Convert degrees to percent slope grade or percent to degrees, find slope from rise and run, or turn a pie-chart angle into a percent of 360°.
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Calculator Functions
| Function | What it does | What it requires |
|---|---|---|
| Percent grade from angle | Converts a slope angle to percent grade | Requires the slope angle in degrees or radians, strictly between −90° and 90° |
| Angle from percent grade | Converts a percent grade to the slope angle in degrees | Requires the percent grade |
| Slope from rise and run | Finds the percent grade and the angle of a slope from two measured distances | Requires the vertical rise (negative for a fall) and the horizontal run |
| Percent of a full circle (pie chart) | Converts an angle to its share of a 360° circle, as used for pie chart sectors | Requires an angle from 0° to 360° |
Rise and run can be entered in meters, feet, inches, centimeters or millimeters, and the two do not have to match. The three slope functions each show the angle in degrees and radians, the percent grade and the slope as a 1-in-N ratio. An optional setting limits results to 0 to 4 decimal places, with 2 as the default.
Degrees to Percent Formula
| P | Percent grade, also called percent slope (%) |
| θ | Slope angle measured up from horizontal (degrees) |
| tan | Tangent, which equals the rise divided by the run |
Percent grade is the rise over the run, multiplied by 100, and the slope angle is the inverse tangent of that ratio [1]. Going the other way, the tangent of the angle gives the ratio back.
Percent grade from angle
The formula above applies directly. A 45° slope has a tangent of 1, so it is a 100% grade, and a 10° slope is tan(10°) × 100 = 17.63%. An angle in radians is used as entered. Angles of 90° or more are rejected, because a vertical slope has no run.
| Slope angle | Percent grade | Slope angle | Percent grade |
|---|---|---|---|
| 0.5° | 0.87% | 10° | 17.63% |
| 1° | 1.75% | 12° | 21.26% |
| 2° | 3.49% | 15° | 26.79% |
| 3° | 5.24% | 20° | 36.40% |
| 4° | 6.99% | 25° | 46.63% |
| 5° | 8.75% | 30° | 57.74% |
| 6° | 10.51% | 35° | 70.02% |
| 7° | 12.28% | 40° | 83.91% |
| 8° | 14.05% | 45° | 100% |
| 9° | 15.84% | 60° | 173.21% |
Angle from percent grade
The percent is divided by 100 and the inverse tangent is taken. A 60% grade is arctan(0.60) = 30.96°, and an 8% grade is 4.57°. A negative grade gives a negative angle, which is a downhill slope.
Slope from rise and run
Rise is the vertical change and run is the horizontal distance, not the length measured along the slope. A rise of 8 ft over a run of 40 ft is 8 ÷ 40 × 100 = 20%, which is an angle of 11.31° and a ratio of 1 in 5.
Percent of a full circle (pie chart)
This is a share of a full turn and has nothing to do with slope. A 90° sector is 25% of the circle and a 57.6° sector is 16%. To turn a percentage back into degrees, multiply it by 3.6, so 40% of a pie chart is a 144° sector.
| Degrees of a circle | Percent of the circle | Degrees of a circle | Percent of the circle |
|---|---|---|---|
| 18° | 5% | 108° | 30% |
| 36° | 10% | 144° | 40% |
| 45° | 12.5% | 180° | 50% |
| 54° | 15% | 216° | 60% |
| 72° | 20% | 270° | 75% |
| 90° | 25% | 360° | 100% |
How to Convert Degrees to Percentage
The answer depends on what the degrees measure. For a slope, road grade or roof pitch, take the tangent of the angle and multiply by 100, so 5° is 8.75%. For a pie chart or any other share of a circle, divide the degrees by 360 and multiply by 100, so 5° is 1.39%. To convert a percentage into degrees, reverse the same step: arctan(percent ÷ 100) for a slope, or percent × 3.6 for a circle.
Limits of This Estimate
- Percent grade is not a share of 90°. A 100% grade is 45°, not vertical, and the percentage grows without limit as the angle approaches 90°.
- The slope functions take the angle from horizontal and the run as a horizontal distance. A distance taped along the sloping surface is longer than the run and gives a grade that is too low.
- The result is geometry only. It does not show whether a ramp, road or roof meets a building code or accessibility rule.
Sources
- National Wildfire Coordinating Group, Firefighter Math, Section 4.5 Slope: slope percent as rise over run times 100, and slope angle as the inverse tangent of the slope percent written as a decimal
