Formula solvers starting with R — 113 solvers

R-Value of a Snow Wall

R = \frac{L}{0.138 - 1.01\,\rho^{*} + 3.233\,\rho^{*2}}

Snow & IceHeat TransferThermodynamicsArea-specific thermal resistance of a snow wall, straight from its thickness and its density, with Sturm's conductivity fit folded in so you never have to look k up. It answers the question a shelter builder actually asks — how much wall is worth building — and nothing whatsoever about whether the shelter is safe.

R-Value of an Insulation Layer (R = L/k)

R = \frac{L}{k}

Heat TransferHVAC & HydronicsThermodynamicsArea-specific thermal resistance of one layer, from its thickness and thermal conductivity — the RSI or R-value quoted on every insulation label.

Rabinowitsch Shear-Rate Correction

\dot{\gamma}_{true} = \dot{\gamma}_{app}\, \frac{3n + 1}{4n}

Polymers & Plastics ProcessingRabinowitsch's 1929 correction turns the apparent wall shear rate — the one worked as if the melt were Newtonian — into the true one. A shear-thinning fluid has a blunter velocity profile than a Newtonian fluid, so the real velocity gradient at the wall is steeper than 4Q/πR³ admits, and the factor (3n+1)/4n is exactly how much steeper.

Radiated Energy from Surface-Wave Magnitude

\log_{10} E = 4.8 + 1.5\,M_s

SeismologyThe classical estimate of the seismic energy an earthquake radiates as waves, from its surface-wave magnitude, with E in joules. A 1950s calibration that modern broadband measurements only loosely agree with — useful for scale, not for precision.

Radiator Output at Non-Rated Temperature

\dot{Q} = \dot{Q}_r \left(\frac{\Delta T}{\Delta T_r}\right)^{n}

HVAC & HydronicsThermodynamicsCorrects a radiator or baseboard's catalogue output to the actual water-to-air temperature difference using the emitter exponent n.

Radioactive Activity (A = λN)

A = \lambda N

Modern PhysicsChemistryPhysicsDecays per second: the decay constant times the number of undecayed nuclei.

Radius from Degree of Curve (Arc Definition)

R = \frac{5729.578}{D}

Civil & SurveyingGeometryConverts degree of curve to radius using the arc definition, where D is the central angle subtending one 100 ft station of arc.

Radius of Gyration (r = √(I/A))

r = \sqrt{\frac{I}{A}}

Strength of MaterialsMechanicsGeometryRadius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a column really is.

Rainfall Depth from Intensity and Duration

P = i \, t

Water & WastewaterCivil & SurveyingFluid MechanicsThe bridge between the two ways rainfall is quoted: a depth is an intensity held for a duration. An IDF curve gives an intensity, the rational method consumes an intensity, and the curve-number and soil-loss methods consume a depth — this is the one line that connects them.

Range (Max minus Min)

R = x_{\max} - x_{\min}

StatisticsAlgebraThe simplest measure of spread in a data set: the distance from the smallest observation to the largest.

Rankine Active Earth Pressure Coefficient

K_a = \tan^{2}\!\left(45^\circ - \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of active earth pressure for a smooth vertical wall retaining level cohesionless backfill that has yielded away from the soil.

Rankine Cycle Thermal Efficiency

\eta = \frac{\left(h_1 - h_2\right) - \left(h_4 - h_3\right)}{h_1 - h_4}

ThermodynamicsPhysicsNet work over heat added for a steam power cycle, taken straight off the four state-point enthalpies: turbine inlet h₁, turbine exhaust h₂, condensate h₃ and feedwater leaving the pump h₄. Enter all four in the same unit; the answer is a ratio either way.

Rankine Passive Earth Pressure Coefficient

K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of passive earth pressure, the resistance mobilised when a wall or footing is pushed into level cohesionless soil.

Raoult's Law

P = x \, P^{0}

ChemistryGives the vapor pressure of a solvent above an ideal solution as its mole fraction times the pure solvent's vapor pressure.

Rate of Climb from Excess Power

RoC = \frac{P_a - P_r}{W}

Aerospace & FlightHow fast an aircraft climbs: the power left over after drag has been paid for, divided by the weight it has to lift. Climb is not a thrust question but a POWER question, and this is why.

Rational Method Peak Runoff

Q = C \, i \, A

Water & WastewaterCivil & SurveyingFluid MechanicsPeak runoff from a small catchment: the runoff coefficient times the rainfall intensity times the drainage area, in the SI form that takes intensity in mm/h and area in square metres.

