The angle of elevation is the angle measured upward from the horizontal to a line of sight; the angle of depression is the angle measured downward from the horizontal. Both create a right-angled triangle with a known horizontal or vertical distance, so SOHCAHTOA or the tangent ratio solves most questions in one step.
What is the angle of elevation?
If you stand on level ground and look up at the top of a building or a bird in the sky, the angle your line of sight makes with the horizontal is the angle of elevation. It is always measured from the horizontal upward.
Picture a right-angled triangle lying on its side:
- The horizontal side is the adjacent side (the distance from you to the base of the object).
- The vertical side is the opposite side (the height of the object above your eye level).
- The angle of elevation is at your position, in the bottom left corner.
What is the angle of depression?
If you look down from a height — from a cliff to a boat, or from a window to a car below — the angle your line of sight makes with the horizontal is the angle of depression. It is always measured from the horizontal downward.
Key fact: the angle of elevation from A to B equals the angle of depression from B to A. These are alternate angles (parallel horizontal lines are cut by the line of sight). This relationship often lets you redraw the problem from a more convenient viewpoint.
How do you set up the right-angled triangle?
Step 1 — Read the question carefully and identify: the angle, the side you know, and the side you want.
Step 2 — Draw a right-angled triangle with the horizontal as the base.
Step 3 — Label the sides relative to the angle: opposite (vertical), adjacent (horizontal), hypotenuse (line of sight).
Step 4 — Choose the correct SOHCAHTOA ratio: tan = opp/adj is most common when you have the horizontal distance and vertical height.
Step 5 — Solve for the unknown.
How do you find a height using an angle of elevation?
Worked example 1: find the height of a tower
A student stands 50 m from the base of a tower. The angle of elevation to the top of the tower is 32°. Find the height of the tower.
Label: angle = 32°, adjacent = 50 m, opposite = height h (unknown).
Use tan: tan 32° = h ÷ 50.
h = 50 × tan 32° = 50 × 0.6249… ≈ 31.2 m (3 s.f.)
Worked example 2: find the distance from a building
The angle of elevation from point P to the top of a 40 m building is 55°. How far is P from the base of the building?
tan 55° = 40 ÷ d
d = 40 ÷ tan 55° = 40 ÷ 1.4281… ≈ 28.0 m (3 s.f.)
How do you find a distance using an angle of depression?
Worked example 3: angle of depression from a cliff
A lighthouse keeper at the top of a 60 m cliff sees a boat at sea. The angle of depression to the boat is 18°. How far is the boat from the base of the cliff?
The angle of depression from the top of the cliff is 18°. The alternate angle of elevation from the boat to the top of the cliff is also 18°.
Draw the right triangle with: angle = 18°, opposite = 60 m (height of cliff), adjacent = d (horizontal distance).
tan 18° = 60 ÷ d
d = 60 ÷ tan 18° = 60 ÷ 0.3249… ≈ 184.6 m (4 s.f.)
Answer: approximately 185 m from the base of the cliff.
What if the question involves two angles?
Worked example 4: two angles of elevation to the same point
From point A, the angle of elevation to the top of a tree is 40°. From point B, 20 m closer to the tree, the angle of elevation is 58°. Find the height of the tree.
Let the height = h and the distance from B to the tree base = d.
From B: tan 58° = h ÷ d → h = d tan 58°
From A: tan 40° = h ÷ (d + 20) → h = (d + 20) tan 40°
Set equal: d tan 58° = (d + 20) tan 40°
1.6003d = (d + 20) × 0.8391
1.6003d = 0.8391d + 16.782
0.7612d = 16.782
d = 22.04 m
h = 22.04 × tan 58° ≈ 22.04 × 1.6003 ≈ 35.3 m (3 s.f.)
Summary of key points
| Angle type | Measured from | Direction | Common trig ratio |
|---|---|---|---|
| Elevation | Horizontal | Upward | tan (opposite/adjacent) |
| Depression | Horizontal | Downward | tan (opposite/adjacent) |
| Relationship | Elevation from A = depression from B | Alternate angles | — |
Frequently asked questions
Why is tan the most common ratio in these problems?
Most elevation and depression questions give a horizontal distance (adjacent) and ask for a vertical height (opposite), or vice versa. tan = opposite ÷ adjacent connects those two sides directly without needing the hypotenuse. Use sine or cosine when the line-of-sight distance (hypotenuse) is involved instead.
Does eye height matter in these problems?
In GCSE questions, you can usually assume the angle is measured from ground level or the stated height — read the question carefully. If the observer's eye height is given (e.g. "from a window 5 m above the ground"), include that in your diagram. If not mentioned, assume the measurement starts at ground level.
What if the angle is measured from the vertical rather than the horizontal?
This is unusual in GCSE problems. If an angle is given from the vertical, the complementary angle from the horizontal = 90° minus that angle. Redraw with the horizontal angle and proceed as normal.
How do I check my answer is reasonable?
Compare the height or distance to the given information. A 60 m cliff with an 18° depression angle should give a large horizontal distance (the boat is far away) — 185 m makes sense. If your answer gives a building taller than the Grand Canyon, recheck whether you used tan, sin, or cos correctly.
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