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.
Referenced across multiple areas rather than tied to one tool.
| Constant / standard | Value used | Source |
|---|---|---|
| Cooking a steak is transient heat conduction: time to a target core temperature scales with roughly the square of the thickness, and beef has a thermal diffusivity near 1.2×10-7 m²/s, varying with fat content | Standard transient-conduction treatment (Incropera & DeWitt, Fundamentals of Heat and Mass Transfer) applied to published thermal properties of beef; the page solves the heat equation numerically rather than using the one-term Heisler approximation, which is only valid for Fourier numbers above ~0.2 and so does not cover most of a sear. Doneness temperature bands follow standard culinary practice | Steak Timer |
| UK student loans repay 9% of income above the plan threshold (6% for postgraduate loans, charged concurrently), with any balance cancelled after 25–40 years depending on plan | GOV.UK student finance repayment guidance and the Student Loans Company repayment plan rules; thresholds shown are 2024/25 and are uprated each April, which is why the tool leaves them editable | Student Loan Calculator |
| Rod-side (annulus) area, retract force and speed, and the 2:1 cylinder | Annulus area π(D²−d²)/4 for a single-rod cylinder; force = P·A and speed = Q/A on each face. Standard fluid-power practice (Parker and Eaton design handbooks; ISO 6020/6022 cylinder series). hydraulic_model.py checks that bore = annulus + rod exactly, that rod = bore/√2 gives an area ratio of precisely 2, and — the identity that ties it together — that force × speed equals P×Q in both directions and in regeneration, so the model can never invent power | Hydraulic Force Calculator |
| Regeneration net area equals the rod area; return-line flow exceeds supply on retract | With the rod side tied back to the cap side both faces see the same pressure, so the net area is bore minus annulus — identically the rod area, which hydraulic_model.py asserts to 1e-18. Retract return flow is supply × (bore area / annulus area); sizing the return for the supply flow instead is a common error that builds back-pressure on the annulus face and subtracts directly from the delivered force | Hydraulic Force Calculator |
| Sequence-of-returns risk reverses sign between saving and drawing down | With no cash flows the order of returns cannot matter at all — a product of growth factors is invariant under reordering, which fire_model.py verifies exactly over 40 random returns. Flows are what make order matter, and the direction of the flow sets the sign: contributions into a slump buy in cheap, withdrawals from one sell a larger share of a shrunken pot. Verified on a case checkable by hand — 100 through +50%/−50% contributing 10 a year ends at 90 and reversed at 100; withdrawing 10 instead it is 60 and 50. The underlying drawdown treatment follows Bengen (1994) and the Trinity study | FIRE Calculator |
| Design heat loss = ΣU·A·ΔT + 0.33·n·V·ΔT; emitter output scales with the 1.3 power of the water-to-air temperature difference against its ΔT50 rating | Standard steady-state building heat loss method (CIBSE Guide A / BS EN 12831); 0.33 Wh/m³K is the volumetric heat capacity of air. The radiator exponent of ~1.3 is the conventional value used in EN 442 output correction | Heat Loss & Heat Pump Sizing |
| Standard gravity, g | 9.80665 m/s² | NIST CODATA — exact by international definition (CGPM, 1901) |
| Gravitational constant, G | 6.674 × 10⁻¹¹ N·m²/kg² | NIST CODATA |
| Universal gas constant, R | 8.314 J/(mol·K) | NIST CODATA |
| Standard atmosphere model | ICAO Standard Atmosphere | ICAO Doc 7488 — the standard reference for pressure/temperature vs. altitude |
| SI unit conversion factors | exact, by definition | NIST SI units guide |
| Claim | Source | Tool |
|---|---|---|
| Psychrometric formulas (Magnus/Arden Buck approximation) | Buck, A.L. (1981), J. Applied Meteorology 20(12) — standard vapour-pressure approximation | Psychrometer |
| Wind chill formula | NWS/Environment Canada 2001 joint formula | Feels Like Temperature |
| Heat index formula | Rothfusz, L.P. (1990), NWS Technical Attachment SR 90-23 | Feels Like Temperature |
| Frostbite time / heat-risk bands | NWS wind chill chart; NWS heat index guidance | Feels Like Temperature |
| The five constant-acceleration equations of motion (SUVAT) | Standard Newtonian kinematics — all five follow from a = dv/dt held constant, and appear in every introductory mechanics text with no single citable origin. Standard gravity is the CGPM defining value 9.80665 m/s². Solving for time uses a numerically stable quadratic; both roots are reported when both are non-negative, and both signs of a square root are reported when solving for a velocity. All ten three-known solve paths are cross-checked in suvat_model.py against 400 randomised motions each | SUVAT Calculator |
| Solar position, sunrise/sunset algorithm | Meeus, J., Astronomical Algorithms (2nd ed.) — standard reference for solar/lunar position calculations | Sun & Moon |
| Arctic/Antarctic Circle latitude (66.56°) and the polar day/night terminator condition | Standard spherical astronomy — the terminator condition sin(elevation)=0 solved per latitude; the 66.56° threshold follows directly from Earth's 23.44° axial tilt (90° − tilt) | Sun & Moon |
| IANA time zone database | IANA tz database, read live through the browser's Intl API — so zone rules and DST transitions are as current as the device. The world map's day/night terminator uses the NOAA solar-position approximation (Spencer's Fourier fit for declination and the equation of time), the same one behind Sun & Moon, with the terminator taken as the locus where solar elevation is zero. Cross-checked in timezone_model.py against Python's own copy of the tz database. Exchange trading hours are each venue's published continuous session, excluding pre- and post-market; Tokyo, Shanghai and Hong Kong lunch breaks included. Session times change occasionally and holidays are not modelled — check the exchange for anything that depends on it | Time Zone Converter |
| Residential electricity rates by country | Ember / Our World in Data electricity price series — representative national averages, edit to match your own bill | Energy Cost Calculator |
| Grid carbon intensity by country | Ember Yearly Electricity Data | Energy Cost Calculator |
| Typical appliance wattages | U.S. DOE Energy Saver appliance energy guide — starting points, fully editable | Energy Cost Calculator |
| Solar panel output model, standard test conditions | IEC 61215 (1000 W/m², 25°C cell temperature) | Solar PV Estimator |
| Everest / cruising altitude / Titanic / Challenger Deep reference points | National Geographic; NOAA Ocean Exploration (Challenger Deep depth, ~10,935 m). The atmosphere model is ISO 2533 / ICAO Standard Atmosphere in full — all seven layers to the 84,852 m ceiling the standard itself defines; the Kármán line at 100 km lies above it and is marked as such rather than clamped to fit | Altitude & Depth |
| Claim | Source | Tool |
|---|---|---|
| Beam deflection formulas | Roark's Formulas for Stress and Strain; Euler-Bernoulli beam theory | Beam Deflection & Stress Calculator |
| Darcy friction factor (Colebrook / Swamee-Jain) | Swamee, P.K. & Jain, A.K. (1976), J. Hydraulics Division ASCE — explicit approximation of the Colebrook equation | Pipe Flow |
| Bolt torque-tension relationship | Shigley's Mechanical Engineering Design, ch. on threaded fasteners; K-factor method | Bolt Torque |
| Bolt property classes and proof stresses (metric) | ISO 898-1 — mechanical properties of carbon and alloy steel fasteners. Class 8.8 is diameter-dependent: proof 580 N/mm², yield 640, tensile 800 for d ≤ 16 mm, and 600/660/830 above. The tensile stress area As = π/4 (D − 0.9382P)² is from the same standard. | Bolt Torque |
