloudless sky, from air temperature and dew point, using the Berdahl-Martin (1984) clear-sky emissivity. This is why frost forms on a clear night and not a cloudy one: a clear sky radiates like a body tens of degrees colder than the air.

Closed Loop Slug Dose Volume

V_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.

CMOS Dynamic Switching Power

P = \alpha C V^2 f

Semiconductors & ChipsPower a CMOS chip burns charging and discharging its own capacitance: the activity factor, the switched capacitance, the supply voltage squared, and the clock frequency. The squared voltage is the reason every process generation chased a lower rail.

CO₂ to Enrich a Sealed Room

m = V \, (c_2 - c_1) \, \rho

Indoor & GreenhouseMass of carbon dioxide needed to raise a sealed growing space from ambient to a target concentration, from the room volume and the density of CO₂ at room conditions.

Cobb–Douglas Production Function

Y = A\,K^{\alpha}\,L^{\beta}

Prices & MarketsOutput produced from capital and labour, each raised to its own exponent, times a productivity factor A. Charles Cobb and Paul Douglas fitted it to US manufacturing from 1899 to 1922 and published it in 1928.

Coefficient of Determination (R²)

R^{2} = r^{2}

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

Coefficient of Performance (COP)

\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.

Coefficient of Restitution

e = \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.

Coefficient of Variation

CV = \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.

Coffin-Manson Strain-Life Relation

\dfrac{\Delta\varepsilon_p}{2} = \varepsilon_f' \left( 2 N_f \right)^{c}

Strength of MaterialsManson's and Coffin's independent 1953-54 discovery that plastic strain amplitude against reversals to failure is a straight line on log-log axes. It is the tool for low-cycle fatigue — thermal cycling, seismic demand, forming operations, solder joints — where the part yields every cycle and a stress-based method has nothing to say.

Cohen's d (Effect Size)

d = \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.

Cohen's Kappa (Inter-Rater Agreement)

\kappa = \frac{p_o - p_e}{1 - p_e}

Epidemiology & Diagnostic AccuracyCohen's 1960 measure of how far two raters agree beyond what chance alone would have produced, given how often each of them uses each label.

Colebrook–White Friction Factor

\frac{1}{\sqrt{f}} = -2 \log_{10}\!\left(\frac{\varepsilon}{3.7 D} + \frac{2.51}{Re \sqrt{f}}\right)

Fluid MechanicsHVAC & HydronicsPhysicsThe reference equation for turbulent friction in a rough pipe, and the curve every Moody diagram is drawn from. It has f on both sides, so this page solves it by iteration rather than by any closed formula.

Column Material Balance (Distillate and Bottoms Split)

D = F\,\frac{z_F - x_B}{x_D - x_B}

Chemical EngineeringHow much of the feed leaves overhead and how much out the bottom, from nothing but the three compositions. Two balances — total flow and light key — solved together, and the answer is a lever rule.

Combinations (nCr)

{}_{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.

Combine Harvest Loss from a Kernel Count

L = \frac{n \, w_{1000}}{1000}

Crop ProductionTurns kernels counted on the ground behind a combine into the yield being left there. The check that decides whether a machine needs adjusting, and the only one that costs nothing but a hoop and five minutes.

Combined Axial and Bending Stress

\sigma = \frac{P}{A} + \frac{M c}{I}

Strength of MaterialsCivil & SurveyingMechanicsExtreme-fibre stress where an axial force and a bending moment act together, as in an eccentrically loaded column or a beam-column. Enter a negative c for the relieved face.

Combined Convection and Radiation Coefficient

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.

Combined Gas Law

\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.

Combining Sound Levels

L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)

Acoustics & NoiseDecibels do not add. Two 60 dB machines running together measure 63 dB, not 120 — the energies add and the logarithm is taken again at the end. This is the most misused arithmetic in noise control, and the fix is one line long.

Combustion (Stack) Efficiency — Siegert Formula

\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.

Common Rafter Length from Run and Pitch

L = R\sqrt{1 + \left(\frac{p}{12}\right)^2}

Trades & ConstructionGeometryThe line length of a common rafter: the level run from ridge to wall plate, stretched by the slope factor for a pitch of p in 12.