Rayleigh Equation (Simple Batch Distillation)

\ln\frac{B_0}{B_1} = \frac{1}{\alpha - 1}\ln\!\left[\frac{x_0}{x_1}\left(\frac{1 - x_1}{1 - x_0}\right)^{\!\alpha}\right]

Chemical EngineeringA pot still with no column and no reflux: vapour is drawn off as fast as it forms, so what is left behind gets steadily poorer in the light component. This says how much charge remains once the pot has fallen to a given composition.

Rayleigh Number

\mathrm{Ra} = \mathrm{Gr} \, \mathrm{Pr}

Heat TransferGrashof multiplied by Prandtl, and the number that actually decides whether a still fluid starts to move. Natural-convection correlations turn out to depend on this product rather than on either factor alone, which is a finding about the physics and not merely a definition.

Rayleigh Range (z_R = πw₀²/λ)

z_R = \frac{\pi w_0^{\,2}}{\lambda}

OpticsThe distance from the waist at which a Gaussian beam's radius has grown by √2 — that is, at which its area has exactly doubled. It is the honest working definition of how far a beam stays collimated, and because it goes as the SQUARE of the waist radius, a beam twice as wide stays collimated four times as far.

RC Capacitor Discharge

V = V_{0} \, e^{-t/\tau}

Electricity & MagnetismPhysicsExponential decay of the voltage on a capacitor discharging through a resistor.

RC Cutoff Frequency

f_{c} = \frac{1}{2\pi R C}

Electricity & MagnetismElectrical TradeThe −3 dB corner of a resistor-capacitor filter, where the output has fallen to 0.707 of the input and the phase shift is 45°.

RC Time Constant

\tau = R C

Electricity & MagnetismPhysicsCharacteristic charging/discharging time of a resistor-capacitor circuit.

Reaction Quotient Q (aA + bB ⇌ cC)

Q = \frac{[\mathrm{C}]^{c}}{[\mathrm{A}]^{a}\,[\mathrm{B}]^{b}}

ChemistryReaction quotient Q from any concentrations, raised to stoichiometric powers — the number you compare with K to predict which way a mixture shifts.

Reaction Yield

Y = \frac{n_D}{n_{A0} - n_A}

Chemical EngineeringMoles of the desired product formed per mole of limiting reactant actually consumed. Yield is the honest partner to conversion: a reactor can consume all its feed and still make very little of what you wanted.

Reactive Power (Power Triangle)

Q = \sqrt{S^{2} - P^{2}}

Electrical TradeElectricity & MagnetismReactive power in vars from the power triangle, where apparent power is the hypotenuse over real and reactive legs.

Readily Available Water from MAD

RAW = AWC \times MAD

Crop ProductionThe depth of water a manager is willing to let the crop draw down before irrigating — the available water capacity multiplied by the management allowable depletion fraction. This, not the full reserve, is the size of an irrigation.

Real Interest Rate (Fisher Equation)

r_{\text{real}} = \frac{1 + i}{1 + f} - 1

Money & BusinessWhat a return is worth after inflation is taken out, done exactly rather than by the rough subtraction that overstates it.

Real Value from a Nominal Value (Deflating by an Index)

V_{\text{real}} = V_{\text{nom}}\,\frac{I_{\text{base}}}{I_t}

Prices & MarketsRestates an amount of money measured in one year's prices in another year's prices, by multiplying by the ratio of the two index values. The base year is whichever year you want the answer expressed in.

Recipe Scaling (Ingredient for a New Yield)

Q_2 = Q_1 \cdot \frac{S_2}{S_1}

Everyday & HealthIngredient quantity rescaled from the yield a recipe was written for to the yield you actually want.

Recombination Frequency and Map Distance

RF = 100 \, \frac{n_r}{n}

Plant Breeding & GeneticsThe share of offspring showing a recombinant genotype, which for closely linked markers is read directly as map distance in centimorgans.

Rectangle Area

A = l \cdot w

GeometryArea of a rectangle as length times width.

Rectangle Diagonal

d = \sqrt{l^2 + w^2}

GeometryDiagonal of a rectangle from its length and width.

Rectangle Perimeter

P = 2(l + w)

GeometryPerimeter of a rectangle from its length and width.

