Slope Calculator: Percent Grade from Rise and Run

What does this quantity describe?

Slope s is the ratio of vertical rise Δh to horizontal run L: s = Δh/L. Multiplied by 100 it gives the percentage. The corresponding slope angle follows from α = arctan(Δh/L), since slope and angle are related through the tangent function rather than linearly.

A road ramp with a 10% slope rises 10 m over 100 m of horizontal distance. That corresponds to an angle of only about 5.7° to the horizontal — considerably flatter than the percentage figure might suggest at first glance, because percent and degrees follow different mathematical functions.

Formula and variables

Slope [%] = 100 · Δh / L

• s = Δh / L
• s [%] = 100 · Δh / L
• α = arctan(Δh / L)
• Δh = L · tan(α)

Choose the inputs correctly

Enter two of the four quantities: slope, angle, rise or horizontal run — noting that slope and angle already fully describe the same incline and only combine with a run length to give a concrete rise. Check whether a percentage or a ratio value is expected.

How to use the calculator

Select the target quantity. For a known incline, slope or angle alone is enough; a concrete rise additionally needs the horizontal run length. Check whether the horizontal run L or the actual (sloped) travelled distance is meant, since the two differ noticeably on steep inclines.

Worked example

Given Δh = 1 m and L = 10 m, substitution gives s = 1/10 = 0.1, or 10%. The angle follows as α = arctan(0.1) ≈ 5.71°. At 100% slope (Δh = L), the angle would instead be α = arctan(1) = 45°.

Understand the result and units

A 10% slope means the path rises 10 m over 100 m of horizontal distance. The corresponding 5.71° angle shows that percentage figures look considerably larger than the actual angle of incline for moderate slopes.

Slope is given dimensionlessly as a ratio or as a percentage, the angle in degrees or radians. Exactly: 100% slope corresponds to a 45° angle, not 90°, since tan(45°) = 1.

Why does 100% slope equal only 45°, not 90°?

The slope percentage is the ratio Δh/L, multiplied by 100. At 100%, Δh equals L, forming an isosceles right triangle, so the angle to the horizontal is arctan(1) = 45°. A vertical rise (90°) would instead correspond to a horizontal run L of zero, where the percentage figure mathematically tends to infinity. This nonlinear relationship is why slopes of roughly 30–40% (corresponding to 17–22°) already count as very steep in road and ramp design, even though the percentage figure may look moderate at first glance.

Typical applications

Slope calculations are used in road and ramp planning, railway construction, conveyor belt inclination, surveying, and roof pitch.

Assumptions, limits and common mistakes

The formulas apply to a straight, uniformly inclined path. For curved alignments or varying grade, slope is only a local or averaged value. On very steep inclines, note additionally that the actual (sloped) travelled distance is longer than the horizontal run L.

Common mistake: Do not confuse 100% slope with a 90° angle (a vertical wall) — 100% actually corresponds to a 45° angle, because the percentage is proportional to the tangent, not the angle itself. Also, do not equate the horizontal run L with the actual (sloped) travelled distance, especially on steep ramps.