Compactive Viscosity of Snow (Kojima)

\eta = \eta_0 \, e^{f \rho_s}

Snow & IceSoil MechanicsStrength of MaterialsThe compactive viscosity of snow, rising exponentially with its own density — Kojima's 1967 relation, in the form Anderson's SNTHERM and most land-surface schemes have used since. It is why snow settles fast on the first night and then almost stops: the act of compacting is what makes it resist compacting.

Compass to True Heading (Variation and Deviation)

T = C + D + V

Navigation & PositionThe TVMDC chain in one line: a compass heading corrected by deviation for the ship's own iron and by variation for the earth's field, giving the true heading a chart is drawn in. East corrections are positive and are added going this way.

Competitive Inhibition — Apparent K_m

K_m^{app} = K_m \left( 1 + \frac{[I]}{K_i} \right)

Chemical EngineeringA competitive inhibitor binds the same active site the substrate wants, so the two compete and the enzyme simply needs more substrate to reach any given rate. K_m appears to rise by the factor (1 + [I]/K_i). V_max does not move at all — enough substrate always wins — and that unchanged V_max is the signature that identifies the mechanism.

Complement Rule

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.

Completing the Square: the Constant Needed

k = \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.

Composite Density from the Rule of Mixtures

\rho_c = \rho_f V_f + \rho_m (1 - V_f)

Strength of MaterialsThe one rule of mixtures that is not a bound but an identity: mass adds whichever way the load runs, so a void-free composite's density is exactly the volume-weighted average of its constituents. That makes it the standard check on a laminate — weigh the panel, measure it, and the shortfall against this number is porosity.

Composite Transmission Loss

TL_c = 10\log_{10}\!\left(\frac{S_w + S_d}{S_w\,10^{-TL_w/10} + S_d\,10^{-TL_d/10}}\right)

Acoustics & NoiseA wall with a door or a window in it, rated as one partition. The two paths add by AREA AND TRANSMISSION, never by decibels, and the answer is the most sobering arithmetic in building acoustics: the weak element sets the rating almost by itself.

Compound Gear Train Value

e = \frac{N_2 \, N_4}{N_1 \, N_3}

Machine DesignThe overall ratio of a two-stage compound gear train: the product of the driven tooth counts over the product of the driver tooth counts. The rule that lets a 40:1 reduction be built from two ordinary meshes instead of one enormous wheel.

Compound Interest (Periodic)

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

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

Compressibility Factor (Z = PV/nRT)

Z = \frac{P V}{n R T}

ChemistryThermodynamicsHow far a real gas departs from ideal behaviour, as a single multiplier on the ideal gas law. Z = 1 is ideal; below 1 attraction dominates, above 1 the molecules' own volume does.

Compression Ratio

\mathit{CR} = \frac{S_u}{S_c}

Computer ScienceHow many times smaller a file got: original size divided by compressed size. A ratio of 4 means the file shrank to a quarter, which is a 75% saving, not a 25% one.

Compressive Strength of Sintered Snow

\sigma_c = \sigma_i \left( \frac{\rho_s}{\rho_i} \right)^{\! n}

Snow & IceStrength of MaterialsSoil MechanicsUnconfined compressive strength of sintered snow as a power law in density, normalised to solid ice. It is an empirical fit with very wide scatter, not a law, and it leaves out the variable that matters as much as density does: how long the snow has been sitting undisturbed.

Concrete Bags from Volume and Bag Yield

N = \frac{V}{y}

Trades & ConstructionCounts bags of premixed concrete: the volume of the pour divided by the yield printed on the bag, typically 0.6 ft³ for an 80 lb bag or 13.5 L for a 30 kg bag.

Concrete Batch Yield

Y = \dfrac{\sum m}{\rho_{fresh}}

ConcreteThe volume of concrete a batch actually produced: everything that went into the mixer, weighed, divided by the density of the fresh concrete that came out. Mass is conserved and volume is not, which is why yield is checked by weighing rather than by adding up what the mix design said.

Concrete Volume with Waste Allowance

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.

Condensate Return Percentage

\%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.

