Sources & References

Where the numbers come from
← Rules of Thumb

Every calculator on this site rests on either a standard formula, a published physical constant, an engineering/regulatory standard, or a specific real-world data point used for calibration. This page lists the primary source for each, grouped by area. Individual tools also link back to the relevant row here from their own footer.

Constants & standards bodies Environmental Engineering Finance Control Systems Computers Physics Simulation DIY

Constants & standards bodies

Referenced across multiple areas rather than tied to one tool.

Constant / standardValue usedSource
Standard gravity, g9.80665 m/s²NIST CODATA — exact by international definition (CGPM, 1901)
Gravitational constant, G6.674 × 10⁻¹¹ N·m²/kg²NIST CODATA
Universal gas constant, R8.314 J/(mol·K)NIST CODATA
Standard atmosphere modelICAO Standard AtmosphereICAO Doc 7488 — the standard reference for pressure/temperature vs. altitude
SI unit conversion factorsexact, by definitionNIST SI units guide
On formulas without a link: named physical laws (Ohm's law, Hooke's law, Kepler's laws, Bernoulli's equation, the Tsiolkovsky rocket equation, Euler's buckling formula) are foundational, centuries-old results with no single "source" — they're cited by name and can be checked against any mechanical/electrical engineering or physics textbook (e.g. Shigley's Mechanical Engineering Design, Halliday & Resnick's Fundamentals of Physics).

Environmental

ClaimSourceTool
Psychrometric formulas (Magnus/Arden Buck approximation)Buck, A.L. (1981), J. Applied Meteorology 20(12) — standard vapour-pressure approximationPsychrometer
Wind chill formulaNWS/Environment Canada 2001 joint formulaFeels Like Temperature
Heat index formulaRothfusz, L.P. (1990), NWS Technical Attachment SR 90-23Feels Like Temperature
Frostbite time / heat-risk bandsNWS wind chill chart; NWS heat index guidanceFeels Like Temperature
Solar position, sunrise/sunset algorithmMeeus, J., Astronomical Algorithms (2nd ed.) — standard reference for solar/lunar position calculationsSun & Moon
IANA time zone databaseIANA tz database, via the browser's Intl APITime Zone Converter
Residential electricity rates by countryEmber / Our World in Data electricity price series — representative national averages, edit to match your own billEnergy Cost Calculator
Grid carbon intensity by countryEmber Yearly Electricity DataEnergy Cost Calculator
Typical appliance wattagesU.S. DOE Energy Saver appliance energy guide — starting points, fully editableEnergy Cost Calculator
Solar panel output model, standard test conditionsIEC 61215 (1000 W/m², 25°C cell temperature)Solar PV Estimator
Everest / cruising altitude / Titanic / Challenger Deep reference pointsNational Geographic; NOAA Ocean Exploration (Challenger Deep depth, ~10,935 m)Altitude & Depth

Engineering

ClaimSourceTool
Beam deflection formulasRoark's Formulas for Stress and Strain; Euler-Bernoulli beam theoryBeam Bench
Darcy friction factor (Colebrook / Swamee-Jain)Swamee, P.K. & Jain, A.K. (1976), J. Hydraulics Division ASCE — explicit approximation of the Colebrook equationPipe Flow
Bolt torque-tension relationshipShigley's Mechanical Engineering Design, ch. on threaded fasteners; K-factor methodBolt Torque
Otto/Diesel/Brayton/Carnot cycle efficiencyCengel & Boles, Thermodynamics: An Engineering Approach — standard air-standard cycle analysisPV Diagram
ISO 281 bearing life (L10)ISO 281:2007 — rolling bearing dynamic load ratings and rating lifeBearing Calculator
O-ring groove dimensionsParker O-Ring Handbook (ORD 5700); AS568 standard sizesO-Ring Calculator
Circlip / retaining ring dimensionsDIN 471 (external), DIN 472 (internal) standard seriesCirclip Calculator
ISO limits and fits tolerance gradesISO 286-1/286-2Limits & Fits
Thermal expansion coefficients by materialASM Engineered Materials Reference BookLimits & Fits
Fluid density & viscosity vs. temperatureNIST Webbook (water); Cannon/CRC Handbook of Chemistry and Physics (other fluids)Fluid Properties
Unit conversion factors (imperial/metric)NIST Handbook 44, appendix C — exact conversion factorsUnit Converter
Resistor 4-band colour codeIEC 60062 — colour coding of resistance and capacitance valuesElectrical Circuit
Wire gauge current-carrying capacityGeneral-purpose reference figures adapted from NEC (NFPA 70) ampacity tables — not a substitute for a proper electrical code lookupElectrical Circuit
Spring rate & Wahl stress-correction formulaShigley's Mechanical Engineering Design, ch. on mechanical springs; Wahl, A.M. (1944), Mechanical SpringsSpring Rate Calculator
Common spring wire material propertiesSAE J1123/J1124; Associated Spring/Barnes Group design handbookSpring Rate Calculator
Euler buckling & Johnson's parabolic formulaTimoshenko, S.P., Theory of Elastic Stability; Shigley's Mechanical Engineering Design, ch. on columnsColumn Buckling Calculator
Structural material yield strengthsASTM A36 (steel), ASTM B209 (aluminium) material standardsColumn Buckling Calculator

