Applications

Have you ever seen a warning sign on a roadway announcing a steep section ahead?

Percents are used to describe the angle of a slope and in other applications where we use right triangles. What does the 14% on this sign mean? Can you spot the right triangle on the sign? If so, which angle in the triangle corresponds to 14%? If we were to measure that angle in degrees, what would it be?

Slope As a Percent

• A line that is increasing in height has a positive slope.
• A line that is decreasing in height has a negative slope.^[2]
• The slope of a horizontal line is 0; it is not increasing or decreasing.

When we are working in the field on land that isn’t flat the distance we walk is the slope distance. The distance we measure on a map is the horizontal distance. We can find the vertical distance by counting elevation contours on a topographic map.

Calculating Slope Angle in Percent

A slope may be expressed as a ratio, a decimal, or a percent. For example, consider this loading ramp with a vertical distance of 23.5 inches and a horizontal distance of 132 inches.

As a Ratio

We express slope as a ratio as VD:HD or [latex]\frac{VD}{HD}[/latex]

For the ramp illustrated above 23.5:132 or [latex]\frac{23.5}{132}[/latex]. We can reduce this ratio to 1:5.62.

As a Decimal

If we divide the vertical distance by the horizontal distance, we get slope as a decimal. In the above ramp [latex]\frac{23.5}{132} = 0.18[/latex].

As a Percent

Expressing slope as a percent means describing the vertical distance that is gained (or lost) when the horizontal distance is set to 100. We take slope as a decimal, and multiply it by 100 to express slope as a percent. In the above ramp…

[latex](\frac{23.5}{132} = 0.18)*100 \to 18[/latex]%.

…or 18 inches of vertical gain for every 100 inches of horizontal distance.

Finding Vertical, Horizontal and Slope Distance

If we know the slope angle and one of the sides of the slope triangle, we can readily find the others. We can use this proportion showing the relationship between vertical distance, horizontal distance, and percent slope.

[latex]\frac{VD}{HD}=\frac{Slope Angle}{100}[/latex]

…or, we can use similar triangles. When we know the slope angle, we know what the vertical distance is when the horizontal distance is 100. For example, if my slope angle is 7%, I can make a triangle that represents the situation when the horizontal distance is 100 feet.

The Pythagorean Theorem gets us the slope distance (hypotenuse). Now, if I’m on a slope of 7% and I know one measurement, I can find the others.

For example, if I’ve walked the slope and measured a slope distance of 25 feet, I can find the horizontal distance as follows:

[latex]\frac{100}{100.24}=\frac{HD}{25}[/latex]

So, the horizontal distance is 24.94 feet for every 25 feet I walk on a slope of 7%.

Relating Slope Angle in Percent to Slope Angle in Degrees

Let’s use the triangle we drew for a 7% slope to illustrate how slope in percent is related to slope in degrees.

You’ve already got trigonometry skills, right? How would you find the angle (\Theta) above in degrees?

[latex]\Theta=\tan^{-1}(\frac{7}{100})[/latex]

To find the slope in degrees, given the slope in percent, use the following:

[latex]\Theta in degrees =\tan^{-1}(\frac{slope angle in percent}{100})[/latex]

Slope in percent is the vertical distance (opposite side) divided by the horizontal distance (adjacent side) – so it’s the tangent of the slope angle in degrees!

To find the slope in percent, given the slope in degrees, use the following:

[latex]\Theta in percent=\tan(\theta in degrees)*100[/latex]

You probably have an intuitive feeling for angles from 0° to 90°. As a percent, these would range from 0% to over 5,270% as the vertical distance (rise) becomes longer than the horizontal distance (run). (We actually can’t take tan 90° because it is undefined. Because the tangent function is defined as sin θ/cos θ, when cos θ becomes zero, the tangent function becomes undefined.) A slope of 45° is equivalent to a 100% slope because the rise is equal to the run. Slopes over 45° will be greater than 100%.