Condenser Water Flow Rate

\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.

Conditional Probability

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.

Conduction Through a Pipe Wall

\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.

Conductor Resistance Temperature Correction

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.

Conduit Fill

F = \frac{n A_{c}}{A_{i}}

Electrical TradeShare of a raceway's internal cross-section taken up by the conductors pulled through it, from the area of one conductor and how many there are.

Cone Frustum Volume (Truncated Cone)

V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)

GeometryVolume of a cone with its tip cut off parallel to the base — the shape of a bucket, a lampshade or a hopper.

Cone Lateral Surface Area

A = \pi r l

GeometryCurved surface of a right circular cone from its base radius and slant height, excluding the base disc.

Cone Slant Height

l = \sqrt{r^2 + h^2}

GeometrySlant height of a right circular cone by Pythagoras on its axial cross-section, from base radius and vertical height.

Cone Total Surface Area

A = \pi r (r + l)

GeometryWhole surface of a solid right circular cone: the curved side pi*r*l plus the base disc pi*r^2.

Cone Volume

V = \frac{1}{3} \pi r^{2} h

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

Confidence Interval Lower Limit

L = \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 = \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.

Conservation of Momentum (Two Bodies)

m_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.

Considère Criterion and Uniform Elongation

\varepsilon_u = n, \qquad e_u = e^{\,n} - 1

Metallurgy & Heat TreatmentConsidère's 1885 argument: a tensile bar necks the moment the metal stops hardening faster than the section shrinks. Put a Hollomon curve into that condition and the answer is startlingly simple — the true strain at the onset of necking equals the strain-hardening exponent. This page turns that n into the uniform elongation a tensile certificate reports.

Consolidation Settlement of Normally Consolidated Clay

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.

Consumer Surplus (Linear Demand)

\mathit{CS} = \tfrac{1}{2}\,(P_{\max} - P)\,Q

Prices & MarketsArea of the triangle between a straight-line demand curve and the price actually paid: what buyers would have been willing to pay, less what they did pay. P_max is the choke price, where the demand line meets the vertical axis and the quantity demanded reaches zero.

Continuity Equation (A₁v₁ = A₂v₂)

A_1 v_1 = A_2 v_2

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

Continuous Compounding

A = P \, e^{r t}

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

Convection Film Resistance

R = \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.

Conversion in n Equal CSTRs in Series

X = 1 - \frac{1}{\left(1 + \mathrm{Da}\right)^{n}}

Chemical EngineeringConversion through a cascade of n identical stirred tanks, where Da is the Damköhler number of a single tank. Splitting one vessel into several buys conversion for nothing but partition walls, and as n grows the cascade converges on plug flow.

Cooling Time t8/5 (Thick Plate)

t_{8/5} = \frac{F_3 \, H}{2 \pi \lambda} \left( \frac{1}{500\,^\circ\mathrm{C} - T_0} - \frac{1}{800\,^\circ\mathrm{C} - T_0} \right)

Welding & JoiningThe seconds a weld spends falling from 800 °C to 500 °C — the range in which austenite decides what it is going to become. This is the number that actually governs the hardness and toughness of the heat-affected zone, and it is what heat input, preheat and thickness are all really arguing about. The three-dimensional form, for heat running away into thick plate.

Cooling Tower Approach

A = 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 Drift Loss

D = \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.

Cooling Tower Evaporation Rate

E = 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.

Cooling Tower Heat Rejection

Q = 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).

Cooling Tower Makeup Water Rate

M = 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 Range

\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.

Cooper-Jacob Drawdown

s = \frac{2.303 \, Q}{4 \pi T} \log_{10}\!\left(\frac{2.25 \, T \, t}{r^{2} S}\right)

Water & WastewaterTransient drawdown in a confined aquifer by the Cooper-Jacob (1946) straight-line approximation: the Theis solution with its infinite series truncated after two terms, valid once the dimensionless time u has fallen below 0.01.

Cooper-Jacob Validity Parameter u

u = \frac{r^{2} S}{4 T t}

Water & WastewaterThe dimensionless argument of the Theis well function. Cooper-Jacob's straight-line approximation is only true once u has fallen below 0.01, and this is the calculation that tells you whether you have waited long enough.