| Bolt grades and proof loads (imperial) | SAE J429 — grade 2 is diameter-dependent (55,000 psi to ¾", 33,000 above); grades 5 and 8 hold across the sizes offered. | Bolt Torque |
| Nut factor K by finish and lubrication | Bickford, An Introduction to the Design and Behavior of Bolted Joints; standard published nut factors, with the ±25% scatter the page states | Bolt Torque |
| Otto/Diesel/Brayton/Carnot cycle efficiency | Cengel & Boles, Thermodynamics: An Engineering Approach — standard air-standard cycle analysis | PV Diagram |
| ISO 281 bearing life (L10) | ISO 281:2007 — rolling bearing dynamic load ratings and rating life | Bearing Calculator |
| O-ring groove dimensions | Parker O-Ring Handbook (ORD 5700); AS568 standard sizes | O-Ring Calculator |
| Circlip / retaining ring dimensions | DIN 471 (external), DIN 472 (internal) standard series | Circlip Calculator |
| Retaining ring thrust load model | Groove capacity as bearing area × yield strength, the form published by Smalley and Rotor Clip. The 0.83 open-ring factor is derived from the published DIN load column, not quoted from a catalogue. Ring capacity is reproduced as published and is not computed. | Circlip Calculator |
| Structural steel yield strengths (S235/S275/S355) | EN 1993-1-1 §3.2.6 nominal values, t ≤ 40 mm | Circlip Calculator |
| ISO limits and fits tolerance grades | ISO 286-1/286-2 | Limits & Fits Calculator |
| Thermal expansion coefficients by material | ASM Engineered Materials Reference Book | Limits & Fits Calculator |
| Fluid density & viscosity vs. temperature | NIST Webbook (water); Cannon/CRC Handbook of Chemistry and Physics (other fluids) | Fluid Properties |
| Unit conversion factors (imperial/metric) | NIST Handbook 44 Appendix C (customary–SI factors). SI base units: see the SI units guide row above (not Handbook 44). All 351 factors are generated by unit_model.py rather than typed: they are built from a handful of exact anchors (the inch is 0.0254 m, a US gallon is 231 cubic inches, a nautical mile is 1852 m, standard gravity is 9.80665 m/s²) and checked against definitional identities and against their own parent units, so area cannot drift from length squared. The historical temperature scales (Delisle, Newton, Réaumur, Rømer) are from their published fixed points; the unusual units behind the toggle are conventional values with no single standards body — the smoot and the beard-second are jokes with an agreed size, not measurements | Unit Converter |
| Spring rate & Wahl stress-correction formula | Shigley's Mechanical Engineering Design, ch. on mechanical springs; Wahl, A.M. (1944), Mechanical Springs | Spring Rate Calculator |
| Common spring wire material properties | SAE J1123/J1124; Associated Spring/Barnes Group design handbook | Spring Rate Calculator |
| Spring surge (natural) frequency, both ends seated | A compression spring between two seats is a fixed-fixed elastic continuum, so its fundamental is the bar result f₁ = c/2L, reducing to ½·√(k/m) with the full active-coil mass — as given for springs between parallel plates in Shigley’s Mechanical Engineering Design, ch. on mechanical springs. spring_model.py derives it twice (from k and m, and in closed form) and requires the two to agree | Spring Rate Calculator |
| The 13× surge margin guidance | Standard spring-design guidance that the surge frequency should be at least about thirteen times the forcing frequency; quoted in Shigley’s and in SMI (Spring Manufacturers Institute) design practice. It is a chosen threshold, not a derived limit — dual springs and progressive pitch deliberately break the resonance up, so a single-frequency check understates a well-designed valve train | Spring Rate Calculator |
| Solid height and buckling slenderness thresholds | Solid height = wire diameter × total coils for squared-and-ground ends. Buckling slenderness L₀/D guidance of about 5.2 between parallel plates and about 2.6 with an end free to pivot, per Shigley’s treatment of spring stability. Both are geometry limits reached before any stress limit | Spring Rate Calculator |
| Euler buckling & Johnson's parabolic formula | Timoshenko, S.P., Theory of Elastic Stability; Shigley's Mechanical Engineering Design, ch. on columns | Column Buckling Calculator |
| Structural material yield strengths | ASTM A36 (steel), ASTM B209 (aluminium) material standards | Column Buckling Calculator |
| Gear-mesh kinematics (ratio, speed, torque, mesh efficiency) | Shigley's Mechanical Engineering Design, ch. on spur/helical gears — standard gear-train kinematics | Gear Train Calculator |
| Planetary (epicyclic) ratios from the Willis equation; planet tooth count and equal-spacing assembly constraint | Willis relation (ωs−ωc)/(ωr−ωc) = −Nr/Ns as given in Shigley's Mechanical Engineering Design and Norton, R.L., Design of Machinery, ch. on epicyclic trains. The three closed-form arrangements, the two-degrees-of-freedom residual and the 24/72 textbook case are each re-derived and checked in gear_model.py before reaching this page | Gear Train Calculator |
| Pitch diameter d = m·z, centre distance m(z₁+z₂)/2, and the 2/sin²α undercut limit | ISO 53 / ISO 54 standard basic rack and module series; undercut limit is the standard full-depth result quoted in Shigley's and Norton — 17.1 teeth at 20°, 31.9 at 14.5°, 11.2 at 25°, each verified in gear_model.py. It is a geometry limit on the generating cutter, not a strength criterion, and profile shift defeats it | Gear Train Calculator |
| Slider-crank piston kinematics; 4-stroke cam/valve timing; 2-stroke piston-port scavenging sequence | Slider-crank kinematics x(θ)=r·cosθ+√(L²−r²sin²θ) — Norton, R.L., Design of Machinery; engine cycle and valve-event treatment follows Heywood, J.B., Internal Combustion Engine Fundamentals. Port timing represents a typical piston-ported, crankcase-scavenged 2-stroke layout, not any single manufacturer's spec | 2 & 4-Stroke Engine Simulator |
| Mean piston speed limits (20 m/s production, 20–25 competition, ~25 for cast pistons) | Heywood, J.B., Internal Combustion Engine Fundamentals — mean piston speed as the governing design limit. Band figures cross-checked against published engine-design references; the Formula 1 anchor is the FIA regulation 80 mm bore × 53 mm stroke, which gives 26.5 m/s at the 15,000 rpm limit by 2 × stroke × rpm. | 2 & 4-Stroke Engine Simulator |
| BMEP ranges by engine type (8–12 bar NA petrol, up to 25 turbo diesel) | Published brake-mean-effective-pressure ranges for naturally aspirated and turbocharged petrol and diesel engines. BMEP is an input on this page rather than a derived figure — it stands in for the breathing and combustion the model does not simulate. | 2 & 4-Stroke Engine Simulator |
| Ideal air-standard Otto efficiency 1 − r1−γ | Standard cold-air-standard cycle analysis at γ = 1.4 (Cengel & Boles, Thermodynamics). Shown as a ceiling no real engine reaches, not a prediction — the same treatment the PV Diagram tool uses. | 2 & 4-Stroke Engine Simulator |
| Hydraulic force (Pascal's law, F = P·A) | Standard fluid statics — Pascal's principle, cross-referenced across multiple fluid mechanics references | Hydraulic Force Calculator |
| Claim | Source | Tool |
|---|---|---|
| Resistor 4-band colour code | IEC 60062 — colour coding of resistance and capacitance values | Resistor Colour Code Calculator · also Electrical Circuit |
| E-series preferred values (E6, E12, E24, E96) are geometric progressions stepping by 10^(1/n) per decade, chosen so consecutive tolerance bands overlap — which is why 4.7k exists and 5k does not | IEC 60063 preferred number series. Verified: each published series tracks 10^(1/n) to within 5%, and the ideal step is smaller than the ±tolerance span ratio (1+t)/(1−t) for E6 at 20%, E12 at 10% and E24 at 5%, so the bands overlap by design | Resistor Colour Code |