Rectangular Prism Space Diagonal

d = \sqrt{l^2 + w^2 + h^2}

GeometryThe corner-to-opposite-corner distance through the inside of a box — the longest straight object it can hold.

Rectangular Prism Surface Area

A = 2(lw + lh + wh)

GeometryTotal surface area of a box — three pairs of matching rectangular faces, from length, width and height.

Rectangular Prism Volume

V = l \cdot w \cdot h

GeometryVolume of a box as the product of its length, width, and height.

Rectifying Operating Line (McCabe–Thiele)

y = \frac{R}{R + 1}\,x + \frac{x_D}{R + 1}

Chemical EngineeringThe straight line above the feed that ties the vapour rising off a stage to the liquid falling onto it. Slope R/(R+1), and it always passes through the point (x_D, x_D) on the 45° line — which is how it gets drawn.

Recurrent Parent Recovery in a Backcross

R = 1 - \left(\tfrac{1}{2}\right)^{n+1}

Plant Breeding & GeneticsThe expected share of the recurrent parent's genome recovered after a number of backcrosses — the arithmetic behind moving a single trait into an established variety.

Reed-Solomon Correctable Errors

t = \left\lfloor \frac{n - k}{2} \right\rfloor

Cryptography & Key SizingSymbol errors a Reed-Solomon code RS(n, k) can correct: half the parity symbols, rounded down. Reed and Solomon's 1960 construction meets the Singleton bound exactly, so the n − k parity symbols do the most any code of that size could — they correct t errors at unknown positions, or twice as many erasures at known ones.

Reflux Ratio

R = \frac{L}{D}

Chemical EngineeringThe liquid sent back down the column divided by the product drawn off the top. One number that sets the slope of the rectifying line, the height of the tower, and most of the energy bill.

Refrigerant Mass Flow Rate

\dot{m} = \frac{\dot{Q}}{\Delta h}

HVAC & HydronicsThermodynamicsRefrigerant circulated per unit time from the cooling capacity and the enthalpy change across the evaporator, the basis of compressor sizing.

Refrigerant Subcooling

\mathrm{SC} = T_{sat} - T_{liquid}

HVAC & HydronicsThermodynamicsDegrees of subcooling at the condenser outlet: how much colder the liquid is than its saturation temperature at the same pressure.

Refrigerant Superheat

\mathrm{SH} = T_{suction} - T_{sat}

HVAC & HydronicsThermodynamicsDegrees of superheat at the compressor suction: how much warmer the vapour is than its saturation temperature at the same pressure.

Refrigeration COP from Enthalpies

COP = \frac{h_1 - h_4}{h_2 - h_1}

ThermodynamicsHVAC & HydronicsCoefficient of performance of a vapour-compression cycle read straight off a pressure-enthalpy chart: refrigerating effect over compressor work. h₁ is the compressor suction, h₂ the discharge, h₄ the enthalpy leaving the expansion device.

Regression Line Intercept

a = \bar{y} - b \bar{x}

StatisticsAlgebraThe y-intercept of a least-squares line, fixed by the requirement that the line pass through the point of averages.

Regression Slope from Correlation

b = r \frac{s_y}{s_x}

StatisticsAlgebraThe least-squares slope of a regression line, recovered from the correlation coefficient and the two standard deviations.

Regular Polygon Area (from Side Length)

A = \frac{n s^2}{4 \tan(\pi/n)}

GeometryArea of a regular polygon with n equal sides of length s — pentagon, hexagon, octagon and beyond.

Regular Polygon Perimeter

P = n s

GeometryPerimeter of a regular polygon: number of sides times side length.

Regular Tetrahedron Surface Area

A = \sqrt{3}\,s^2

GeometryTotal surface area of a regular tetrahedron: four equilateral triangles of edge s, which sum to exactly √3 s².

Regular Tetrahedron Volume

V = \frac{s^3}{6\sqrt{2}}

GeometryVolume of a regular tetrahedron — four identical equilateral triangles — from its single edge length.

Reinforced Concrete Nominal Moment Capacity

M_n = A_s f_y \left(d - \frac{a}{2}\right)

Civil & SurveyingStrength of MaterialsNominal flexural strength of a singly reinforced concrete section: the tension force in the rebar times the internal lever arm to the centroid of the Whitney compression block.

Relation Between E, G and ν

E = 2G(1 + \nu)

Strength of MaterialsMechanicsPhysicsThe isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and Poisson's ratio in one step.