Coriolis Parameter from Latitude

f = 2\,\Omega \sin\varphi

Air Quality & DispersionThe Coriolis parameter is not an abstract constant — it is a place on the Earth. Twice the planet's rotation rate times the sine of the latitude, zero on the equator and maximum at the poles, and it is the f that the Rossby and Ekman numbers both divide by.

Corn Yield Estimate (Yield Component Method)

Y = \frac{E \, k}{K} \, w_b

Crop ProductionThe standing-crop yield estimate every extension service teaches: count the ears on a measured strip, count the kernels on a representative ear, and divide by the kernels it takes to fill a bushel.

Corrected Calcium for Albumin

\mathrm{Ca}_{corr} = \mathrm{Ca} + 0.8\,(4.0 - \mathrm{Alb})

Biomedical & ClinicalTotal serum calcium adjusted upward for low albumin, since roughly 40 % of calcium travels bound to albumin and is not biologically active.

Corrosion Rate from Coupon Weight Loss

P = \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.

Cost of Production per Unit

c = \frac{C}{Y}

Farm OperationsWhat a tonne of the crop cost to grow: total cost per hectare divided by yield per hectare. The one figure that can be laid beside a market price and compared without further arithmetic.

Cost of Water Over a Period

C_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.

Cost per Unit of Nutrient

c_n = \frac{c_p}{f}

Crop ProductionWhat a nutrient actually costs, from the product price and its analysis. The only honest way to compare two fertilisers, since the price per tonne of product compares nothing.

Coulomb Friction Force

F = \mu N

Machine DesignFriction force is the coefficient of friction times the normal force, and it does not depend on the apparent contact area or on the sliding speed. It is the oldest relation in tribology and it is an APPROXIMATION rather than a law — an empirical summary that works well in the middle of its range and fails at both ends.

Coulomb's Law

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

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

Courant Number (CFL condition)

C = \frac{v \, \Delta t}{\Delta x}

Computer ScienceHow far information travels in one time step, measured in grid cells. For an explicit scheme it must not exceed one cell per step, because a scheme that only looks at its neighbours cannot see something that has already gone past them.

Cramer's Rule for x (2×2 System)

x = \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 = \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.

Creatinine Clearance (Cockcroft-Gault)

\mathrm{CrCl} = \frac{(140 - a)\,m\,F}{72\,S_{cr}}

Biomedical & ClinicalCockcroft and Gault's 1976 estimate of creatinine clearance in mL/min from age, body mass, serum creatinine and a sex factor.

Crest Vertical Curve Length for Sight Distance

L = \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.

Crimp and Take-up Conversion

TU = \frac{100\,C}{100 + C}

Textiles & FabricCrimp and take-up describe the same waving of yarn in cloth and differ only in what they are measured against: crimp is the excess yarn as a fraction of the FABRIC length, take-up is the same excess as a fraction of the YARN length. Take-up is therefore always the smaller number, and the two are never interchangeable.

Crimp Percentage

c = \frac{L_y - L_f}{L_f}

Textiles & FabricA yarn in a woven cloth does not run straight — it waves over and under the threads crossing it, so a metre of cloth swallows more than a metre of yarn. Crimp is that excess, expressed as a fraction of the cloth length. It is measured by unravelling a marked length, straightening the yarn under a standard tension, and measuring what comes off.

Critical Angle for Total Internal Reflection

\sin\theta_c = \frac{n_2}{n_1}

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

Critical Depth in a Rectangular Channel

y_c = \left(\frac{Q^2}{g \, b^2}\right)^{1/3}

Fluid MechanicsCivil & SurveyingWater & WastewaterCritical depth of a rectangular channel from the discharge and the bed width — the depth of minimum specific energy, where the Froude number is exactly one.

Critical Fibre Length

l_c = \frac{\sigma_f \, d}{2 \, \tau_i}

Strength of MaterialsThe shortest fibre that can be loaded to its own breaking stress before it pulls out of the matrix. Load enters a discontinuous fibre only through shear at its surface, so a short fibre simply slides; below this length it never breaks, never carries its full share, and the composite around it is weaker for it.