| The published values are rounded, and rounding opens small real gaps in every series — E12’s worst step is 1.25 (12 to 15) against a ±10% span of 1.222 | Measured directly from the published value tables rather than from the idealised step: the worst consecutive ratio exceeds the tolerance span in E6, E12, E24, E48 and E96 alike. A 1.34k target sits more than 10% from both 1.2k and 1.5k, so no single E12 part reaches it | Resistor Colour Code |
| SMD marking: 3-digit (473 = 47k), 4-digit (4702 = 47k), R for the decimal point (4R7 = 4.7Ω), and EIA-96 (01C = 10k, 68X = 49.9Ω) | Standard surface-mount resistor marking conventions. EIA-96 uses a two-digit index into the E96 table followed by a letter multiplier (Z 0.001 through F 100000), so code 68 is E96 value 499 and X multiplies by 0.1 | Resistor Colour Code |
| Standard power ratings and 2× derating — 5 V across 220Ω dissipates 0.114 W and wants a 1/4 W part, not the 1/8 W the arithmetic alone allows | P = V²/R, against the standard rating ladder (1/16, 1/10, 1/8, 1/4, 1/2, 1, 2, 3, 5 W). The 2× factor is a working allowance rather than a standard: manufacturer ratings assume free air at 25°C, which an enclosure does not provide | Resistor Colour Code |
| Wire gauge current-carrying capacity (advisory only) | Rough illustrative bands for the circuit demo only — not a full NEC ampacity lookup. For design values use the dedicated Wire Gauge Calculator, which implements NEC Table 310.16 / Ch.9 Table 8 / 240.4(D) as cited there | Electrical Circuit |
| UK DNO reactive-power charging threshold (0.95 PF) | UK Power Networks connection terms; DNO reactive-power methodology statements | Power Factor Calculator |
| Capacitor sizing formula, kVAR = kW·(tanφ₁−tanφ₂) | Power-triangle relationship kVAR = kW·(tanφ₁−tanφ₂) — IEEE Std 141 (Red Book), Recommended Practice for Electric Power Distribution for Industrial Plants, power-factor correction chapter | Power Factor Calculator |
| Typical uncorrected equipment power factors (motors, lighting, welding) | Illustrative reference ranges compiled from multiple electrical-engineering references — genuinely equipment- and load-dependent, not fixed constants | Power Factor Calculator |
| Ideal transformer turns-ratio and ampere-turns relationships | Ideal-transformer relationships V₁/V₂ = N₁/N₂ and N₁I₁ = N₂I₂ — Fitzgerald, Kingsley & Umans, Electric Machinery, transformer chapter | Transformer Calculator |
| Transformer full-load efficiency by size class (94-99.75%) | Compiled from multiple transformer manufacturer and industry engineering references; distribution transformers 97-99%, large power transformers up to 99.75% | Transformer Calculator |
| Maximum transformer efficiency occurs where copper loss equals core loss | Standard transformer loss theory, confirmed across multiple power-systems engineering references — typically 40-70% of rated load for distribution transformers | Transformer Calculator |
| NEC Table 310.16 copper ampacity (60/75/90°C) and Chapter 9 Table 8 conductor resistance | National Electrical Code (NFPA 70), cross-referenced across multiple published NEC ampacity/resistance charts | Wire Gauge Calculator |
| NEC 240.4(D) small-conductor overcurrent protection limits (15A/20A/30A for 14/12/10 AWG) | National Electrical Code (NFPA 70) — overrides raw ampacity-table values for these three gauges | Wire Gauge Calculator |
| Ambient-temperature correction factors for conductor ampacity | NEC 310.15(B)(1). The closed form the published table is built from — correction = √((Tc − Ta) / (Tc − 30)) — is used here rather than a lookup, so there is no interpolation argument at a range boundary. wire_model.py asserts it reproduces fourteen published rows across the 60, 75 and 90 °C columns exactly | Wire Gauge & Voltage Drop Calculator |
| Adjustment factors for more than three current-carrying conductors | NEC 310.15(C)(1) — 80% for 4–6 conductors, 70% for 7–9, 50% for 10–20, 45% for 21–30, 40% for 31–40, 35% above. Applied multiplicatively with the ambient correction, both verified in wire_model.py | Wire Gauge & Voltage Drop Calculator |
| Terminal temperature limitation (why 90 °C cable is capped at the lug rating) | NEC 110.14(C). Derating is applied to the conductor’s own insulation column and the result is then capped at the terminal column — doing it in the other order derates the cable twice and makes 90 °C insulation look pointless when it is exactly what rescues a hot or crowded run | Wire Gauge & Voltage Drop Calculator |
| Claim | Source | Tool |
|---|---|---|
| Regional mortgage model differences (compounding, rate reversion, mortgage insurance, escrow) | Market conventions rather than a single citation: Canadian semi-annual compounding is required by the Interest Act; UK and Canadian lending fixes for 2–5 years against a longer amortisation; US PMI cancels at 80% LTV under the Homeowners Protection Act while Canadian CMHC and Australian LMI premiums are capitalised into the loan. Rates shown per market are indicative typical new lending, not quotes | Mortgage Calculator |
| UK stamp duty (SDLT) bands and first-time buyer relief | England & Northern Ireland residential rates as at April 2025 — 0% to £125k, 2% to £250k, 5% to £925k, 10% to £1.5m, 12% above; first-time buyer relief 0% to £300k then 5% to £500k, withdrawn entirely above £500k. Thresholds change with budgets and differ in Scotland (LBTT) and Wales (LTT); the tool says so | Mortgage Calculator |
| Affordability tests by market (LTI, DTI, GDS, debt-service ratio) | Typical lender or regulatory limits: UK loan-to-income around 4.5×, US debt-to-income around 36%, Canadian gross debt service around 39%, and the French HCSF 35% debt-service cap. Individual lenders apply their own stress tests, so the tool presents these as a rule of thumb rather than a decision | Mortgage Calculator |
| Fee drag modelled period-by-period on the balance | Method rather than a citation: charges are levied pro rata each compounding period on the opening balance, which is how platform and fund fees accrue. There is no closed form once fees and contributions interact, so the tool runs the schedule. The reported cost is the difference against the same plan with no charge, which is larger than the fees paid because the money taken would also have compounded | Compound Interest Calculator |
| Long-run reference returns marked on the rate slider | Rounded long-run nominal annual averages for orientation, not forecasts: cash ~2%, inflation ~3%, bonds ~4.5%, global equities ~7%, US equities ~10%. Realised returns over any particular period differ substantially from long-run averages | Compound Interest Calculator |
| Compound interest formula | Time-value-of-money compounding A = P(1 + r/n)^(nt) — Brealey, Myers & Allen, Principles of Corporate Finance, time-value chapter | Compound Interest Calculator |
| Mortgage amortisation formula | Standard fixed-rate amortisation mathematics; matches the method used by CFPB-regulated lenders | Mortgage |
| Car loan / hire-purchase amortisation; PCP balloon (optional final payment) | Same fixed-rate amortisation as mortgages; UK product shapes follow common FCA-regulated HP/PCP structures (deposit + amortising credit ± guaranteed future value). Illustrative only — not a regulated quote. | Car Finance |
| Currency conversion from ECB reference rates | Frankfurter open API publishing European Central Bank euro foreign exchange reference rates — mid-market, not a retail quote | Currency Converter · also Unit Converter Currency category |
| "4% rule" safe withdrawal rate | Bengen, W.P. (1994), Journal of Financial Planning — the original Trinity-study-adjacent safe withdrawal rate research | FIRE Calculator |
| UK National Living Wage / National Minimum Wage rates | GOV.UK, Low Pay Commission — current as of April 2026 | Salary Converter |
| US federal minimum wage | U.S. Department of Labor — $7.25/hr, unchanged since 2009 | Salary Converter |