Relative Compaction (Percent Proctor)

R = \frac{\gamma_{d,field}}{\gamma_{d,max}}\times 100

Soil MechanicsStrength of MaterialsRelative compaction of placed fill as the field dry unit weight divided by the Proctor maximum dry unit weight, in percent.

Relative Density (Per Cent of Theoretical)

\rho_{rel} = \frac{\rho_m}{\rho_{th}}

Additive ManufacturingThe measured density of a built part divided by the density the same alloy has when it is fully solid. The single number the whole parameter-development exercise is trying to move, quoted everywhere as a percentage — and a number that hides more than it reports, because how the porosity is arranged matters far more than how much of it there is.

Relative Density of a Granular Soil

D_r = \frac{e_{max} - e}{e_{max} - e_{min}}\times 100

Soil MechanicsStrength of MaterialsRelative density of a sand or gravel, placing its in-situ void ratio on the scale between its loosest and densest laboratory states.

Relative Humidity from a Sling Psychrometer

\varphi = \frac{p_{ws}(t_{wb}) - A\,p\,(t_{db} - t_{wb})}{p_{ws}(t_{db})}

HVAC & HydronicsThermodynamicsAir Quality & DispersionRelative humidity from a dry-bulb and a wet-bulb reading, using the psychrometric equation with A = 6.66×10⁻⁴ K⁻¹ — the calculation behind every paper psychrometric slide rule.

Relative Humidity from Vapour Pressure

\varphi = \frac{p_v}{p_{ws}}

HVAC & HydronicsThermodynamicsAir Quality & DispersionRelative humidity as what it actually is: the vapour pressure present divided by the saturation vapour pressure at the same temperature — a ratio to a target that moves whenever the air is heated.

Relative Risk (Risk Ratio)

\mathrm{RR} = \frac{R_{e}}{R_{u}}

Epidemiology & Diagnostic AccuracyThe risk in the exposed group divided by the risk in the unexposed group. A multiplier: 3 means three times the risk, and it says nothing at all about how large either risk was.

Relative Volatility (Binary)

\alpha = \frac{y\left(1 - x\right)}{x\left(1 - y\right)}

ChemistryThe single number that says how easy a binary mixture is to distil: the ratio of the light component's vapour-to-liquid enrichment against the heavy component's. α = 1 means the two boil identically and no simple column will part them.

Reorder Point

R = d\,L

Industrial EngineeringStock level that should trigger the next order: the quantity consumed during the supplier's lead time, so the last unit leaves the shelf as the replacement arrives.

Resistance of a Wire (R = ρL/A)

R = \frac{\rho L}{A}

Electricity & MagnetismPhysicsResistance from material resistivity, length, and cross-sectional area.

Resultant of Two Vectors at an Angle

R = \sqrt{A^2 + B^2 + 2AB\cos\theta}

Vectors & MatricesTrigonometryPhysicsFinds the magnitude of the sum of two vectors from their lengths and the angle between them, the parallelogram rule in one equation.

Return Activated Sludge Rate

Q_r = \frac{Q \, X}{X_r - X}

Water & WastewaterWater TreatmentReturn sludge flow needed to hold a target mixed liquor concentration, from a solids balance around the aeration basin.

Return on Investment (ROI)

\mathit{ROI} = \frac{G - C}{C}

Money & BusinessProfit expressed as a fraction of what was spent: total value returned minus cost, divided by cost.

Return Period from Rank (Weibull)

T = \frac{n + 1}{m}

Water & WastewaterCivil & SurveyingProbabilityThe Weibull plotting position: rank the annual maxima from largest to smallest and the event at rank m in a record of n years is assigned a return period of (n+1)/m years. This is where return periods come from before any distribution is fitted — they are counted, not calculated.

Revenue Response to a Price Change

\%\Delta R \approx \%\Delta P\,(1 + E_d)

Prices & MarketsApproximate percentage change in total revenue caused by a small percentage change in price. The sign of the bracket decides everything: with E_d between −1 and 0 a price rise RAISES revenue, and below −1 it CUTS it.

Reynolds Number

Re = \frac{\rho v D}{\mu}

Fluid MechanicsPhysicsThe dimensionless ratio of inertial to viscous forces that decides laminar versus turbulent flow.

Rhombus Area (from Diagonals)

A = \frac{d_1 d_2}{2}

GeometryArea of a rhombus as half the product of its two diagonals.

