Percent grade and degrees describe the same slope in two ways

Percent grade is the rise over the run, multiplied by 100. The angle in degrees is the arctangent of the same ratio. A 45° slope is therefore a 100% grade. Converting between the two needs one function on a calculator in degree mode, and the table below covers the values that appear on maps, road signs and in the energy and terrain tools on this site.

Two formulas for one right triangle

Picture a slope as a right triangle. The run is the flat base, the rise is the vertical side, and the ground surface is the hypotenuse. The angle sits between the base and the surface, and its tangent is rise over run. The percentage grade is that ratio times 100. Wikipedia’s entry on grade describes the same relation: the grade is the tangent of the angle of inclination, multiplied by 100.

grade (%) = tan(angle) × 100
angle (°) = arctan(grade ÷ 100)
ratio     = 1 in n, where n = 100 ÷ grade

The ratio form states the same slope as one unit of rise for every n units of run. A 20% grade is 1 in 5, which means 1 m of rise for every 5 m of run.

Conversion table for common grades

The table gives the angle and the ratio for grades from 5% to 100%. Angles are rounded to two decimal places and ratios to one.

Going the other way, 30° is 57.7%, 35° is 70.0% and 40° is 83.9%. Above 45° the percentage rises quickly. A 60° slope is 173.2%, because tan 60° is 1.732.

What a 100% grade means

A 100% grade is 45°, where the rise equals the run. It is not a vertical slope. Wikipedia describes the scale the same way: the numbers run from 0 for flat, to 100% at 45 degrees, to infinity at vertical. As the slope approaches 90°, the percentage grows without limit, so a vertical face has no finite grade.

The percentage is the most common way to state a slope in Europe and the United States, and Wikipedia notes that road gradients are expressed as percentages on European signage. A 10% sign describes a slope of 5.71°. It does not describe a 10° slope, which is 17.6%.

Converting a grade from a map or a sign

A grade from a map or a sign is usually a percentage, and the conversion is a single arctangent. The worked example uses an 8% grade, which is a rise of 8 m for every 100 m of run.

angle = arctan(8 ÷ 100) = arctan 0.08 = 4.57°
ratio = 100 ÷ 8 = 12.5, so 1 in 12.5
check: tan 4.57° × 100 = 8.0%

A percentage applies to the run, not to the walked distance. A trail description that gives 20% over a 500 m run therefore means a climb of 0.20 × 500 = 100 m. The distance walked along the surface is √(500² + 100²) = √260,000 = 510 m, which is 2% more than the run. The same rule applies to any grade, so the surface distance grows faster on steeper ground.

If the grade came from a contour map, the slope from contour lines guide shows how to calculate it from the interval and the map distance before you convert.

The small-angle shortcut and its error

For small angles the tangent is close to the angle in radians. One radian is 57.2958°, so the angle in degrees is close to 57.3 × grade ÷ 100. The shortcut is quick to do in your head, and it is accurate up to about 10°, which is roughly 18%.

shortcut angle (°) ≈ 57.3 × grade ÷ 100

20%: 57.3 × 0.20 = 11.46°   exact 11.31°   error 0.15° (1.3%)
50%: 57.3 × 0.50 = 28.65°   exact 26.57°   error 2.08° (7.8%)

The error grows with the slope. At 20% the shortcut is 0.15° high, which rarely matters on a map. At 50% it is more than 2° high. Use the exact arctangent when the angle matters for a decision.

Where each unit shows up

Each unit suits a different job, and the tools on this site use them in different ways.

• Road gradients and trail signs are usually percentages, as the Wikipedia entry notes for European signage.
• The Loaded miles tool takes uphill grade as a percentage from 0% to 15%, and the Pandolf equation guide covers the energy model behind it.
• The map slope tool reports the grade as a percentage and the angle in degrees for the same leg.
• The Tobler walking speed in the map slope tool uses the signed ratio S, which is rise over run. A 10% uphill leg enters the formula as S = 0.10, and a 10% downhill leg as S = −0.10. The Tobler hiking function guide covers the formula.

When you enter a grade into a tool, check which unit the field expects. The Tobler formula and the energy model both take a percentage or ratio, not an angle.

Two mistakes when mixing units

The first mistake is to read an angle as a grade. A 45° slope is 100%, and a 45% grade is about 24.2°. The second is to enter a degree value into a percent field. A 10° slope is 17.6%, so entering 10 in a percent field understates the slope by 7.6 percentage points.

Both mistakes change the answer by a large amount, so check the unit on any figure before it goes into a plan. Keep the units in your notes. Write the grade and the angle together, for example “20% (11.3°)”, so that a later reader can see which figure a tool used. Naismith’s rule does not use a slope unit at all, and the Naismith’s rule guide covers the time estimate that goes with a route.