A circle has 360 degrees, so if you want to express an angle in terms of a percentage, just divide the angle measurement (in degrees) by 360 and multiply by 100. In reverse, divide the percentage by 100 and multiply by 360. Things get a bit more complicated when you want to convert a slope angle from a percentage to a number of degrees and back again. A slope angle can be expressed as the ratio of the vertical rise to the horizontal run from the apex of the angle to a point directly under the highest point. Multiply by 100 to get the slope percentage. The angle is equal to the arc tangent of the ratio, which is the percentage divided by 100.
Convert the percent of a circle to degrees by remembering that a circle has 360 degrees. When discussing slopes or inclines, convert the slope percentage to a ratio and look up the ratio in a tangent table.
When calculating percentage, you have to know what constitutes 100 percent, and for the number of degrees in a circle, that's 360. If you're discussing sectors of a pie chart, and one sector covers x percentage of the chart, its angle is then x/360 • 100. For example, a section that covers 62 percent of the chart corresponds to an angle of (62/100 • 360) = 223.2 degrees. You can make things even easier by remembering that x/100 corresponds to a two-place decimal. So 62/100 = 0.62.
The maximum number of degrees isn't always 360. For example, if you want to express the height of a celestial object above the horizon as a percentage, you have to remember that the arc of the visible sky has 180 degrees from horizon to horizon. A 62 percent angle in this context would correspond to 0.62 • 180 = 111.6 degrees.
Converting Slope Percentage
When mapmakers are preparing detailed topographical maps of an area, they often express slopes as percentages. To do this, they first calculate the rise (y) and run (x) of a slope using sophisticated surveying equipment. They then express the slope as a ratio of the rise to the run (y : x or y/x) and multiply by 100 to get a percentage for the rise. They can then calculate the angle of the slope by recognizing that the ratio y/x is the tangent of the angle.
If you know the percentage of the angle, you can take a shortcut. Simply divide the percentage by 100 to produce a fraction, and then relate that fraction to an angle by looking it up in a chart of tangents.
The rise and run of a slope are 10 and 15 feet respectively. What are the percentage and angle of the slope?
The ratio of rise to run is 10/15 or 0.67. The percentage of the slope is 0.67 • 100 = 67 percent. According to a tangent table, this ratio corresponds to an angle of 34 degrees.
What is the angle of a slope with a slope percentage of 15 percent?
Divide the slope percentage by 100 to get a slope ratio of 0.15. Look up the angle in a tangent table. That angle is 8.5 degrees.