Choose a solve mode to calculate slope between two points, convert grade and inclination, solve for rise or run, or find a second point from a straight-line distance and slope or angle.

Slope is change in y per unit change in x. Angle, grade and distance assume both axes use the same length unit and scale.

Average Slope Formula

The following formula is used to calculate the Average Slope (average rate of change) between two points. 

AS = (yโ‚‚ - yโ‚) / (xโ‚‚ - xโ‚)
  • Where AS is the average slope (ฮ”y/ฮ”x, i.e., rise/run)
  • (xโ‚, yโ‚) is the first point
  • (xโ‚‚, yโ‚‚) is the second point

To calculate an average slope between two points, compute the rise (yโ‚‚ โˆ’ yโ‚) and the run (xโ‚‚ โˆ’ xโ‚), then divide rise by run. Identical points do not define a unique line; a nonzero rise with zero run is vertical and has undefined slope. Geometric grade, angle, and distance require equal length units and scales on both axes. If you are averaging slopes across multiple segments with different run lengths, the correct โ€œoverallโ€ average slope is total rise divided by total run (a run-weighted average), not a simple arithmetic mean of slope values.

How to Calculate Average Slope?

The following examples use explicit solve modes. Grade uses horizontal run, not sloped distance. Rise/run modes convert units automatically. For a second point, distance is measured along the line and both coordinate axes must share its unit. Inclinations range from โˆ’90ยฐ to 90ยฐ; forward increases x except at vertical angles, where it follows the angleโ€™s sign. Backward reverses that direction.

Example Problem #1:

  1. First, determine the first point (xโ‚, yโ‚).
    • The first point is given as: (2, 3).
  2. Next, determine the second point (xโ‚‚, yโ‚‚).
    • The second point is provided as: (8, 15).
  3. Finally, calculate the Average Slope using the equation above: 

AS = (yโ‚‚ โˆ’ yโ‚) / (xโ‚‚ โˆ’ xโ‚)

The values given above are inserted into the equation below and the solution is calculated:

AS = (15 โˆ’ 3) / (8 โˆ’ 2) = 12 / 6 = 2 (units of y per unit x)

Example Problem #2: 

For this problem, the variables needed are provided below:

rise = 6

run = 8

This example problem is a test of your knowledge on the subject. Use the calculator above to check your answer. 

AS = rise / run = 6 / 8 = 0.75, or 75% grade, when rise and run use the same unit.