Critical Radius of Insulation

r_{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.

Critical Shear Stress for Sediment Entrainment

\tau_c = \theta_c \, (s - 1) \, \rho \, g \, d

Erosion & SedimentThe Shields relation turned around and used the way a designer uses it: pick a critical Shields number, multiply by the submerged weight of a grain, and you have the shear stress at which the bed begins to move. Below it a channel is stable; above it, it is not. This is the tractive force method behind stable channel design.

Critical Speed of a Shaft

N_c = \frac{1}{2\pi} \sqrt{\frac{48 E I}{m L^{3}}}

Machine DesignThe running speed at which a shaft's own bending resonance is reached and it begins to whirl: the first critical of a simply supported shaft carrying a central rotor. A speed to pass through quickly, or to stay well away from.

Crop Evapotranspiration

ET_c = K_c \times ET_0

Crop ProductionThe water a specific crop uses on a given day, from a reference evapotranspiration that describes the weather and a crop coefficient that describes the canopy. The FAO-56 two-step method, which separates what the atmosphere demands from what the plant does about it.

Crop Revenue per Unit Area

R = Y \, p

Farm OperationsWhat a hectare of the crop is worth: yield multiplied by price. The top line of every crop budget, and the only line on it that two people can agree on before harvest is over.

Crop Share as an Equivalent Cash Rent

r = s \, Y \, p

Farm OperationsWhat a landlord's share of the crop is worth per hectare, so a share agreement and a cash rent can be compared in the same currency. The arithmetic behind every kitchen-table conversation about a lease.

Crop Yield per Unit Area

Y = \frac{m}{A}

Crop ProductionTotal harvest divided by the area it came off. The simplest equation on the site and the one most often quoted from the wrong area — gross field, net cropped, or the strip the yield monitor happened to see.

Cross Product Magnitude

|\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

(\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.

Cross Track Error

e_{xt} = R\arcsin\left(\sin\frac{d_{13}}{R}\,\sin\left(\theta_{13}-\theta_{12}\right)\right)

Navigation & PositionThe perpendicular distance from the intended track to where you actually are, on a sphere: the number a plotter shows on its deviation bar. Positive is right of track, negative is left.

CSTR Conversion (First Order)

X = \frac{\mathrm{Da}}{1 + \mathrm{Da}}

Chemical EngineeringHow far a first-order reaction gets in a single perfectly mixed tank, as a function of the Damköhler number alone. The curve rises steeply and then flattens: it passes 50% at Da = 1, needs Da = 9 for 90%, and needs an infinite tank for 100%.

CSTR Design Equation (First Order)

V = \frac{v_0\,X}{k\left(1 - X\right)}

ChemistryVolume of a perfectly mixed continuous tank reactor for a first-order liquid-phase reaction at a target conversion. The whole vessel sits at the outlet concentration, which is why the volume runs away as conversion approaches 100%.

CT Achieved (Disinfectant Residual × Contact Time)

\text{CT} = C \, T_{10}

Water TreatmentThe disinfection CT: the residual concentration a pathogen is exposed to multiplied by the time it is exposed for, returned in the regulatory unit of mg·min/L. This is the CT your plant ACHIEVED. Whether it is enough is a separate question, answered by your regulator's table for your pathogen, disinfectant, temperature and pH — never by this page.

Cube Face Diagonal

d = s\sqrt{2}

GeometryDiagonal measured across one square face of a cube — shorter than the space diagonal, which cuts through the interior.

Cube Space Diagonal

d = a \sqrt{3}

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

Cube Surface Area

S = 6 a^{2}

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

Cube Utilisation

U_v = \frac{V_c}{V_a}

Freight & ShippingThe share of the available space a load actually occupies. The volume half of the pair of numbers every load planner watches, and the one that quietly reveals how much air is being shipped.

Cube Volume

V = a^{3}

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

Cube-Out Crossover Density

\rho_x = \frac{W_{\max}}{V_u}

Freight & ShippingThe density at which a piece of equipment runs out of space and runs out of payload at the same moment. Freight lighter than this cubes out; freight heavier weighs out. It is the same quantity a dimensional-weight divisor describes, arrived at from the equipment rather than from the tariff.

