# How to Calculate the Grade of a Slope

Print

In elementary school mathematics, when students learn to graph simple linear functions, they are introduced to the concept of a slope.

A linear function is just one with a graph represented by a straight line of some sort, with its placement and direction in relation to the x- and y-axes depending on the function's properties.

A linear equation has the form

y=mx+b

Where y is the dependent variable, m is the slope, and b is a quantity called the y-intercept, the point the line crosses on the y-axis.

But you may have also heard of a mathematical construct called a grade, or a percent grade. Muddled, ambiguous terms like "slope ratio" and "grade of slope" don't help.

Are slopes and grades related? They are indeed, and both are indispensable in mathematics and engineering.

## What is Slope?

In everyday terms, a slope is a steady, sustained climb or descent. That's what it means in mathematics as well, but in a more formal way. The slope of a line is the change in vertical (y) distance per one-unit change in horizontal (x) distance.

For example, if a point in a coordinate system moves 11 units in the positive x-direction and four units in the negative y-direction, the slope is (–4)/(11) = –0.364. The minus sign means the line angles "downhill" in relation to the horizontal x-axis.

A horizontal line such as the function y = 5, in which there is no vertical change throughout, has a slope of 0. A vertical line, such as x = −3, has an undefined slope as there is no horizontal change and dividing by zero is not permitted in mathematics.

## The Point-Slope Formula

The point-slope formula is helpful for determining the equation of a line when either two points or one point and the slope are known. It has the form

y − y_0 = m(x − x_0)

If you were given coordinates (12, −7) and told that the graph of the function had a slope of 1.25, you could determine the general equation:

(y − (−7)) = 1.25(x − 12) \\ (y + 7) = 1.25x −15 \\ y = 1.25x − 22

Grade, or percent grade, is just the slope expressed as a percentage. It is often used in real-life situations involving the construction of roads, the very steepest of which have surprisingly low slope values.

For example, the Pennsylvania Turnpike in the Eastern U.S. has a maximum slope of 0.03, meaning it rises or falls no more 3 feet for every 100 horizontal feet traveled over any segment. The percent grade in this instance is 100 × 0.03 = 3 percent.

In trigonometry, y/x, or "rise over run," is also the tangent of the angle formed by the ascending or descending line and the horizontal. This means that the inverse tangent (tan −1 or arctan on a calculator) of the slope equals this angle.

• In the grueling Tour de France, a three-week race through the mountains of Western Europe featuring the best male cyclists in the world, grades that reach 13 percent are considered extraordinarily fierce.

## Slope Distance Calculator

If you know the slope of a line, you can calculate horizontal distance traveled as a function of vertical distance, or the other way around. Say you know you're walking up a 4 percent grade. If you walk for 30 minutes and your horizontal position changes at a rate of 4 miles per hour, how much elevation have you gained?

4 mph for 30 min (1/2 hr) is 2 miles, and if the percent grade is 4, the slope is 4/100 = 0.04. Since slope is rise over run and in this case the "run" is 2 miles, the vertical gain can be found as follows:

\begin{aligned} 0.04 &= \frac{y}{2 \;\text{miles}} \\ y &= 0.04×2 \\ &= 0.08 \;\text{miles, or about} \\ &0.08 \;\text{mi}×5,280 \;\text{ft/mi} = 422 \;\text{ft} \end{aligned}