Slope, Grade and Pitch: Converting Between Ratios, Percentages and Degrees

A 10 percent hill, a 1 in 10 gradient and 5.71 degrees are the same slope. How to convert between all three, and why 100 percent is 45 degrees rather than vertical.

Introduction

A slope can be written three ways, and they look nothing alike. A road sign says 10 percent, a railway engineer says 1 in 10, and a surveyor says 5.7 degrees. All three describe the same hill. Mixing them up is easy, and one particular confusion — treating percent as though it were degrees — produces answers that are wildly wrong at the steep end.

The Core Mathematics

Every form starts from the same two measurements: the rise (how far you go up) and the run (how far you go along, horizontally).

The conversions follow directly:

So a 10 percent grade is 1 in 10, and arctan(0.10) = 5.71 degrees. The percentage and the ratio are just rearrangements of each other. Only the angle needs trigonometry, which is precisely why it gets confused with the other two.

The 100 Percent Trap

The single most useful thing to know: a 100 percent grade is 45 degrees, not vertical.

It feels wrong, because 100 percent sounds like a maximum. But percent grade is rise divided by run, and at 45 degrees the rise equals the run, so the figure is 1, or 100 percent. Go steeper and the percentage keeps climbing past 100 with no ceiling. A vertical cliff has infinite percent grade, since the run is zero and you cannot divide by it.

This is why percentages and degrees diverge so sharply as slopes get steeper:

At shallow angles the two are nearly proportional, which is what makes the error so easy to miss: a 5 percent grade really is about 2.9 degrees, and treating percent as roughly "degrees times 1.75" works passably up to about 10 degrees. Beyond that it falls apart completely.

Roof Pitch and the American /12 Convention

Roofing in the United States uses its own version of the ratio, quoting the rise per 12 inches of run rather than per unit. A "6 in 12" or 6/12 pitch rises 6 inches for every 12 along.

Converting is the same arithmetic with 12 as the run:

The 12/12 case is a neat confirmation of the rule above: equal rise and run, 100 percent, 45 degrees.

Ramps and Accessibility

This is where slope conversion has consequences beyond arithmetic, and where the ratio form dominates because the standards are written in it.

Accessible ramps are typically limited to 1:12, which is 8.33 percent or 4.76 degrees, and gentler gradients are required as a ramp gets longer — commonly 1:15 (6.67 percent) or 1:20 (5 percent) for longer runs. The exact figures and the maximum rise between landings vary between jurisdictions, so the governing building regulations are the authority rather than a conversion table.

What the conversion makes clear is how shallow these slopes are. A compliant ramp is under 5 degrees. Anyone eyeballing a slope and guessing "about 10 degrees, that looks fine" is describing something close to 18 percent, more than twice the permitted maximum.

Roads, Railways and Drainage

Different fields settled on different forms, largely for historical reasons.

Roads use percent on modern signage, though older British signs used ratios. A 25 percent sign is 1 in 4, or 14 degrees. The steepest residential streets in the world reach around 35 percent, which is still only about 19 degrees — a reminder of how steep a small number of degrees can feel underfoot.

Railways stick to 1 in N, because the numbers involved are so small that percentages would all be decimals. Steep main-line gradients are around 1 in 38, which is 2.65 percent, or 1.5 degrees. Trains are limited by adhesion between steel wheel and steel rail, so what looks like a negligible slope is genuinely demanding.

Drainage also uses ratios, and the falls are shallower still. A common fall for soil pipe is 1:40, which is 2.5 percent or 1.43 degrees. Too shallow and solids are left behind; too steep and water runs away from them. That is a case where the precision of the ratio form earns its place.

Run or Slope Length?

A subtlety worth knowing. Percent grade is defined against the horizontal run, not the distance travelled along the slope surface. Those are different lengths, since the slope is the hypotenuse of the triangle.

For shallow slopes the difference is irrelevant: at 5 percent the slope is only 0.12 percent longer than the run, and at 20 percent still under 2 percent longer. At 100 percent, though, the slope surface is 41 percent longer than the run, so the distinction genuinely matters on steep ground.

This is also why a measurement taken with a tape laid along a hillside is not the run and cannot be used directly. Our guide to significant figures and rounding covers how to keep that kind of error from compounding.

Quick Reference Conversion Table

Common Mistakes Worth Avoiding

Reading a percentage as an angle

A 20 percent hill is 11.3 degrees, not 20. The two only resemble each other near zero, and the gap widens fast.

Assuming 100 percent is the steepest possible

It is 45 degrees. Percent grade has no upper limit, and vertical is undefined rather than 100.

Using the slope length as the run

Grade is rise over horizontal run. On steep ground, measuring along the surface understates the gradient.

Converting a ratio by multiplying

1:20 is 5 percent, not 20. The ratio inverts when it becomes a percentage, so divide 100 by N.

Conclusion

Ratio, percentage and angle are three descriptions of one slope. Divide 100 by N to turn a ratio into a percentage, and take the arctangent of rise over run to get an angle. Above all, remember that 100 percent is 45 degrees: percentages and degrees are close enough to confuse at shallow gradients and completely different once a slope gets steep.

Try our degrees to radians converter for working with the resulting angle in other units.

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