Percent 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 percent slope means you have gone up 10 feet after traveling 100 feet in horizontal distance and a 100 percent slope means you have climbed 100 feet, the slope in degrees has only increased from 6 degrees to 45 degrees. Conversions from percent slope to degrees are easy using trigonometric functions.
Calculate the "rise" of the slope, which is the distance climbed. Assume a "run, " which is the horizontal distance traveled, of 100 feet. Multiply the slope, in percent, by the 100 feet of run to get the rise. A slope of 10 percent will yield a rise of 10 percent times 100 feet, or 10 feet. A slope of 25 percent will yield a rise of 25 feet.
Note that the run is the theoretical horizontal distance traveled and is slightly less than the distance traveled along the road going up the slope. Calculate the distance traveled 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 percent 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.
Divide the "rise" by the "run." For a 10 percent slope, the result is 10 feet divided by 100 feet equals 0.1. For a 25 percent 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 percent slope, the corresponding value in degrees is 5.7 degrees. For a 25 percent slope, the value is 14 degrees.
Check the result by changing the degrees back to the percent 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 percent slope. Enter 14 degrees into the calculator and get 0.25. Multiply by 100 to get a 25 percent slope. Enter 5.7 degrees and multiply the result by 100 to get a 10 percent slope.