| Regional tipping norms (US/Canada 18–25%, UK service charge often included, Western Europe optional 5–10%, Australia/NZ ~10%, Japan/South Korea not customary) | Cross-referenced across multiple 2026 sources: Autopilot Travel's tipping guide, Matador Network, Capital One, Global Rescue, and World Population Review's country tipping rankings — individual figures vary a few points by source, the broad regional pattern does not | Tip Calculator |
| US federal/FICA and UK income tax/NI bands | IRS; GOV.UK income tax rates — simplified, single-filer approximation, not personalised tax advice | Salary Converter |
| Debt avalanche vs. snowball payoff simulation | Standard amortisation mathematics — month-by-month interest accrual and payment allocation, verified against a hand-checked two-debt divergent case | Debt Payoff Calculator |
| Snowball method's behavioural completion-rate advantage | Gal, D. & McShane, B.B. (2012), "Can Small Victories Help Win the War? Evidence from Consumer Debt Management", Journal of Marketing Research 49(4) — closing accounts in ascending balance order predicted higher completion rates, despite avalanche minimising total interest | Debt Payoff Calculator |
| Rent-vs-buy opportunity-cost comparison methodology | Standard approach used by major rent-vs-buy calculators (e.g. New York Times) — net cost of buying vs. renting-and-investing the difference | Rent vs Buy Calculator |
| Claim | Source | Tool |
|---|---|---|
| PID control theory | Åström & Murray, Feedback Systems (Princeton, freely available) — standard modern reference | PID Control |
| Ziegler–Nichols, Cohen–Coon and lambda/SIMC tuning rules | Ziegler & Nichols reaction-curve rules (1942): Kp = 1.2τ/(Kθ), Ti = 2θ, Td = 0.5θ. Cohen & Coon (1953) for dead-time-heavy loops. Lambda/SIMC per Skogestad, with λ = 3θ as the robust default. All three as given in Åström & Murray, Feedback Systems and Seborg et al., Process Dynamics and Control. Each is re-derived in pid_model.py, which also asserts the identities they must all obey — doubling process gain halves Kp, more dead time backs every method off, and larger λ always detunes | PID Control |
| θ/τ as the measure of loop difficulty | The ratio of dead time to time constant is the standard controllability measure for FOPDT loops; above about 1 the loop is dead-time dominant and PID stops being the right tool, which is the same conclusion the site’s own “dead time beats PID” rule reaches. Bands used here: <0.1 easy, 0.1–1 normal, >1 dead-time dominant | PID Control |
| Nyquist-Shannon sampling theorem | Shannon, C.E. (1949), Proc. IRE 37(1), "Communication in the Presence of Noise" | Sampling & Aliasing |
| Ball-and-beam double-integrator dynamics | Standard control-theory teaching example — see Åström & Murray, Feedback Systems, or any undergraduate controls course | Ball and Beam |
| Bode plot construction, gain/phase margin | Nyquist, H. (1932), Bell System Technical Journal; Bode, H.W. (1945), Network Analysis and Feedback Amplifier Design | Bode Plot |
| Claim | Source | Tool |
|---|---|---|
| RAID levels and fault-tolerance mathematics | Patterson, Gibson & Katz (1988), "A Case for Redundant Arrays of Inexpensive Disks (RAID)" — the original paper defining RAID | RAID Calculator |
| IPv4 subnetting (CIDR) | RFC 4632 — Classless Inter-Domain Routing | Subnet Calculator |
| Two's complement, IEEE-754-style bitwise semantics | Standard computer architecture reference — Patterson & Hennessy, Computer Organization and Design | Bit Calculator |
| Binary vs. decimal prefixes (KiB vs. kB) | IEC 60027-2 — the standard defining kibi-, mebi-, gibi- prefixes | Bit Calculator |
| Netflix 4K/HD minimum bandwidth (25 Mbps / 5 Mbps) | Netflix Help Center's own published internet speed recommendations | Internet Speed Test |
| Typical movie file sizes (~15GB 4K, ~5GB 1080p, two-hour runtime) | Cross-referenced streaming/file-size references (GoBrolly, Dark Skies, Fortra) | Internet Speed Test |
| GTA V modern install size (~100GB, Enhanced edition with Online) | Cross-referenced across Rockstar/Steam listings and multiple gaming-press size reports | Internet Speed Test |
| Country-level median internet speeds worldwide | Ookla Speedtest Global Index, cross-referenced 2025-2026 snapshots — a representative sample, not a live feed | Internet Speed Test |
| Typical download:upload ratios by connection type | Cross-referenced ISP/connection-type technical references (fiber symmetric; cable/DSL asymmetric) | Internet Speed Test |
| Capacitor sizing Q = 2πfCV²; a delta bank needs one third the capacitance of a star bank for the same kVAR | Direct consequence of each capacitor seeing line voltage in delta and line/√3 in star, with reactive power scaling as V²; standard in power-factor correction practice and manufacturer bank design guides | Power Factor Calculator |
| Terminal velocity vt = √(2mg / ρCdA); free-fall time t = √(2h/g) in vacuum | Standard results from Newton's second law with quadratic drag (F = ½ρCdAv²); the vacuum case is Galileo's own, demonstrated on the Moon by Apollo 15 in 1971 | Galileo Drop Experiment |
| Equivalent dynamic load P = XFr + YFa; for single-row deep-groove ball bearings e and Y are functions of Fa/C₀ alone | ISO 281 rolling bearing dynamic load ratings; the e/Y table is reproduced identically in the SKF, NSK, NTN and FAG general catalogues, confirming it is type-dependent rather than part-dependent | Bearing Calculator |
| Kirchhoff's current law (ΣI = 0 at a node) and voltage law (ΣV = 0 around a loop) | Gustav Kirchhoff, 1845 — consequences of conservation of charge and of energy respectively; standard in every circuit-analysis text | Electrical Circuit |
Password entropy is length × log₂(alphabet) for random strings and words × log₂(list) for passphrases; a 7,776-word Diceware list gives 12.9 bits per word against 6.6 for a character from a 95-symbol set | Shannon entropy applied to a uniform generator. Diceware list size is Reinhold’s original 6⁵ = 7,776. NIST SP 800-63B withdrew composition rules and mandatory rotation, and recommends length over character-class requirements, for the reasons the tool demonstrates | Password Strength & Crack Time |
Sample size per variant for two proportions is (zα/2+zβ)²[p₁(1−p₁)+p₂(1−p₂)]/(p₂−p₁)², so halving the detectable effect roughly quadruples the sample | Standard two-sample proportion power calculation with a normal approximation to the binomial; the inverse normal uses Acklam’s rational approximation. The 5% baseline / 10% relative lift case returns ~31,000 per arm, matching published sample-size calculators | A/B Test Sample Size Calculator |
| Converting money between two years is the ratio of consumer price index values, and £100 in 1970 is about £1,609 in 2025 money | UK CPI (ONS, 2015 = 100), annual averages to 2025 plus the latest published month, with pre-1988 values from the ONS CPI-consistent modelled historical series and pre-1950 values from the ONS long-run composite price index, chained onto CPI at 1950 so that nothing from 1950 onward is affected; US CPI-U (BLS, 1982–84 = 100). Recent UK years are derived from the published annual-average rates (1.8, 0.9, 2.6, 9.1, 7.3, 2.5%), and the resulting series reproduces the Bank of England calculator's landmarks: £1 in 1970 → £16.09, in 1990 → £2.61, in 2000 → £2.00. The US series reproduces the BLS figure of $100 (1980) → $380.70 and $1 (1913) → $31.69 | Inflation Calculator |
| Rest between sets of roughly 3–5 minutes for strength, 60–90 seconds for hypertrophy and 30–60 seconds for muscular endurance; one-rep-max estimated from a submaximal set; barbell plate loading and session volume load | Rest bands follow standard strength-and-conditioning guidance (ACSM and NSCA position stands on resistance training). One-rep max uses three published regressions shown side by side rather than one: Epley (1985), Brzycki (1993) and Lombardi (1989). Their divergence is computed rather than asserted — within ~5 kg at five reps on a 100 kg lift, 76.8 kg at twenty; Epley and Brzycki cross exactly at ten reps, and Epley reads 3.3% high at a true single because it is not calibrated there. Working-weight percentages are the inverse of Epley. Plate loading is a greedy per-side fill that never overshoots. All re-derived in gym_model.py (48 checks, including an exhaustive sweep of every half-kilo from 20–200 kg). General guidance for healthy adults, not medical advice | Gym Companion |
