Example: 1 m rise over 20 m run is exactly 5%

Both measurements use metres, so their units cancel directly. Linear representations remain exact; only the arctangent result is approximate.

Divide rise by horizontal run

s = 1 m ÷ 20 m = 1/20 = 0.05 exactly.

Scale the exact fraction

0.05 × 100 = 5%, and 0.05 × 1000 = 50‰. The reciprocal gives the exact ratio 1:20.

Apply arctangent last

θ = arctan(0.05) ≈ 2.862405°. The rounded angle should not be used to claim more measurement precision than the inputs support.

Check the result against the original geometry

A 5% slope changes height by five units for every 100 horizontal units. Over 20 metres that is one metre, so the converted percentage returns the starting rise without introducing a new assumption.