How steep is too steep?
Tonight's idea
Builders, road engineers and roofers all measure steepness, and each trade says it differently. A common rule for wheelchair ramps is "no steeper than 1 in 12": 1 unit up for every 12 along the level. A road sign says 10%: 10 up for every 100 along. A roofer says "6 in 12". A surveyor gives an angle.
All of them describe one right triangle. The height gained is the rise and the level distance is the run. Rise ÷ run is the slope, and it is also the tangent of the angle \(\theta\) the slope makes with the ground: opposite over adjacent.
\(\tan\theta = \dfrac{\text{rise}}{\text{run}}\). As a percent, multiply by 100. For the angle, use the \(\tan^{-1}\) key, which answers "which angle has this tangent?"
The legal ramp is \(\tan^{-1}(1/12) \approx 4.8°\). The 10% road is \(\tan^{-1}(0.1) \approx 5.7°\), a little steeper than the ramp. Slopes that feel steep are small angles.
Watch one: a roof, both ways
A roof rises 6 for every 12 along. Then \(\tan\theta = 6/12 = 0.5\), which is a 50% slope, and \(\theta = \tan^{-1}(0.5) \approx 26.6°\).
Now backwards. Another roof is at 30°. Then rise ÷ run \(= \tan 30° \approx 0.577\), a slope of 57.7%. For a run of 12 the rise is \(12 \times 0.577 \approx 6.9\), so a roofer would call it about "7 in 12".
Drag the angle and watch opposite ÷ adjacent change: mathdetours.com/demos/trigtriangle.html.
Your turn
- Give each slope as a percent and as an angle to 1 decimal place: (a) a path that rises 1 in 20; (b) a road signed 15%; (c) a roof that rises 4 in 12.
- Each step of a staircase rises 18 cm and goes 27 cm forward. (a) What angle does the staircase make with the floor? (b) What is its slope as a percent? (c) How many times steeper than a 1 in 12 ramp is that, comparing slopes?
- A ramp must be no steeper than 1 in 12. One ramp is 5 m long, measured along its sloping surface, and rises 40 cm. (a) Use Pythagoras to find its run. (b) Is it legal?
- (a) Find the angles of a 10% slope and a 20% slope. Is the second angle about double the first? (b) Do the same for 100% and 200%. (c) Is a 100% slope vertical? What percent would a vertical wall be?
Stretch: which distance?
Some people work out a percent slope as rise ÷ distance travelled along the slope. That ratio is \(\sin\theta\), not \(\tan\theta\). Find both percents for a 5° slope and for a 30° slope. Why do the two methods agree for gentle slopes and disagree for steep ones?
Answers
- (a) \(1/20 = 5\%\); \(\tan^{-1}(0.05) \approx 2.9°\). (b) 15%; \(\tan^{-1}(0.15) \approx 8.5°\). (c) \(4/12 \approx 33.3\%\); \(\tan^{-1}(1/3) \approx 18.4°\).
- (a) \(\tan\theta = 18/27 = 2/3\), so \(\theta \approx 33.7°\). (b) About 66.7%. (c) \(\tfrac23 \div \tfrac1{12} = 8\) times as steep.
- (a) Run \(= \sqrt{5^2 - 0.4^2} = \sqrt{24.84} \approx 4.98\) m. (b) Slope \(= 0.4/4.98 \approx 0.080\), under \(1/12 \approx 0.083\), so it is legal, at an angle of about 4.6°. A quicker check: a 40 cm rise needs a run of at least \(12 \times 0.4 = 4.8\) m, and this ramp has 4.98 m.
- (a) 5.7° and 11.3°: close to double. (b) 45° and 63.4°: nowhere near double. Angle and slope keep step only when both are small. (c) No. 100% means rise equals run, which is 45°. A vertical wall has no run at all, so its percent is not a number: as the angle approaches 90° the percent grows without limit.
Stretch. At 5°: \(\tan 5° \approx 8.75\%\) and \(\sin 5° \approx 8.72\%\). At 30°: \(\tan 30° \approx 57.7\%\) and \(\sin 30° = 50\%\). The two ratios share the same rise and differ only in what they divide by: the run or the sloping length (the hypotenuse). On a gentle slope those two lengths are almost equal. On a steep one the hypotenuse is much longer than the run.