| Waist under half your height is a single boundary that works for both sexes and from childhood, and catches cases BMI misses in both directions | Waist-to-height ratio with a 0.50 boundary, as used by NICE guidance on assessing central adiposity; the two disagreement cases are worked directly — a 180 cm 95 kg athlete is BMI 29.3 with a ratio of 0.46, a 180 cm 78 kg sedentary person is BMI 24.1 with a ratio of 0.53. Lower cut-offs apply for some ethnic groups | Waist-to-Height Ratio Calculator |
| Induction motors draw 6–8× full-load current at standstill, and that band straddles the trip window of an ordinary breaker | Full-load current from P/(√3·V·PF·η) with the plate rating as shaft output; locked-rotor multiples from motor nameplate code letters. Magnetic trip windows of 3–5×, 5–10× and 10–20× for type B, C and D are the standard IEC 60898 curves. A 7.5 kW 400 V motor computes to 14.2 A running, 85 A at 6× and 113 A at 8× against a 20 A type C window of 100–200 A | Motor Starting & Inrush |
| Reduced-voltage starting cuts current in proportion to voltage and torque with the square of it, so star-delta’s symmetric one-third is a coincidence of the connection | Induction motor torque is proportional to the square of applied voltage at fixed slip, current to the first power. The delta reconnection divides line current and torque by exactly three; an autotransformer tap a divides both by a²; a solid-state starter divides current by a and torque by a². A drive holds volts-per-hertz constant and so is not governed by the square law at all | Motor Starting & Inrush |
| A PI controller driving a saturating actuator overshoots substantially, and anti-windup removes it without changing a single gain | Simulated directly: a first-order plant (τ = 10 s, unity gain) under a parallel-form PI controller (Kc = 2, Ti = 8 s) stepped to 10 with the output clipped at 12 overshoots 12.8% unguarded and 0.0% with conditional integration. Back-calculation follows Åström and Hägglund’s formulation. With the clip raised to 200 all three behaviours agree to within half a point, confirming the effect is saturation rather than tuning | Integral Windup & Anti-Windup |
| On-off cycling rate peaks near 50% duty, and halving the deadband roughly doubles the number of cycles | Classical bang-bang result for a first-order plant: on-time and off-time are the exponential crossing times of the deadband, and because the heating ceiling is ambient plus the unit’s authority, both extremes of load send one of the two to infinity. Manufacturer limits of roughly six starts per hour and ten minutes minimum run are typical across compressor and boiler documentation | On-Off Control & Hysteresis |
| A first-order-plus-dead-time step response reaches 63.2% of its final value one time constant after the delay ends, and 98.2% after four; open-loop tuning constants follow from K, τ and L alone | 63.2% is 1 − e−1 by definition. Tuning relations are the published open-loop rules: Ziegler & Nichols (1942) reaction curve, Cohen & Coon (1953), and lambda / internal model control tuning as given in Seborg, Edgar & Mellichamp | First-Order Lag & Dead Time |
Delivered capacity falls with discharge current per Peukert’s law t = H(C/IH)k, with k ≈ 1.1–1.3 for lead-acid and ≈ 1.05 for LiFePO₄; usable depth of discharge is about 50% and 80–90% respectively | Peukert (1897), still the standard high-rate model; exponent and depth-of-discharge ranges are the values consistently given across lead-acid and LiFePO₄ manufacturer datasheets. The formula returns the rated time exactly at the rated current for any exponent, which is the internal check that it is being applied correctly | Battery Sizing & Runtime |
| Sweat rate is body mass change plus fluid taken in, divided by duration, taking 1 kg as 1 L; losing about 2% of body mass measurably impairs endurance performance, and rates range roughly 0.5–2.5 L/h between individuals | American College of Sports Medicine position stand on exercise and fluid replacement; the mass-balance method and the 2% threshold are standard in it. Guidance on drinking to thirst rather than ahead of it follows the exercise-associated hyponatraemia consensus statements, which reversed earlier advice after fatalities in endurance events | Sweat Rate & Hydration |
| Maximum heart rate falls with age at roughly 0.7 beats/year (208 − 0.7 × age); the older 220 − age overestimates in the young and underestimates in the old, crossing it at age 40. Zones by heart rate reserve exceed the same percentage of maximum by resting × (1 − percentage) | Tanaka, Monahan & Seals (2001), meta-analysis of 351 studies / 18,712 subjects; Fox (1971) for 220 − age; Nes et al. (2013) HUNT study; Gulati et al. (2010) for the women-derived equation. The reserve identity follows directly from Karvonen’s definition | Heart Rate Zones & Max HR |
| DC charging holds near peak power to a knee around 50–60% state of charge then tapers, so the last 20% can take as long as the first 70%; charging losses are roughly 12% on home AC and 6% on DC | Constant-current / constant-voltage lithium-ion charging — taper is required to hold cell voltage inside a safe window as the battery fills. Loss figures are consistent across published metered charging tests; AC is worse because rectification happens in the onboard charger | EV Charging & Running Cost |
| ISO week 1 is the week containing the year's first Thursday (equivalently 4 January); weeks run Monday–Sunday and an ISO year has 53 weeks when 1 January is a Thursday, or a Wednesday in a leap year | ISO 8601-1:2019 §4.2.2, date and time representations for information interchange — the 53-week condition follows directly from the week-1 rule rather than being separately defined | Week Number Calculator |
| Clock hand angle = |30H − 5.5M|; hands coincide every 720/11 minutes (11 times in 12 hours) | Direct consequence of the hand rates — minute hand 6°/min, hour hand 0.5°/min — giving a 5.5°/min closing speed; standard result in recreational and competition mathematics | Clock Angle Calculator |
| 60 pixels per degree is the resolving limit of 20/20 vision; thresholds scale with the Snellen ratio | Snellen definition of visual acuity (20/20 resolves 1 arcminute; 60 arcminutes per degree), consistent with the ~30 cycles/degree contrast-sensitivity limit of the healthy eye | Screen Size & Viewing Distance |
| Recommended horizontal field of view 30° (SMPTE) to 40° (THX) | SMPTE EG-18 cinema viewing recommendation and THX screen placement guidance | Screen Size & Viewing Distance |
| Transfer time = size / throughput (Mbps vs MB/s factor of 8) | Definitional relationship between bit-rate and byte-rate; same estimates as Internet Speed Test use-cards | Data Transfer Time Calculator · also Speed Test |
| Ethernet protocol overhead is 5.07% at a 1500-byte MTU and 0.86% with jumbo frames, so a gigabit link carries at most 949 Mbps of payload | Frame arithmetic, not an estimate: 20 B IP + 20 B TCP headers inside the frame, and 14 B Ethernet header + 4 B FCS + 8 B preamble + 12 B inter-frame gap outside it — 1460 B of payload per 1538 B on the wire, and 8960 B per 9038 B at MTU 9000 | Data Transfer Time Calculator |
| TCP throughput is capped at window ÷ round-trip time, so a 64 KB window over a 10 ms link cannot exceed 52 Mbps however fast the line is | A sender may have only one window in flight per round trip; the ceiling is the same statement as the bandwidth-delay product, and parallel streams multiply it because each carries its own window. Verified against the identity that a window equal to the BDP exactly fills the link | Data Transfer Time Calculator |