Rhombus Perimeter

P = 4s

GeometryPerimeter of a rhombus — four equal sides, like a leaning square.

Rhumb Line Distance

d = R\sqrt{\left(\varphi_2-\varphi_1\right)^{2} + q^{2}\left(\lambda_2-\lambda_1\right)^{2}}

Navigation & PositionThe length of the track that crosses every meridian at the same angle: the route you actually steer when you set one course and hold it. Longer than the great circle, and the stretching factor q comes from the Mercator latitude.

Richardson Number (Buoyancy against Shear)

Ri = \frac{g L \, \Delta\rho}{\rho v^{2}}

Fluid MechanicsBuoyancy trying to keep a stratified fluid in layers, against shear trying to stir it back together. Large Ri and the stratification survives — a salt wedge holds in an estuary, a temperature inversion traps a plume, a chilled supply layer stays on the floor. Small Ri and turbulence wins and the layers mix. It is the inverse square of the densimetric Froude number, which is the same statement in the other trade's language.

Right-Triangle Cosine Ratio (CAH)

\cos\theta = \frac{a}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its adjacent side and the hypotenuse.

Right-Triangle Sine Ratio (SOH)

\sin\theta = \frac{o}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its opposite side and the hypotenuse.

Right-Triangle Tangent Ratio (TOA)

\tan\theta = \frac{o}{a}

TrigonometryGeometryRelates an acute angle of a right triangle to the ratio of its opposite and adjacent legs.

Ring Crush to Edge Crush (Whitsitt Relation)

ECT = 0.8 \sum RCT + 12 \;\mathrm{lbf/in}

Packaging & Box StrengthAn empirical regression that estimates the edge crush value of combined board from the ring crush values of the liners and the medium that went into it. Useful when the components were tested and the combined board was not — and carrying one honest ugliness: unlike McKee's constants, this one has an ADDITIVE term, so it is unit-bound in a way no choice of exponent can fix.

RL Cutoff Frequency

f_{c} = \frac{R}{2\pi L}

Electricity & MagnetismElectrical TradeCorner frequency of a resistor-inductor filter, reached when the inductive reactance 2πfL has grown to equal the resistance.

RL Time Constant (τ = L/R)

\tau = \frac{L}{R}

Electricity & MagnetismPhysicsHow quickly current builds or decays in an inductor-resistor circuit.

RMS and Peak Voltage

V_{rms} = \frac{V_{peak}}{\sqrt{2}}

Electricity & MagnetismPhysicsFor a sine wave, the effective (heating-equivalent) voltage is the peak divided by √2.

Rock Avalanche Volume-Mobility Relation

\frac{H}{L} = k\,V^{-n}

Soil MechanicsRock MechanicsThe empirical power law behind large-landslide mobility: the Heim ratio H/L falls as the event volume rises, so bigger rock avalanches travel disproportionately further than friction allows. Both coefficients are regional fits and are entered by the reader.

Rock Mass Deformation Modulus (Hoek–Diederichs)

E_{rm} = 100000 \left[ \frac{1 - D/2}{1 + e^{(75 + 25D - GSI)/11}} \right] \ \mathrm{MPa}

Rock MechanicsThe stiffness of a jointed rock mass, from Hoek and Diederichs' 2006 simplified expression: a sigmoid in GSI, knocked down by the disturbance factor. This is the number that governs how much a tunnel converges or a foundation settles — strength decides whether the ground fails, stiffness decides how much it moves before it does.

Rock Quality Designation (RQD)

RQD = \frac{\sum L_{i \ge 100\,\mathrm{mm}}}{L_t} \times 100\%

Rock MechanicsDeere's 1964 index: of the core you pulled out of a run, what fraction came up as sound pieces at least 100 mm long. It is the oldest and cheapest measure of rock quality still in daily use, it feeds both the Q-system and RMR, and it is far cruder than the confidence usually placed in it.

Rocket Mass Ratio from Delta-v

MR = e^{\Delta v / v_e}

Aerospace & FlightThe ratio of loaded mass to burnout mass a vehicle needs in order to deliver a given delta-v. The rocket equation turned inside out, and the fastest way to see how brutally the exponential punishes an ambitious mission.

Rocket Thrust from Mass Flow

F = \dot m \, v_e

Aerospace & FlightThe momentum term of the thrust equation: mass leaving per second multiplied by the speed it leaves at. Newton's third law with a stopwatch on it, and the whole of the thrust when the nozzle is perfectly expanded.

