Percent slope answers one question: how much does the ground rise for every hundred units you travel horizontally? It is the unit that drainage, roads, ramps and earthworks are specified in, and it is also the one where the most common mistake hides — measuring the run along the sloping surface instead of horizontally. This guide gives the formula, works it both ways, and shows how the same gradient looks as a percentage, an angle, a ratio and a roof pitch.
Rise is the vertical change between two points, run is the horizontal distance between
them, and both must be in the same unit before you divide — the units cancel, which is why a slope
percentage has no unit at all. A rise of 1 over a run of 12 is 1 ÷ 12 × 100 = 8.33%. A
rise of 6 over a run of 100 is 6%. A rise of 2.5 feet over a run of 20 feet is
2.5 ÷ 20 × 100 = 12.5%.
Percent slope is the tangent of the slope angle, expressed in hundredths. So converting between the two is a tangent and an arctangent:
A 100% slope is not vertical, it is 45° — a useful fact to hold on to, because a 100% slope sounds
like a cliff and is actually a steep but walkable hillside. A 6% grade, the sort you see on a highway
sign, is arctan(0.06) = 3.43°. A 1:12 ramp is 8.33%, or
arctan(0.08333) = 4.76°.
| Ratio (1 : n) | Percent | Degrees | Roof pitch (x : 12) | Typical use |
|---|---|---|---|---|
| 1 : 48 | 2.08% | 1.19° | 0.25 : 12 | Minimum fall in a large drainage pipe |
| 1 : 20 | 5.00% | 2.86° | 0.6 : 12 | Walkway rather than a ramp |
| 1 : 12 | 8.33% | 4.76° | 1 : 12 | Accessible ramp limit |
| 1 : 6 | 16.67% | 9.46° | 2 : 12 | Minimum pitch for standard shingles |
| 1 : 4 | 25.00% | 14.04° | 3 : 12 | Low-slope roof, embankment batter |
| 1 : 2 | 50.00% | 26.57° | 6 : 12 | Common residential roof pitch |
| 1 : 1 | 100.00% | 45.00° | 12 : 12 | Rock cut, steep roof |
Slope is defined on the horizontal projection of a line, not on the line itself. If you pace or tape along a sloping surface and use that figure as the run, your percentage comes out too low, and the steeper the slope the bigger the error. On a 1:1 slope the surface distance is 41% longer than the horizontal distance, so a slope measured as 100% from the surface length is really closer to 71%.
The relationship between the three lengths is the familiar right triangle:
slope length = √(rise² + run²). If what you measured was the slope length, recover the
run first — run = √(slope length² − rise²) — and only then divide. Where a slope is
gentle the two are nearly equal and the error is small; where a client is asking about a ramp or a
roof, it is not small enough to ignore.
On a contour map, slope percentage comes out of the contour interval and the horizontal spacing:
slope % = (contour interval ÷ spacing) × 100. With a 10 m interval and contours
100 m apart, the ground falls at 10%. Halve the spacing to 50 m and the slope doubles to
20%. Because contour spacing is measured perpendicular to the contours, the same rule gives the
steepest direction at that point — which is why a slope map drawn from a DEM looks the way it does.
Two cautions apply. Contour maps carry vertical exaggeration in the sense that a 10 m interval can only resolve slope in steps, so a slope read off a single pair of contours is an average over that interval and not a local value. And on a published map the horizontal distances are true only if you account for the map scale and any projection distortion — reading distances off a small-scale map with a ruler compounds both errors.
Drainage falls are specified as a minimum percentage so that water keeps moving: 1% to 2% for a surface channel is common, and piped systems often go flatter. Accessible ramps are capped at 1:12 in the ADA standard, with landings required at intervals, and a slope of 1:20 or flatter is treated as a walkway rather than a ramp. Roof pitch is quoted over 12 units of run, so a 4:12 roof is a 33.3% slope. Earthwork batters are usually quoted as ratios — a 2:1 batter is 50% — and a cut slope in soil is often limited to something in the 1:1 to 1:2 range, while rock can stand much steeper.
Check the direction of the fall before you check the number. A drainage line with the correct gradient running the wrong way is a common and expensive error, and it survives every arithmetic check. Confirm which end is low, and confirm it on the drawing rather than in the calculation.
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