Ramp Slope, Pythagorean Length, and a Switchback Proposal
The entrance to the Maple Street Community Center has a door threshold 30 inches above the sidewalk. The center plans a straight access ramp along the front of the building, where only 24 feet of horizontal space is available before a fire hydrant. The city's design guideline for this project says the slope may not be steeper than 1:12, meaning 1 inch of rise for every 12 inches of horizontal run. A contractor proposes a switchback with two 12-foot runs, each rising 15 inches to a landing, arguing that 'splitting the height in half doubles the safety.' The board also wants the actual length of the ramp surface for the 24-foot option, because the handrail supplier charges by the foot, and it wants the percent grade for a sign that warns pedestrians. Measurements are exact to the inch, and the ramp must rise the full 30 inches from sidewalk to threshold.
1. Using the 1:12 guideline, compute the minimum horizontal run for a 30-inch rise, in inches and in feet. Compare it with the 24 feet available and state the actual slope ratio for a 24-foot run.
Read the worked answer
At a 1:12 slope, each inch of rise needs 12 inches of run. For a 30-inch rise: 30 × 12 = 360 inches of run, which is 360 ÷ 12 = 30 feet.
Only 24 feet (288 inches) is available, so the straight ramp is 6 feet short of the guideline. The slope actually produced by a 24-foot run is 30 ÷ 288 = 1/9.6, which is steeper than 1/12 because the same rise is packed into less run. Equivalently, 288 ÷ 30 = 9.6 inches of run per inch of rise, while the guideline requires 12. The ramp would not meet the 1:12 guideline in the available space.
2. The contractor's plan uses two 12-foot runs, each rising 15 inches. Does splitting the rise fix the slope problem? Explain using ratios rather than intuition.
Read the worked answer
For each segment, slope = 15 inches of rise ÷ 144 inches of run = 1/9.6. That is exactly the same ratio as the single 24-foot run (30/288 reduces to 1/9.6). Splitting both the rise and the run in half does not change their quotient, so each segment is still steeper than 1:12.
To meet 1:12, each 15-inch rise needs 15 × 12 = 180 inches = 15 feet of run. Two such segments need 30 feet of run, plus landing space, which is more than the 24 feet available. The switchback only helps if the total run is increased or the total rise is reduced; it does not create a gentler slope by itself.
3. For the 24-foot straight run, find the ramp surface length to the nearest inch using the Pythagorean theorem. Then compute the percent grade and compare it with the 1:12 guideline.
Read the worked answer
Treat the rise and run as the legs of a right triangle, with the ramp surface as the hypotenuse. Run = 24 feet = 288 inches; rise = 30 inches.
Surface = √(288² + 30²) = √(82,944 + 900) = √83,844 ≈ 289.56 inches. To the nearest inch, that is 290 inches, or 24 feet 2 inches. For handrail ordered by the foot, 290 inches is about 24.17 feet, so 25 feet should be ordered to allow for waste.
Percent grade = (rise ÷ run) × 100 = (30 ÷ 288) × 100 ≈ 10.42%. The 1:12 guideline is (1 ÷ 12) × 100 ≈ 8.33%, so the 24-foot ramp is about 2.1 percentage points steeper than allowed. The Pythagorean length is only slightly longer than the run because the rise is small compared with the run.