Rocket Thrust with the Pressure Term

F = \dot m \, v_e + (p_e - p_a) A_e

Aerospace & FlightThe complete thrust equation: the momentum the exhaust carries away, plus whatever the gas pressure at the nozzle exit gains or loses against the ambient pressure outside. The reason an engine's thrust rises as it climbs.

Rockfall Kinetic Energy at a Barrier

E = \tfrac{1}{2} m v^{2}

Soil MechanicsMechanicsTranslational kinetic energy of a falling or bouncing block at the point it reaches a barrier — the number a rockfall net, fence or embankment is selected against, and the number that barrier ratings are published in.

Rolled Throughput Yield

\mathit{RTY} = y^{\,n}

Industrial EngineeringFraction of units that pass every step of an n-step process with no rework, when each step has the same first-pass yield: the yield raised to the number of steps.

Rolling Cue Ball Deflection (the real 30° rule)

\tan\alpha = \frac{5 \sin\varphi \cos\varphi}{2 + 5\sin^{2}\varphi}

Billiards & Cue SportsMechanicsWhere a ROLLING cue ball actually ends up after a cut. It leaves along the tangent line at 90° to the object ball, but it is still carrying its forward roll, so friction bends it forward again until it is rolling once more. The final path sits α off the original line of travel — and α hovers near 32° for every cut angle between 20° and 40°, which is the whole reason the 30° rule exists.

Roof Area from Footprint and Pitch

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

Trades & ConstructionGeometryTurns the plan area a roof covers into the sloping area you actually have to shingle, using the pitch multiplier for a pitch of p in 12.

Roof Pitch to Slope Angle

\theta = \arctan\!\left(\frac{p}{12}\right)

Trades & ConstructionGeometryConverts a roof pitch quoted the framing way, p inches of rise per 12 inches of level run, into the slope angle from horizontal, and back.

Rope Tension When Lifting a Mass

T = m\left(g + a\right)

MechanicsPhysicsTension in a rope lifting a mass with an upward acceleration, exceeding the static weight by the factor (g + a).

Rosin–Rammler Passing Fraction

P(x) = 1 - \exp\!\left[ -\left( \frac{x}{x_c} \right)^{n} \right]

Rock MechanicsRosin and Rammler's 1933 distribution for broken and ground material: the fraction of a sample that passes a screen of size x, from just two parameters — a characteristic size and a uniformity index. It is the second half of Kuz-Ram, it is how a P80 is defined, and it is the curve behind every screen analysis in mineral processing.

Rossby Number

\mathrm{Ro} = \frac{v}{f \, L}

Air Quality & DispersionInertia against the Coriolis force. Small Rossby numbers mean rotation runs the flow, which is why weather systems circle their low-pressure centres instead of filling them; large Rossby numbers mean rotation is irrelevant, which is why your bath does not care which hemisphere it is in.

Rotating Unbalance Force

F = m_u \, e \, \omega^{2}

Machine DesignThe rotating force a small unbalanced mass throws off, growing with the SQUARE of speed. The reason a rotor that runs smoothly at half speed can tear its mounts apart at full speed.

Rotational Kinetic Energy

KE_{rot} = \tfrac{1}{2} I \omega^{2}

MechanicsPhysicsKinetic energy stored in rotation: half the moment of inertia times angular velocity squared.

Rotational Power (P = τω)

P = \tau \omega

MechanicsPhysicsMechanical power delivered by a torque turning at angular velocity ω.

Rotor Disc Loading

DL = \frac{T}{A}

Aerospace & FlightThrust carried per unit of disc area — the rotor's answer to wing loading. One number, and it fixes the downwash, the hover power, the brownout, and whether the machine is a helicopter or a jet-lift fighter.

Rotor Figure of Merit

FM = \frac{P_{ideal}}{P_{actual}}

Aerospace & FlightHow close a real rotor gets to the momentum-theory floor in hover: ideal power divided by the power actually required. A ratio between 0 and 1 that quietly contains everything the actuator disc refused to model.

Rotor Solidity

\sigma = \frac{N_b c}{\pi R}

Aerospace & FlightWhat fraction of the disc is actually blade: total blade planform area divided by the area of the circle it sweeps. For rectangular blades this collapses to number of blades times chord over π times radius — usually somewhere between 5% and 12%.