| Typical sustained sequential storage speeds by device class — USB 2.0 35, SD card 90, 7200 rpm hard disk 120, USB 3.0 400, SATA SSD 550, NVMe 3500 MB/s | Representative figures for each class of device, not measured for any specific product. Used only to establish which ceiling binds: a 7200 rpm disk at 960 Mbps is just above the ~930 Mbps a gigabit link delivers as payload, so it does not bind, while an SD card at 720 Mbps does | Data Transfer Time Calculator |
| Claim | Source | Tool |
|---|---|---|
| Damped harmonic oscillator equations | Standard 2nd-order ODE result — Halliday & Resnick, Fundamentals of Physics, ch. on oscillations | Mass-Spring-Damper |
| Multi-link pendulum equations of motion (RK4 integration) | Lagrangian mechanics, standard derivation — e.g. Taylor, J.R., Classical Mechanics | Pendulum Simulator |
| Coulomb friction model (static/kinetic) | Standard introductory mechanics result; friction coefficients from Marks' Standard Handbook for Mechanical Engineers | Inclined Plane |
| A rolling body accelerates at g·sinθ/(1+k) where k = I/mr², so a hoop gets exactly half a frictionless slide and a solid sphere five sevenths — independent of mass and radius | Standard rigid-body result from combining Newton's second law along the slope with the torque equation about the contact point; k is 2/5 for a solid sphere, 1/2 for a solid cylinder, 2/3 for a hollow sphere and 1 for a hoop. Verified: mass and radius cancel algebraically, and the ordering sphere < cylinder < hollow sphere < hoop holds at every angle | Inclined Plane |
| Rolling without slipping requires μ ≥ tanθ × k/(1+k) — 0.17 for a sphere at 30°, 0.29 for a hoop | The friction force needed to supply the angular acceleration, divided by the normal force. Below it the body slips and reverts to the sliding case, arriving sooner because it no longer spends energy on rotation | Inclined Plane |
| Pushing a load up a ramp needs mg(sinθ + μcosθ), giving a mechanical advantage of 1/sinθ at best and an efficiency of sinθ/(sinθ + μcosθ) | Force balance along the slope at constant velocity. Verified that with μ = 0 the work done equals the lifting work exactly, so a ramp trades force for distance and never saves energy; a 1:12 ramp at μ 0.2 is 29% efficient | Inclined Plane |
| The angle of repose is the critical angle, so μ = tan(repose): dry sand 34°, dry soil 35°, gravel 45° | Typical published repose angles for granular materials; the identity with the critical angle is definitional, since a heap stands at exactly the angle where the surface grains are on the point of sliding | Inclined Plane |
| Newton's law of cooling; convective/radiative heat transfer | Incropera & DeWitt, Fundamentals of Heat and Mass Transfer | Beverage Cooling Simulator |
| 3-iron, 5-iron and gap-wedge launch figures | Interpolated within the same PGA Tour Trackman ladder as the other clubs in this tool — each sits between its neighbours on ball speed, launch angle and spin. Not separately measured averages; they complete a standard 14-club bag for comparison purposes | Projectile Motion & Golf Ball Flight |
| Driver ball-speed distributions by standard of player | Approximate cohort means and standard deviations drawn from published launch-monitor datasets — Trackman tour averages for the tour row, and club-fitting / Arccos amateur distributions for the rest. Rounded deliberately: they describe populations, not individuals, and should not be read as a ranking | Projectile Motion & Golf Ball Flight |
| Whole-bag estimation from a single calibrated club | Method, not a citation: ball speed is scaled by the reader's ratio to the tour figure for their selected club, and spin at 55% of that ratio. It assumes bag ratios are consistent within a player, which holds far better than absolute numbers but breaks down for a genuinely uneven bag — stated in the tool's field notes | Projectile Motion & Golf Ball Flight |
| Quadratic drag equation, drag coefficients | Hoerner, S.F., Fluid-Dynamic Drag — standard reference for drag coefficient by shape | Galileo Drop Experiment, Projectile Motion |
| Apollo 15 hammer-and-feather drop | NASA Apollo 15 mission transcript/video — Commander David Scott, 2 August 1971 | Galileo Drop Experiment |
| Golf ball flight — Magnus effect, launch conditions | TrackMan/Titleist Performance Institute published launch-condition data; USGA equipment rules for legal driver COR | Projectile Motion & Golf |
| PGA Tour average driving distance | PGA Tour official statistics | Projectile Motion & Golf |
| Kepler's laws, vis-viva equation | Kepler, J. (1609/1619), Astronomia Nova / Harmonices Mundi; modern treatment in Curtis, H.D., Orbital Mechanics for Engineering Students | Orbital Mechanics |
| ISS orbital altitude, period, velocity | NASA ISS Facts and Figures — ~400 km altitude, ~90–93 min period | Orbital Mechanics |
| Standard gravitational parameters (μ) by body | NASA JPL Solar System Dynamics | Orbital Mechanics, Rocket Equation |
| Hohmann transfer Δv and transfer time between circular orbits | Standard two-impulse result from vis-viva: Δv₁ = √(μ/r₁)(√(2r₂/(r₁+r₂))−1), Δv₂ = √(μ/r₂)(1−√(2r₁/(r₁+r₂))), transfer time π√(a³/μ) for a = (r₁+r₂)/2. Curtis, Orbital Mechanics for Engineering Students; Vallado, Fundamentals of Astrodynamics and Applications. orbital_model.py reproduces the published LEO–GEO figures (2.455 + 1.477 = 3.932 km/s over 5.26 h) and checks that the transfer ellipse’s periapsis and apoapsis speeds match the post- and pre-burn speeds to machine precision | Orbital Mechanics |
| Bi-elliptic transfer becomes cheaper past a radius ratio of about 11.94 | Classic three-impulse result; the 11.938765… crossover for an infinitely distant intermediate apoapsis is given in Curtis and Vallado. orbital_model.py locates it numerically by bisection and agrees to 5×10⁻³. It is the standard counter-example to “Hohmann is the minimum-energy transfer”, which is stated more confidently than it deserves | Orbital Mechanics |
| Tsiolkovsky rocket equation | Tsiolkovsky, K.E. (1903), "The Exploration of Cosmic Space by Means of Reaction Devices" | Rocket Equation & Delta-v Budget |
| Falcon 9 / Saturn V / Electron vehicle specifications | SpaceX, NASA (Saturn V Flight Manual SA-503), Rocket Lab public specifications — approximate figures assembled for illustration, not manufacturer data sheets | Rocket 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 estimates | Rocket Equation & Delta-v Budget |
| 1D/2D coefficient-of-restitution collision equations | Standard classical mechanics — any introductory physics or engineering dynamics textbook (e.g. Halliday & Resnick, Fundamentals of Physics) | Collision Simulator |
| Newton's Cradle "N in, N out" behaviour from equal-mass elastic collisions | Standard result of solving simultaneous momentum and kinetic-energy conservation for equal masses — verified numerically in this tool against the textbook signature | Collision Simulator |
| Claim | Source | Tool |
|---|---|---|
| RYB subtractive colour mixing model | Gossett, N. & Chen, B. (2004), "Paint Inspired Color Mixing and Compositing for Visualization" — the RYB trilinear interpolation approach this tool is based on | Paint Colour Mixing |
| Refresh rate measured from animation-frame callbacks | W3C HTML Living Standard — requestAnimationFrame runs once per presented frame. There is no browser API that reports a display’s refresh rate, so this is an inference from presentation timing, not a hardware reading | Screen Refresh Rate Test |
| Why the estimate uses a median rather than an average | Measured, not cited: refresh_model.py runs the estimator against synthetic frame streams of known rate. A 240 Hz stream with 3% dropped frames averages out to about 232 Hz; the median-and-filter estimate returns 240 and reports the drops separately | Screen Refresh Rate Test |
