Rise relates to the change of distance on the vertical Y-axis. In the real world, this could be the difference between elevation points on a hill or the height difference between the top and bottom of your roof. Conversely, run is the change of distance on the horizontal X-axis, such as the map distance between two points or how far out the roof extends from the center. If you divide rise by run, you calculate the slope, which is the ratio of the two measurements.
Using the Cartesian Coordinate System
Determine the plots for two points for which you want to calculate the rise and run. As an example, the first point might be aligned with "2" on the X-axis and "4" on the Y-axis, so the plotted point is at (2,4). You might then find the second point is at (5,9).
Subtract the first X-axis point from the second one to calculate the run. In the example, 5 minus 2 gives you a run of 3.
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Subtract the first Y-axis point from the second one to calculate the rise. Continuing with the example, subtract 4 from 9 to get a rise of 5.
Divide the rise by the run to calculate the slope, which is useful to find the rise and run between other points on the same line. In the example, 3 divided by 5 calculates a slope of 0.6. A positive slope means the line goes up from left to right, but a negative slope means it goes down. If you were asked to express the example slope in percent format, simply multiply 0.6 by 100 to get 60%.
Multiply the slope by the run to calculate the rise between subsequent points. In the example, if you wanted to know the rise given a run of 10, multiply 10 times 0.6 to calculate a rise of 6.
Divide the rise by the slope to calculate the run. In the example, if you had a rise of 12, divide by 0.6 to calculate a run of 20.
Example: Finding Rise, Run and Slope of a Hill
Subtract the difference in elevation between two points on a hill to calculate the rise. The elevation could be determined by an altimeter or you could use a topographic map. As an example, you might read 900 feet at the top of a hill and 500 feet at the bottom, so subtract 500 from 900 to get a rise of 400 feet.
Measure the distance between the top and bottom of the hill to find the run. For example, you could align a map's distance scale to determine the distance. However, you couldn't use a pedometer and walk up the mountain because you'd then be measuring the distance over the slope instead of the true horizontal distance. In the example, if the scale showed that 1 inch equals 500 feet and you measured 1.5 inches on the map, multiply 1.5 times 500 to get a run of 750 feet.
Divide the rise by the run to calculate the slope. In the example, 400 divided by 750 calculates a slope of 0.53. The slope of a hill is important, because it gives insight into how fast water runs off, which affects water contamination, erosion and the danger of flash flooding.
The Cartesian coordinate system is the standard, two-dimensional graph system, which is often called the rectangular coordinate system due to its reliance on a vertical and horizontal scale.
Slope ratio doesn't have to use the same units, because 200 ft/mi, for example, validly explains that a gradual incline adds 200 feet of elevation for every mile of horizontal distance; if you do use different units, keep both units with the slope to make the distinction clear, such as the example's "200 ft/mi." However, slope percentages must use the same units, or the calculation will be wrong. In the later example, you'd convert 100 feet to 0.038 miles and then multiply by 100 to find the slope of 3.8%.