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Learn to compute slope using the vertical and horizontal leg lengths of a right triangle. Practice with clear steps, worked examples, and self-checking exercises.
Slope measures steepness as the ratio of vertical change (rise) to horizontal change (run). When a right triangle is drawn with one vertical leg and one horizontal leg, those leg lengths give the rise and run directly. This lesson shows how to identify each leg, form the ratio rise over run, simplify it, and check your work. You will also see common mistakes and a worked example, then apply the method to four practice tasks. All measurements use the same unit, and any numerical values chosen for illustration are hypothetical examples.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
A slope triangle is a right triangle whose vertical leg shows how much a quantity changes up or down, and whose horizontal leg shows how much it changes left or right. The vertical leg length is the rise. The horizontal leg length is the run. Slope is rise divided by run.
Label the legs before calculating. Put the vertical length on top and the horizontal length on the bottom. Keep units consistent; if both legs are in centimeters, the ratio is unitless. If one leg is in meters and the other in centimeters, convert first so both use the same unit.
After substituting, reduce the fraction by dividing both numbers by their greatest common factor. For example, rise 6 and run 8 gives 6/8, which reduces to 3/4. A slope of 3/4 means 3 units up for every 4 units right.
You can also express slope as a decimal when the division is exact or when a decimal is requested. Keep the fraction form when exactness matters. State the form you used so the answer is clear.
If the triangle rises as you move left to right, the slope is positive. If it falls, the slope is negative. The leg lengths themselves are positive distances, so the sign comes from the direction of the vertical change, not from the length.
When a problem gives only lengths and no direction, report the positive ratio and note that direction was not specified. When direction is shown, attach the correct sign to the rise before dividing.
A right triangle has a vertical leg of 9 centimeters and a horizontal leg of 12 centimeters. The triangle rises from left to right. Compute the slope.
Slope = 9/12 = 3/4, positive. The triangle rises 3 centimeters for every 4 centimeters to the right.
A right triangle has a vertical leg of 6 units and a horizontal leg of 8 units. It rises from left to right. Compute the slope.
3/4
Why: Rise = 6, run = 8, so slope = 6/8. Divide both by 2 to get 3/4. The triangle rises, so the slope is positive.
Check: Multiply 3/4 by 8 to get 6, which matches the rise.
A right triangle has a vertical leg of 10 meters and a horizontal leg of 4 meters. It falls from left to right. Compute the slope.
-5/2
Why: Rise is downward, so use -10. Run = 4. Slope = -10/4 = -5/2. The negative sign shows the downward direction.
Check: Multiply -5/2 by 4 to get -10, matching the signed rise.
A right triangle has a vertical leg of 15 inches and a horizontal leg of 20 inches. It rises from left to right. Compute the slope and write it as a decimal.
3/4 or 0.75
Why: Slope = 15/20. Divide both by 5 to get 3/4. Since 3 divided by 4 is 0.75, the decimal form is 0.75.
Check: Multiply 0.75 by 20 to get 15, matching the rise.
A right triangle has a vertical leg of 7 centimeters and a horizontal leg of 14 centimeters. It rises from left to right. Compute the slope.
1/2
Why: Slope = 7/14. Divide both by 7 to get 1/2. The triangle rises, so the slope is positive.
Check: Multiply 1/2 by 14 to get 7, matching the rise.
A zero run would make division impossible. In a slope triangle, the horizontal leg must be nonzero; if it is zero, the situation is vertical and slope is undefined.
Yes. If rise is 10 and run is 2, slope is 10/2 = 5, a whole number meaning 5 units up per 1 unit right.
Simplifying makes the ratio easier to interpret and compare. Unless a problem asks for an unsimplified ratio, give the reduced form.
Multiply the simplified slope by the run. The result should equal the rise. For example, slope 3/4 with run 8 gives (3/4)×8 = 6, matching a rise of 6.