How to convert a percentage slope to degrees

Written by bert markgraf
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Per cent slopes are generally used on roads to indicate the steepness of hills, but the nonlinearity of these measurements means they must be converted to degrees to get a more intuitive, visual picture of how steep the slope actually is. While a 10 per cent slope means you have gone up 10 feet after travelling 100 feet in horizontal distance and a 100 per cent slope means you have climbed 100 feet, the slope in degrees has only increased from 6 degrees to 45 degrees. Conversions from per cent slope to degrees are easy using trigonometric functions.

Skill level:
Easy

Things you need

• Calculator with trig functions

Instructions

1. 1

Calculate the "rise" of the slope, which is the distance climbed. Assume a "run, " which is the horizontal distance travelled, of 100 feet. Multiply the slope, in per cent, by the 100 feet of run to get the rise. A slope of 10 per cent will yield a rise of 10 per cent times 100 feet, or 10 feet. A slope of 25 per cent will yield a rise of 25 feet.

Note that the run is the theoretical horizontal distance travelled and is slightly less than the distance travelled along the road going up the slope. Calculate the distance travelled along the road by adding the square of the rise to the square of the run and taking the square root of the result. For a 25 per cent slope, this is 25 squared, or 625, plus 100 squared (10,000) for a result of 10,625. The square root is 103 feet, which is slightly more than the 100-foot run.

2. 2

Divide the "rise" by the "run." For a 10 per cent slope, the result is 10 feet divided by 100 feet equals 0.1. For a 25 per cent slope, it is 0.25. Calculate the arctangent or inverse tangent of this result to get the slope in degrees. Set your calculator to calculate in degrees. Enter the amount calculated and press the "inverse" or "INV" key and the tangent key to get the slope in degrees. For a 10 per cent slope, the corresponding value in degrees is 5.7 degrees. For a 25 per cent slope, the value is 14 degrees.

3. 3

Check the result by changing the degrees back to the per cent slope. Enter the number of degrees in the calculator and press the tangent key to get the slope's ratio of "rise" to "run." Multiply by 100 to get the per cent slope. Enter 14 degrees into the calculator and get 0.25. Multiply by 100 to get a 25 per cent slope. Enter 5.7 degrees and multiply the result by 100 to get a 10 per cent slope.

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