Slope % ⇄ Angle ⇄ Ratio Converter

Understanding the slope percent, angle and ratio converter — concept and principle

The steepness of a road, ramp, roof or pipe can be written as a percentage, an angle, a ratio or a per-mille value, and different trades use different forms. This converter takes a slope in percent and gives the angle in degrees, the ratio 1 : n (one unit of rise per n units of horizontal run) and the per-mille value. Typical uses are checking an accessibility ramp, reading a road gradient sign, setting drainage falls and comparing roof pitches.

How it is calculated

• Slope % = rise ÷ run × 100, where run is the horizontal distance, not the length along the slope.
• Angle = arctan(slope % ÷ 100), converted to degrees.
• Ratio 1 : n with n = 100 ÷ slope %.
• Per mille (‰) = slope % × 10.

The principle

Rise and horizontal run form the two legs of a right triangle, and the slope surface is the hypotenuse. The ratio rise ÷ run is, by definition, the tangent of the angle of inclination, so the angle is its arctangent. Because tangent is not linear, percent and degrees are not proportional: 100% is 45° (rise equals run), and an infinitely steep vertical wall would be an infinite percentage, not 200%. For gentle slopes the two are close to proportional, roughly 1° ≈ 1.75%.

Worked example

1. Default input: slope = 8.33%.
2. rise ÷ run = 8.33 ÷ 100 = 0.0833.
3. Angle = arctan(0.0833) = 0.083108 rad × 180/π = 4.761744998°.
4. Ratio: n = 100 ÷ 8.33 = 12.0048, so 1 : 12.00480192. This is the familiar 1 : 12 ramp, since 1 ÷ 12 = 8.333…%.
5. Per mille: 8.33 × 10 = 83.3‰.

How to read the result

Rough benchmarks: drainage falls on flat surfaces are often 1–2%; most main roads stay below about 6–8%; very steep public streets reach 20–35%; 100% is 45°. For wheelchair ramps, the US 2010 ADA Standards set a maximum running slope of 1 : 12 (8.33%). Roof pitch is often given as rise per 12 units of run, so a 4-in-12 roof is 33.3% or about 18.4°.

Common mistakes

• Dividing the rise by the sloped length instead of the horizontal run; that gives the sine, not the tangent. The difference is small for gentle slopes but large for steep ones.
• Assuming percent and degrees are proportional, for example thinking 50% is 22.5° (it is 26.57°).
• Reading 1 : 12 as 12% instead of 8.33%.

Frequently asked questions

What angle is a 10% slope?

arctan(0.10) ≈ 5.71°.

How do I convert an angle to percent?

Slope % = tan(angle) × 100. For example 30° gives tan 30° × 100 ≈ 57.7%.

Can a slope be more than 100%?

Yes. Any angle above 45° is more than 100%; 60° is about 173%.

Sources: U.S. Department of Justice, 2010 ADA Standards for Accessible Design, §405.2 (ramp slope) · James Stewart, Calculus: Early Transcendentals, Appendix on trigonometry (tangent and inclination)