Trigonometry · Ages 14–15
Gradient and inclination
Gradient = vertical ÷ horizontal = tan θ.
How to do it
Method
Gradient = vertical ÷ horizontal = tan θ
1 : 8: tan⁻¹(18) ≈ 7.1°
Another way
Legs 1 and n
1 + 82 = 65
In full
A slope's gradient = vertical distance ÷ horizontal distance, often written 1 : n or as a percentage; its inclination θ is the angle it makes with the horizontal, and tan θ = the gradient. Up 1 m in 8 m across: the gradient is 18 = 1 : 8 = 12.5%, the inclination tan⁻¹(18) ≈ 7.1°. The distance along the slope is the hypotenuse: 200 m up a 10° slope rises 200 × sin 10° ≈ 34.7 m and moves 200 × cos 10° ≈ 197.0 m across. Which slope is steeper: compare the gradients, or the inclinations — the same answer. Another way: see 1 : n as a right-angled triangle's two shorter sides 1 and n, the hypotenuse 1 + n² by Pythagoras.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A slope with a gradient of 1 : 3 is steeper than one inclined at 8°.
True or false?
Show the answer and working
Answer: True
A gradient of 1 : 3 is an inclination of tan⁻¹(13) ≈ 18.4°; compare it with 8°.
Example 2
A slope with a gradient of 1 : 17 runs 130 m across. Correct to 1 decimal place, how long is it along the slope, in m?
Show the answer and working
Answer: 130.2 m
The inclination θ = tan⁻¹(117); along the slope = 130 ÷ cos θ ≈ 130.2 m. Or by Pythagoras: the rise is 13017, the length 1302 + (13017)2.
Example 3
A road sign shows the road ahead has a gradient of 38%. To the nearest 0.1°, what is its inclination?
Show the answer and working
Answer: 20.8°
38% rises 38 m for every 100 m across: tan θ = 38100, θ ≈ 20.8°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A slope runs 5 m across and rises 4 m. What is its gradient, as a fraction in its lowest terms?
Answer in simplest form
Answer:
2.A slope has a gradient of 1 : 5. Going 15 m across, how much does it rise?
- A. 52 m
- B. 75 m
- C. 3 m
- D. 13 m
Choose the right answer
Answer:
3.A slope with a gradient of 1 : 9 is steeper than one inclined at 8°.
True or false?
Answer:
4.A slope has a gradient of 1 : 13. To the nearest 0.1°, what is its inclination?
Answer:
5.A slope with a gradient of 1 : 11 runs 280 m across. Correct to 1 decimal place, how long is it along the slope, in m?
Answer:
All the answers
- 1. 45
- 2. C. 3 m
- 3. False
- 4. 4.4°
- 5. 281.2 m
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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