Geometry, Slope & Unit Conversion
Exact unit conversion (length, area, volume, mass, temperature); right triangle from any two values; slope as x:12, 1:x, percent and degrees; ramp running slope against 1:12; tolerance stack-up by worst case and RSS.
Method
geometry-conversion 1.1
Engine
1.1.0
Output status
Informational
Review
Awaiting review
Formulas
- Exact factors: 1 in = 25.4 mm, 1 ft = 0.3048 m, 1 lb = 0.45359237 kg, 1 U.S. gal = 231 in3; areas and volumes as squares and cubes
- Right triangle: c^2 = a^2 + b^2; a = c sin A; b = c cos A; A + B = 90 deg
- Slope = rise / run; percent = 100 x rise / run; degrees = atan(rise / run)
- Ramp: running slope compared with 1:12 (2010 ADA Standards 405.2)
- Worst case = sum of tolerances; RSS = sqrt(sum of squared tolerances)
Conventions
- Angle A is opposite leg a; the right angle is between the legs.
- Tolerances are symmetric (+/-).
Limits - what it will not do
- The ramp check is a reference check of 405.2 only - not a code compliance determination. Cross slope, landings, rise limits, handrails and local codes are not checked.
- RSS assumes independent, centred, normally distributed dimensions.
Who must review it
- AFS: confirm the units and tolerances the shop and its customers use
Worked example
Inputs 12 ft to m; legs 3 and 4; 1:12 slope; dimensions 10 +/- 0.1, 20 +/- 0.2, 5 +/- 0.05
Expected 3.6576 m; hypotenuse 5, 36.87 / 53.13 deg; 8.33%, 4.76 deg; 35 +/- 0.35 worst case, +/- 0.229 RSS
Checked by an automated test on every change.
Sources
- NIST Special Publication 811 - Guide for the Use of the International System of Units (SI), Appendix B conversion factors - National Institute of Standards and Technology, read 2026-10-16
Exact factors: 1 in = 0.0254 m; 1 ft = 0.3048 m; 1 yd = 0.9144 m; 1 mi = 1609.344 m; 1 lb = 0.45359237 kg; 1 U.S. gallon = 231 in3 = 3.785411784 L; 1 Btu (IT) = 1055.05585262 J; t(degC) = (t(degF) - 32) / 1.8; T(K) = t(degC) + 273.15.
- Right-triangle geometry (Pythagorean theorem and trigonometric ratios) - Method sheets roof-geometry/1.0 and geometry-conversion/1.0, read 2026-10-16
Slope factor = sqrt(1 + (rise/run)^2); a hip or valley between equal-pitch planes meeting at 90 deg in plan runs sqrt(2) x the common run, so its length factor is sqrt(2 + (rise/12)^2) per unit of common run.
- 2010 ADA Standards for Accessible Design, 405.2 Slope - U.S. Department of Justice, read 2026-10-16
Ramp runs shall have a running slope not steeper than 1:12. Used here as a reference check of one provision only, not a code compliance determination.
- Linear tolerance stack-up: worst case and root sum of squares - Method sheet geometry-conversion/1.0, read 2026-10-16
Worst case = sum of the tolerances; RSS = square root of the sum of the squared tolerances, valid for independent, centred, normally distributed dimensions.