How To Convert A Percentage Slope To Degrees

When you're driving in hilly country, you'll come across road signs alerting you to steep grades ahead, and these are always expressed as percentages. For truckers, a 4 percent downgrade is enough to switch into low gear, and if you're driving an older RV with an underpowered engine, you might think twice about trying to climb a grade steeper than 10 percent. But what does slope percentage mean in degrees?

It isn't just drivers who may need to convert percent grade to degrees. Architects, landscapers and environmental managers often express slope as a percentage, but they may find it more convenient to convert to degrees when making calculations. They may have a slope-to-degrees calculator they can use when they're in the office, but in the field, they may have to do the calculations themselves.

A slope percentage, or percent grade, is a fractional expression. It expresses the ratio of difference in altitude between two points on a slope to the horizontal distance between the points, multiplied by 100. For example a 10 percent slope means that, for every 100 feet of horizontal distance, the altitude changes by 10 feet:

\(katex
{10 ft \over 100 ft} × 100 = 10%\)

Suppose a slope changes 25 feet over a distance of 1,000 feet. The slope percentage is then

\({25 \over 1000} × 100 = 2.5%\)

To convert a percent grade to degrees, you need to use a little trigonometry. If you consider the slope between two points as the hypotenuse of a right triangle, the change in altitude between those points is the rise of the triangle, or the length of the side opposite the angle of the slope (θ), and the horizontal distance between the point is the run, or the length of the side adjacent to θ. The rise over the run is the tangent of angle θ.

Here's how to make the conversion to degrees:

1. Express the percent grade as a fraction of 100. For example, a 10 percent grade is 10/100, and a 25 percent grade is 25/100.
2. Convert the fraction to a decimal fraction. In the examples above, 10/100 = 0.1 and 25/100 = 0.25.
3. Look up the decimal fractions in a table of tangents to find the angles to which they correspond. For example, if tanθ is 0.1, θ is a little less than 6 degrees (it's actually 5.7 degrees), and if tanθ is 0.25, θ is a little more than 14 degrees (14.04 degrees). 

At first blush, it's tempting to assume that a slope percentage of 100 corresponds to a 90-degree angle, but it doesn't. When the slope percentage is 100, the rise over any horizontal distance equals the run, and the tangent of the angle is 1, which corresponds to a slope of 45 degrees. A 90-degree angle corresponds to an infinite slope percentage.

If you don't have a tangent table handy, you can use a scientific calculator as a slope to degrees calculator. After converting the percent slope to a decimal fraction, you'll want the calculator to display the inverse tangent, or arctan, which is the angle that corresponds to that fraction.

Some calculators have an (arctan) key, usually denoted by (tan^-1). Enter the fraction, press that key and read out the angle. If yours doesn't have an (arctan) key, press the (2nd) key or the inverse (INV) key, enter the fraction and press the (tan) key to get the corresponding angle in degrees.