Rotor Swept Area

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

Solar & Wind PowerThe area of the disc a horizontal-axis rotor sweeps, from its diameter. Every wind power equation on this site takes this area, and it is the circle the blade tips trace — not the surface of the blades themselves.

Rotor Thrust Coefficient

C_T = \frac{T}{\rho A (\Omega R)^{2}}

Aerospace & FlightRotor thrust made dimensionless on the disc area and the SQUARE OF THE TIP SPEED — with no factor of ½. It is disc loading divided by ρ(ΩR)², and it is the number that lets rotors of different sizes be compared at all.

Round Bin Capacity

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

Farm OperationsThe volume of grain in a round bin, from the inside diameter and the depth of grain. Answered in bushels or cubic metres — the conversion between them is geometry, not agronomy.

Round Duct Air Velocity

v = \frac{4 \dot{V}}{\pi d^{2}}

HVAC & HydronicsFluid MechanicsAir velocity in a round duct from the volume flow and the duct diameter, the check that keeps branches quiet and mains efficient.

Rouse Suspension Number

P = \frac{w_s}{\kappa \, u_*}

Erosion & SedimentHunter Rouse's 1937 ratio: how fast a grain falls, against how strongly the turbulence lifts it. Below about 0.8 the grain travels in full suspension; around 1.2 to 2.5 it is suspended near the bed; above about 2.5 it stays on the bed and rolls, hops and slides. One number decides which of two entirely different transport regimes you are in.

Row Length for a Stand Count

L = \frac{A}{k \, w}

Crop ProductionHow much row to count so that multiplying the count by k gives the population directly. The 1/1000th-acre method is this equation with A one acre and k a thousand: 43.56 square feet of ground, which in 30-inch rows is 17 feet 5 inches of row.

Row Length per Unit Area

L = \frac{A}{w}

Crop ProductionThe total length of row a field contains at a given row spacing — 10,000 divided by the spacing in metres, per hectare. The figure behind drip tube orders, in-row plastic, hand-hoeing time and per-plant treatment costs.

RQD from Volumetric Joint Count

RQD = 115 - 3.3\,J_v

Rock MechanicsPalmstrom's 1982 estimate of RQD for rock you can see but cannot drill: count the joints crossing a cubic metre of the mass, and the correlation returns the RQD a core run would probably have shown. Useful in a face, a trench or an outcrop, and never a substitute for core where core exists.

RSA Modulus

n = p \cdot q

Computer ScienceThe public modulus of an RSA key: the product of the two secret primes. Multiplying is instant; undoing it is the whole of RSA's security.

RSA Primes from Modulus and Totient

p = \frac{S + \sqrt{S^2 - 4n}}{2}, \quad S = n - \varphi + 1

Computer ScienceKnowing both n and φ(n) hands over the primes. Their sum is n − φ + 1 and their product is n, so they are the two roots of a quadratic — no factoring required.

RSA Totient

\varphi(n) = (p-1)(q-1)

Computer ScienceEuler's totient of an RSA modulus, counting the numbers below n that share no factor with it. The private exponent is built from this, which is why it must stay secret.

Rule of 72 (Doubling Time)

n \approx \frac{0.72}{i}

Money & BusinessMental-arithmetic estimate of how many periods it takes money to double at rate i — with the rate written as a percentage R, it is the familiar 72 divided by R.

Rule of Mixtures — Longitudinal Modulus

E_1 = E_f V_f + E_m (1 - V_f)

Strength of MaterialsStiffness along the fibres: each phase carries its own volume share of the load, so the moduli simply add in proportion. It is the upper bound on what any fibre and matrix can give you, it is the one direction where a real lamina nearly reaches the bound, and it is the number people wrongly use for every other direction.

Ryshkewitch-Duckworth Porosity-Strength Relation

\sigma = \sigma_0 \, e^{-bP}

Ceramics & GlassStrength falls exponentially with porosity, and much faster than the missing cross-section alone would explain. Ryshkewitch measured it on alumina and zirconia in 1953 and Duckworth gave it its usual form in the same volume of the Journal of the American Ceramic Society; it has held up across sixty years of sintered materials.

Ryznar Stability Index (RSI)

\mathrm{RSI} = 2\,\mathrm{pH_s} - \mathrm{pH}

Water TreatmentChemistryAn empirical scaling index scaled so that values below about 6 predict scale and values above about 7 predict corrosion in the same water.