Finance

ClaimSourceTool
Compound interest formulaStandard time-value-of-money mathematics — any introductory finance textbookCompound Interest
Mortgage amortisation formulaStandard fixed-rate amortisation mathematics; matches the method used by CFPB-regulated lendersMortgage
"4% rule" safe withdrawal rateBengen, W.P. (1994), Journal of Financial Planning — the original Trinity-study-adjacent safe withdrawal rate researchFIRE Calculator
UK National Living Wage / National Minimum Wage ratesGOV.UK, Low Pay Commission — current as of April 2026Salary Converter
US federal minimum wageU.S. Department of Labor — $7.25/hr, unchanged since 2009Salary Converter
US federal/FICA and UK income tax/NI bandsIRS; GOV.UK income tax rates — simplified, single-filer approximation, not personalised tax adviceSalary Converter

Control Systems

ClaimSourceTool
PID control theoryÅström & Murray, Feedback Systems (Princeton, freely available) — standard modern referencePID Control
Nyquist-Shannon sampling theoremShannon, C.E. (1949), Proc. IRE 37(1), "Communication in the Presence of Noise"Sampling & Aliasing
Ball-and-beam double-integrator dynamicsStandard control-theory teaching example — see Åström & Murray, Feedback Systems, or any undergraduate controls courseBall and Beam
Bode plot construction, gain/phase marginNyquist, H. (1932), Bell System Technical Journal; Bode, H.W. (1945), Network Analysis and Feedback Amplifier DesignBode Plot

Computers

ClaimSourceTool
RAID levels and fault-tolerance mathematicsPatterson, Gibson & Katz (1988), "A Case for Redundant Arrays of Inexpensive Disks (RAID)" — the original paper defining RAIDRAID Calculator
IPv4 subnetting (CIDR)RFC 4632 — Classless Inter-Domain RoutingSubnet Calculator
Two's complement, IEEE-754-style bitwise semanticsStandard computer architecture reference — Patterson & Hennessy, Computer Organization and DesignBit Calculator
Binary vs. decimal prefixes (KiB vs. kB)IEC 60027-2 — the standard defining kibi-, mebi-, gibi- prefixesBit Calculator

Physics Simulation

ClaimSourceTool
Damped harmonic oscillator equationsStandard 2nd-order ODE result — Halliday & Resnick, Fundamentals of Physics, ch. on oscillationsMass-Spring-Damper
Multi-link pendulum equations of motion (RK4 integration)Lagrangian mechanics, standard derivation — e.g. Taylor, J.R., Classical MechanicsPendulum Simulator
Coulomb friction model (static/kinetic)Standard introductory mechanics result; friction coefficients from Marks' Standard Handbook for Mechanical EngineersInclined Plane
Newton's law of cooling; convective/radiative heat transferIncropera & DeWitt, Fundamentals of Heat and Mass TransferBeverage Cooling Simulator
Quadratic drag equation, drag coefficientsHoerner, S.F., Fluid-Dynamic Drag — standard reference for drag coefficient by shapeGalileo Drop Experiment, Projectile Motion
Apollo 15 hammer-and-feather dropNASA Apollo 15 mission transcript/video — Commander David Scott, 2 August 1971Galileo Drop Experiment
Golf ball flight — Magnus effect, launch conditionsTrackMan/Titleist Performance Institute published launch-condition data; USGA equipment rules for legal driver CORProjectile Motion & Golf
PGA Tour average driving distancePGA Tour official statisticsProjectile Motion & Golf
Kepler's laws, vis-viva equationKepler, J. (1609/1619), Astronomia Nova / Harmonices Mundi; modern treatment in Curtis, H.D., Orbital Mechanics for Engineering StudentsOrbital Mechanics
ISS orbital altitude, period, velocityNASA ISS Facts and Figures — ~400 km altitude, ~90–93 min periodOrbital Mechanics
Standard gravitational parameters (μ) by bodyNASA JPL Solar System DynamicsOrbital Mechanics, Rocket Equation
Tsiolkovsky rocket equationTsiolkovsky, K.E. (1903), "The Exploration of Cosmic Space by Means of Reaction Devices"Rocket Equation & Delta-v Budget
Falcon 9 / Saturn V / Electron vehicle specificationsSpaceX, NASA (Saturn V Flight Manual SA-503), Rocket Lab public specifications — approximate figures assembled for illustration, not manufacturer data sheetsRocket Equation & Delta-v Budget
Typical mission delta-v budgets (LEO, GTO, TLI, escape)Curtis, H.D., Orbital Mechanics for Engineering Students — typical, trajectory-dependent published estimatesRocket Equation & Delta-v Budget

DIY

ClaimSourceTool
RYB subtractive colour mixing modelGossett, N. & Chen, B. (2004), "Paint Inspired Color Mixing and Compositing for Visualization" — the RYB trilinear interpolation approach this tool is based onPaint Colour Mixing
Corrections welcome. If a figure here looks out of date or a source is wrong, that's useful to know — every row links to the specific tool it backs, so the claim is easy to check against the source directly.