Culmann Planar Wedge Factor of Safety

FS = \frac{2c'\sin\beta}{\gamma H\sin(\beta-\theta)\sin\theta} + \frac{\tan\phi'}{\tan\theta}

Soil MechanicsMechanicsFactor of safety of a rigid triangular wedge sliding on a single plane beneath a steep cut of height H and face angle β — Culmann's 1866 analysis, the oldest slope calculation still in use and the right one for a steep face where the infinite-slope assumption fails.

Cumulative ESALs with Traffic Growth

W_c = w_1 \cdot \frac{(1+g)^{Y} - 1}{g}

Axle Loads & PavementTotal equivalent single axle loads accumulated over a design period when traffic grows at a compound annual rate. The growth factor is already a SUM over the whole period, which is why multiplying it by the number of years again — a mistake made constantly — inflates the answer by that many times.

Current Divider

I_{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.

Current Sharing Between Parallel Conductors

I_{1} = I_{t} \frac{L_{2}}{L_{1} + L_{2}}

Electrical TradeHow current divides between two conductors of the same size and material run in parallel but to different lengths — the shorter one always takes more.

Cushion Rebound Angle

\tan\theta_{out} = \frac{\tan\theta_{in}}{e}

Billiards & Cue SportsMechanicsWhy a ball does not come off a cushion at the angle it went in. The cushion squashes and pushes back along its own normal, so only the perpendicular part of the velocity is affected — it comes back multiplied by e — while the part parallel to the rail sails through untouched. The result is always a wider angle out than in.

Cushion Rebound Speed

v_{out} = v_{in}\sqrt{e^{2}\cos^{2}\theta + \sin^{2}\theta}

Billiards & Cue SportsMechanicsHow much speed a cushion takes out of a ball. Only the perpendicular component is squeezed and returned at e times its size; the parallel component is untouched. Add the two back as vectors and the answer depends entirely on the angle — a ball hitting the rail square loses the most, and one grazing along it loses almost nothing.

Custom Hire Break-Even Area

A = \frac{C_f}{r - c_v}

Farm OperationsThe area at which owning a machine costs the same as hiring the work done. Below it the custom operator is cheaper; above it the fixed cost is spread thinly enough that ownership wins.

Cut Angle from Ball Fraction

\sin\varphi = \frac{b}{2R}

Billiards & Cue SportsMechanicsThe whole of aiming geometry in one line. Two spheres of radius R touch when their centres are 2R apart, so if the cue ball's centre passes a perpendicular distance b to the side of the object ball's centre, the line of centres at contact — and therefore the object ball's departure direction — sits at sin⁻¹(b/2R) from the cue ball's path. A half-ball hit gives exactly 30°.

Cutting Power from Specific Cutting Energy

P_c = u \, Q

Machining & TurningThe power a cut demands: the volume of metal removed per second multiplied by the energy it takes to remove a unit volume of that material. The check that tells you whether the machine can actually take the cut you have planned.

Cutting Speed and Spindle Speed

V = \pi \, D \, N

Machining & TurningThe bridge between the speed the cutting edge wants and the rpm the machine is set to: circumference times revolutions per minute. The single most confused pair of numbers in machining, and the reason a carbide grade rated at 250 gets run at 250 rpm and ruined.

Cycles of Concentration (COC = M/B)

\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 Chloride

\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.

Cycles of Concentration from Conductivity

\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.

Cyclic Stress Ratio for Liquefaction

\mathrm{CSR} = 0.65\,\dfrac{a_{max}}{g}\,\dfrac{\sigma_v}{\sigma'_v}\,r_d

SeismologyThe seismic demand side of liquefaction assessment, from Seed and Idriss's 1971 simplified procedure: the average cyclic shear stress an earthquake puts into a soil element, normalised by the effective stress holding that element together.

Cylinder Lateral Surface Area

A = 2\pi r h

GeometryArea of a cylinder's curved wall only, with no end caps — the label on a can, the insulation around a pipe.

Cylinder Surface Area

S = 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.

Cylinder Volume

V = \pi r^{2} h

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