Slope and Grade: Rise Over Run, Percent Grade, Roof Pitch, Ramps and the Slope Between Two Points

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Opens on Chapter 4’s example: a 6% grade for 2 miles; a 6-in-12 roof; a 1:12 ramp up 2 feet; a taxi fare from two trips.

1. Rise and run

Rise

Run

Then over a run of

2. A road at a grade

Grade

Distance (miles)

3. A ramp

Ratio 1 :

Rise to climb (feet)

4. The slope between two points

x₁ miles

y₁ fare

x₂

y₂

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How it works

Slope = rise ÷ run, the change in height for each unit across. As a percent it's a grade: 6% means 6 feet up for every 100 feet along. A roof pitch says the rise for 12 inches of run, so 6 in 12 is a slope of 0.5. A negative slope goes downhill; zero is level. Between two points the slope is (y₂ − y₁) ÷ (x₂ − x₁): the rate of change, in the y-units per x-unit. A straight line has the same slope everywhere, so the start value is y₁ − slope × x₁. A curve doesn't: that's the problem calculus solves.

From the book, Chapter 4. 2 miles is 10,560 feet, so a 6% grade climbs 633.6 feet. A 6-in-12 roof has slope 0.5, rising 5 feet over 10. A 1:12 ramp up 2 feet needs 24 feet of run, a grade of 8.3%. The taxi rises $12.50 over 5 miles: $2.50 a mile, starting at $4.00. The tank falls −300 gallons in 50 minutes, a slope of −6 gallons a minute. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.