| Timestamp resolution, and why results are aggregates | Browsers deliberately quantise performance.now() to mitigate timing side-channel attacks (Spectre); the page measures and displays the resolution it actually gets. At 240 Hz on a 1 ms clock every individual interval reads 4 or 5 ms and never 4.17 | Screen Refresh Rate Test |
| The list of standard refresh rates, and the snapping tolerance | Conventional panel rates. The 1.2% tolerance is set by the closest pair in that list (175/180 Hz, 2.78% apart) so no suggestion is ambiguous — a reading further out is reported as measured rather than rounded to a rate the panel may not have | Screen Refresh Rate Test |
| Snell's law, the critical angle, and total internal reflection | Standard geometric optics: n₁ sinθ₁ = n₂ sinθ₂, with θc = arcsin(n₂/n₁) | Refraction & Diffraction |
| Reflectance of a transparent boundary, and Brewster's angle | The Fresnel equations in their real-valued dielectric form, for s- and p-polarised light; Brewster’s angle arctan(n₂/n₁) is where the p-component reaches zero. Not valid for metals or absorbing media, which need a complex refractive index | Refraction & Diffraction |
| Diffraction angles for slits, gratings and circular apertures | Fraunhofer (far-field) diffraction: d sinθ = mλ, read as minima for a single slit of width d and as maxima for a spacing d; the circular aperture uses the 1.22 λ/D Airy/Rayleigh criterion | Refraction & Diffraction |
| Refractive indices of common materials | Conventional textbook values at the sodium D line (589 nm). Real materials vary by formulation and with wavelength — the page states this rather than implying a measurement | Refraction & Diffraction |
| Light transport in the room preview: bounced light, soft shadows and colour bleeding between surfaces | Path tracing after Kajiya, J. (1986), The Rendering Equation, with explicit light sampling (next-event estimation) for the window, sun and ceiling fitting | Room Colour Visualiser |
| How sheen changes a surface, from matt to gloss | GGX/Trowbridge-Reitz microfacet distribution with Smith masking and a Schlick Fresnel term at F0 = 0.04, the standard dielectric value. The roughness assigned to each sheen name is illustrative: paint is sold by gloss units measured at a fixed angle and manufacturers do not agree on where one name stops and the next begins | Room Colour Visualiser |
| Light reflectance value (LRV) quoted for a wall colour | CIE Y tristimulus value under D65, expressed as a percentage — the same quantity the trade quotes on a colour card | Room Colour Visualiser |
| Tone mapping of the rendered image | The ACES filmic approximation (Narkowicz, 2015). Needed because a sunlit wall beside a window spans a far wider range of brightness than a screen can show | Room Colour Visualiser |
| The fourteen tube colours, and their positions in the mixing model | Representative masstone renderings of each pigment, with the RYB coordinates fitted to reproduce them under the model above rather than typed by hand — subject to a cap on each tube’s implicit black content, without which a tube fitted to a dark pigment darkens every mix it joins. Not measured spectral data for any particular manufacturer’s paint, which is what accurate prediction would need | Paint Colour Mixing |
| sRGB transfer function and primaries (hex/RGB conversions, the linear segment near black) | IEC 61966-2-1:1999, Default RGB colour space — sRGB | Paint Colour Mixing |
| CIE L*a*b* under the D65 white point, and ΔE*ab (CIE76) as the colour-difference measure | CIE 15:2004, Colorimetry — CIE76 is a plain Euclidean distance in Lab and is known to overstate differences among saturated blues, which is why the page presents ΔE as a ranking | Paint Colour Mixing |
| Contrast ratios quoted against white and black | W3C WCAG 2.2, relative luminance and contrast ratio definitions | Paint Colour Mixing |
| How much chroma a mix can reach at each hue (the gamut readout and hue chart) | Measured, not cited: colour_model.py takes the maximum chroma in each Lab hue bucket for the mixable set and for the sRGB gamut surface. It is a property of the RYB model above, not of any real paint range — the model places its red and yellow primaries on the sRGB primaries, so the warm end reads as fully reachable by construction | Paint Colour Mixing |
| Paint coverage rates by type (emulsion, masonry, gloss, primer) | Compiled from multiple UK trade decorating references — illustrative, genuinely varies by product and manufacturer | Paint Coverage Calculator |
| Porosity and wastage adjustment factors for paint calculation | UK trade decorating guidance — new plaster/porous surfaces reduce coverage; 10% minimum wastage allowance | Paint Coverage Calculator |
| UK stud spacing standards (400mm/600mm centres) | UK trade joinery and drylining references, tied to plasterboard thickness (9.5mm → 400mm, 12.5mm → 600mm) | Screw & Stud Sizing Selector |
| Plasterboard fixing practical load ceiling (~20kg) | Independent destructive testing of real fixings, finding plasterboard itself fails around 60kg of axial load regardless of fixing type — a third of peak load is the trade-standard safety margin | Screw & Stud Sizing Selector |
| Wall plug diameter bands by load | UK fixing manufacturer and retailer guidance (Toolstation, ManoMano) | Screw & Stud Sizing Selector |
| Tea extraction follows first-order kinetics (exponential approach to a maximum) | Spiro & Jago's steady-state kinetic model (1982), the standard framework for tea infusion kinetics research; confirmed across multiple peer-reviewed studies | Tea Brewing Calculator |
| CTC teabag tea reaches near-maximum extraction by ~6-8 minutes | Published tea extraction studies (De Gruyter 2022; open-access black tea extraction kinetics research) — used to calibrate this tool's teabag rate constant | Tea Brewing Calculator |
| Smaller leaf particle size extracts faster (teabag vs. loose leaf) | Price & Spiro (1985), "effect of leaf size... on the rate of infusion" — decreased particle size increases extraction rate constants | Tea Brewing Calculator |
| Closed-form heat-diffusion solution for boiling-egg cook time | Williams, C.D.H., University of Exeter, "The Science of Boiling an Egg" — verified against all four of the author's own published worked examples | Egg Timer Calculator |
| Egg white and yolk coagulation temperature ranges | Williams (as above); egg white 62-80°C depending on protein fraction, yolk 65-70°C | Egg Timer Calculator |
| Green-ring (ferrous sulphide) formation above ~77°C | Williams (as above); hydrogen sulphide from the white reacting with iron in the yolk at elevated temperature | Egg Timer Calculator |
| Poaching water temperature (82-88°C, below a rolling boil) | Cross-referenced consensus across multiple culinary references (Jessica Gavin, What's Cooking America, American Egg Board) | Egg Timer Calculator |
| Semi-infinite-slab heat conduction (one-sided pan frying) | Semi-infinite solid with a step change in surface temperature, T(x,t) via the complementary error function — Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, transient conduction chapter; reuses this tool's own verified thermal constants | Egg Timer Calculator |
| Typical pan temperatures by fried-egg style (120-165°C) | Cross-referenced culinary references (Food Republic, Tasting Table, Gygi, All-Clad) | Egg Timer Calculator |
| Lumped-capacitance exponential heating model for stirred scrambled eggs | Standard result for a well-mixed thermal mass — the same model form used in this site's Tea Brewing and Beverage Cooling calculators, calibrated against published scrambled-egg timing (All-Clad, Hestan Cue) | Egg Timer Calculator |
| Typical egg masses by type (chicken sizes, duck, quail, goose, turkey) | Compiled from multiple poultry and nutrition references (Meyer Hatchery, Cackle Hatchery, USDA weight classes) | Egg Timer Calculator |
