An angle of elevation is measured upwards from a horizontal line at the observer’s eye to the line of sight. To use it, draw a right-angled triangle: the horizontal line is one leg, the vertical height is the other leg, and the line of sight is the hypotenuse.
The skill connects everyday description with the ratios in right-triangle trigonometry, and it appears in questions about towers, poles, cliffs and ramps.
How do you turn the words into a triangle?
Elevation and depression always start from a horizontal line. So the first job is to draw that line, then the line of sight, then join the two with a vertical.
With depression, the angle is below a horizontal line at the observer’s eye. The ground is a second horizontal line, parallel to the first, so the angle at the bottom of the triangle equals the angle of depression (alternate angles). Work in the triangle at the bottom, where the angle sits inside the right-angled triangle.
- Sketch the ground, the vertical object and the observer.
- Draw the horizontal at eye level.
- Draw the line of sight to the upper point (elevation) or lower point (depression).
- Mark the given angle between horizontal and line of sight.
- Label the sides opposite, adjacent and hypotenuse from that angle.
- Choose the ratio and solve, then adjust for eye height if needed.
Worked example
A student stands 40 m from the base of a tower on level ground. Her eye is 1.6 m above the ground.
The angle of elevation of the pole’s upper end?wer’s summit is 28°. Find the height of the tower.
Step 1, triangle: the horizontal at eye level is 40 m long and is adjacent to the 28° angle. The height above eye level is the opposite side.
Step 2, ratio: opposite and adjacent means tangent. tan 28° = h/40.
Step 3, solve: h = 40 × tan 28° = 40 × 0.5317 = 21.27 m.
Step 4, add the eye height: 21.27 + 1.6 = 22.87 m, so the tower is 22.9 m (3 s.f.).
Check: a 28° climb over 40 m should rise less than the 40 m run, since tan 28° is less than 1. 21.3 m is less than 40 m. The tower is taller than the eye height, as it should be.
Now a depression case. Standing on a cliff edge 45 m above the sea, you see a boat at an angle of depression of 12°. How far is the boat from the cliff base?
The angle at the boat, between the sea and the line of sight, is also 12°. The cliff is opposite that angle and the distance is adjacent, so tan 12° = 45/d and d = 45 ÷ tan 12° = 45 ÷ 0.2126 = 211.7 m, which is 212 m (3 s.f.).
The mistake to watch for
A common slip is forgetting the observer’s eye height, or measuring the angle from the vertical.
Mistaken working: the student writes the tower height as 40 × tan 28° = 21.3 m and stops.
That value is only the height above eye level. The tower is 21.3 + 1.6 = 22.9 m tall.
The correction is to ask, “Where does my triangle start?” The triangle starts at the eye, not the ground. Write a short note on the diagram: “h above eye” and ”+ eye height” as a reminder before you finish.
Check yourself
1. A flagpole reaches 21 m above eye level. The observer stands 50 m away horizontally. Find the angle of elevation to 1 decimal place.
Show answer
Opposite 21, adjacent 50, so tan θ = 21/50 = 0.42 and θ = tan⁻¹(0.42) = 22.8°.
2. A person on a bridge looks at a boat on the water that is 60 m away horizontally. The angle of depression is 18°. How high above the water is the person’s eye? Give your answer to 3 significant figures.
Show answer
The angle at the boat is 18° too. The adjacent side is 60 m, and the height above the water is opposite: 60 × tan 18° = 60 × 0.3249 = 19.5 m.
3. A 15 m pole stands on level ground. You stand 40 m from its base, with your eye at ground level for this question. What is the angle of elevation of the pole’s upper end??
Show answer
tan θ = 15/40 = 0.375, so θ = tan⁻¹(0.375) = 20.6°.
Where this leads next
Next, check a triangle solution against its geometry to catch impossible answers. Mixed questions sit in the right-triangle trigonometry practice set. The non-calculator working trainer is handy for keeping exact fractions such as 15/40 tidy before you use the calculator.
Diagrams are where many students first lose the thread, and a teacher watching you draw one can see it happen. That is part of what we do in online one-to-one Mathematics tuition.