| Standard concrete mix ratios by grade (C20 1:2:4, C25 1:2:3, C30 1:1.5:3), treated as ratios by volume, and the 1.54× dry-volume correction | Cross-referenced across multiple published concrete-calculator and construction references. Splitting the dry volume by the ratio and converting each share at its own bulk density (cement 1440, sand 1600, aggregate 1500 kg/m³) gives 317 kg of cement per m³ for C20, inside the 300–330 kg/m³ band published for nominal 1:2:4, and a wet density of 2,499 kg/m³. The three materials sum to the quoted dry volume exactly | Concrete Mix Calculator |
| Strength falls roughly exponentially with the water-cement ratio — 0.60 keeps about 74% of a 0.50 mix and 0.70 about 55% | Abrams' law, S = A/B^(w/c), fitted to a published 28-day cube-strength table (0.40→50, 0.50→40, 0.60→30, 0.70→22 MPa); the fit tracks that table within 8% across the range. Shown only as a ratio against 0.50, because the absolute constants move widely with cement type, aggregate and compaction | Concrete Mix Calculator |
| Strength gain over time, and the effect of temperature on it — 70% takes about 4.8 days at 20°C but 9.5 days at 5°C | Eurocode 2 strength-development expression βcc(t) = exp{s[1 − √(28/t)]} with s = 0.25 for class N cement, combined with Nurse–Saul equivalent age on a −10°C datum. Reproduces the familiar textbook figures: 50% at about two days at 20°C, and 70% at seven days at 10°C | Concrete Mix Calculator |
| Tap drill size formula (major diameter minus pitch/1÷TPI, ~75% thread engagement) | Standard machinist reference, cross-checked against published tap drill charts for both metric and UNC threads | Screw, Stud & Tap Drill Sizing |
| Bolt proof, yield and ultimate strength by property class | ISO 898-1 — all three values for class 8.8 step at d = 16 mm, which matters here because M20 is offered. Classes 10.9 and 12.9 do not step in this range. | Screw, Stud & Tap Drill Sizing |
| Coffee-to-water ratios by brew method (1:16 pour-over, 1:15 French press, 1:2 espresso, etc.) | Specialty Coffee Association Gold Cup Standard, cross-referenced across multiple brewing guides | Tea & Coffee Brewing Calculator |
| Average human visual reaction time (~250ms, <200ms fast, <150ms exceptional) | Cross-referenced across multiple published reaction-time studies and large-scale benchmarks (Human Benchmark, ~81 million participants) | Precision Timer & Stopwatch |
| World Athletics false-start threshold (100ms) | World Athletics competition rules — reaction below this is ruled a false start on the basis that it precedes possible neurological response to the start signal | Precision Timer & Stopwatch |
| Tabata protocol (20s work / 10s rest × 8 rounds) | Izumi Tabata's 1996 study on high-intensity intermittent exercise, the origin of the widely-used training format | Precision Timer & Stopwatch |
| Web Audio API lookahead scheduling for drift-free timing | Standard technique documented by the Web Audio community (the "Tale of Two Clocks" pattern) for sample-accurate browser audio timing | Precision Timer & Stopwatch |
| Claim | Source | Tool |
|---|---|---|
| Trigonometric ratios and the Pythagorean theorem | Standard right-triangle trigonometry — foundational identities with no single citable origin, cross-referenced across any standard trigonometry reference | Trigonometry & Pythagoras |
| 2D/3D area, perimeter, volume and surface-area formulas | Standard Euclidean geometry formulas, cross-referenced across any standard geometry reference | Geometry — 2D & 3D |
| Function plotting: pixel sampling, discontinuity detection, equal-scale axes | Standard numerical plotting technique — evaluate at each pixel column and break the path where the function is undefined or the value jumps beyond the visible range; no single citable origin | Graphing Calculator |
| Exact distributions for dice sums and for keep-highest / keep-lowest rolls; four non-transitive dice each beating the next with probability 2/3; and a second pair of six-sided dice with the same sum distribution as an ordinary pair | Sums are computed by discrete convolution of the uniform single-die distribution; keep-highest and keep-lowest come from the order statistics of n independent uniform draws, P(max ≤ k) = (k/s)n. The four dice are Efron’s dice, introduced by Bradley Efron and popularised by Martin Gardner in Scientific American; the alternative pair is the Sicherman dice (George Sicherman, again via Gardner), which are provably the only other pair of positive-integer six-sided dice with the sum distribution of 2d6. Fairness uses Pearson’s chi-squared goodness-of-fit test against the uniform distribution at the 5% level. Randomness is the Web Crypto API’s getRandomValues with rejection sampling to eliminate modulo bias. All distributions are re-derived independently in dice_model.py in exact rational arithmetic (77 checks). Ideal dice only — no physical bias, spin or throw technique is modelled | Dice Roller & Coin Toss |
| Normal distribution, standard deviation and z-scores | Standard statistical theory. Quartiles by linear interpolation between order statistics (R type 7, also numpy's default); Tukey's 1.5 × IQR outlier fences; Freedman–Diaconis histogram bin width; Blom plotting positions with Acklam's inverse-normal approximation for the Q–Q plot; adjusted Fisher–Pearson skewness and excess kurtosis; Welch–Satterthwaite degrees of freedom for the two-sample t-test; Galton board as the physical model of the binomial and the central limit theorem. All cross-checked in stats_model.py | Statistics & Probability |
| Percentage arithmetic (X% of Y, reverse %, increase/decrease) | Standard arithmetic — percent means per hundred; same identities used by the site command palette | Percentage Calculator |
| Claim | Source | Tool |
|---|---|---|
| BMI formula and WHO adult category thresholds | World Health Organization / CDC standard adult BMI classification (Underweight <18.5, Normal 18.5–24.9, Overweight 25–29.9, Obese Class I–III) | BMI Calculator |
| Ethnicity-adjusted BMI thresholds for South/East Asian populations | National health body guidance (Japan, Singapore, India) applying lower overweight/obese cutoffs, reflecting measurably different risk curves at the same BMI | BMI Calculator |
| Riegel endurance formula (T₂ = T₁·(D₂/D₁)^1.06) for running race-time prediction | Riegel, P. (1977/1981), American Scientist — validated across tens of millions of race results since; confirmed against multiple independent worked examples | Pace Calculator |
| Elite, competitive, average and beginner running times by distance (5K, 10K, half, marathon) | Cross-referenced across RunRepeat's State of US Marathons 2025, Marathon Handbook's age/ability benchmark guides, and World Athletics elite/world-record data | Pace Calculator |
| Cycling speed bands by rider category (beginner through professional) | Cross-referenced cycling references (BikeTips, Fitness Health, road-bike.co.uk) — recreational 10-15mph through professional 25-31mph sustained | Pace Calculator |
| Swimming pace bands per 100m by ability level | Cross-referenced swimming references (220 Triathlon, Polar, Fitness Health) and current 100m freestyle world records | Pace Calculator |
| Log-normal shape for race-finish-time population distributions | Standard, well-documented statistical model for right-skewed finish-time data; calibrated here against published median/beginner benchmarks rather than a specific event's raw results | Pace Calculator |
| Mifflin-St Jeor BMR equation and standard TDEE activity multipliers (1.2–1.9) | Mifflin MD, St Jeor ST, et al. (1990), American Journal of Clinical Nutrition — cross-referenced against the Frankenfield comparison review and ISSA/NASM activity multiplier tables | TDEE Calculator |
| The naive 3,500 kcal/lb (7,700 kcal/kg) fixed-deficit rule overestimates real-world weight loss, increasingly over longer timeframes | Thomas DM, Martin CK, Lettieri S, et al. (2014), International Journal of Obesity; Hall KD, Sacks G, Chandramohan D, et al. (2011), The Lancet — cross-referenced against a direct simulation comparing the fixed-deficit rule to a weight-dependent BMR recalculation | TDEE Calculator |