DWG NO. 08.5 — Lesson 5 of 5

Angles of Elevation and Depression

Unit 8: Right Triangles and Trigonometry · ~15–30 min

Objective

Set up and solve real-world problems — heights, distances, sightlines — using the angle of elevation or depression and right-triangle trig.

Two names for the same idea

Both angles are measured from a horizontal line at the observer's eye level, never from the ground or a wall.

The angle of elevation is the angle a horizontal line makes with the line of sight when looking up at something.
The angle of depression is the angle a horizontal line makes with the line of sight when looking down at something.

When two observers sight each other along the same line, their horizontal lines are parallel. That makes the angle of elevation from the lower point equal to the angle of depression from the higher point — they're alternate interior angles cut by the same transversal.

The angle of elevation is measured up from a horizontal sightline at the observer's eye.

Worked Example 1 · Height from an angle of elevation

ProblemFrom a point 50 ft from the base of a tower, the angle of elevation to the top is 35°. Find the tower's height h.

1.Height is opposite the angle, distance is adjacent, so use tangent: tan 35° = h/50.

2.tan 35° ≈ 0.7002, so h = 50 · 0.7002 ≈ 35.0.

Tower height ≈ 35.0 ft

Worked Example 2 · Distance from an angle of depression

ProblemA lighthouse keeper 80 ft above sea level sights a boat with an angle of depression of 12°. Find the boat's horizontal distance d from the base of the lighthouse.

1.The angle of depression equals the angle of elevation from the boat, so tan 12° = 80/d.

2.tan 12° ≈ 0.2126, so d = 80/0.2126 ≈ 376.3.

Distance ≈ 376.3 ft

Guided practice

1.From a point 100 ft from a building, the angle of elevation to its roof is 28°. Find the building's height.
Show answer
tan 28° = h/100, so h = 100 · tan 28° ≈ 100(0.5317) ≈ 53.2 ft.
2.A drone hovers 120 ft above the ground. The angle of depression to a parked car is 40°. Find the horizontal distance from the drone's ground point to the car.
Show answer
tan 40° = 120/d, so d = 120/tan 40° ≈ 120/0.8391 ≈ 143.0 ft.
3.A kite string is 200 ft long and makes a 50° angle of elevation with the ground. Find the kite's height above the ground.
Show answer
Here the string is the hypotenuse: sin 50° = h/200, so h = 200 · sin 50° ≈ 200(0.7660) ≈ 153.2 ft.
4.From the top of a 60 ft cliff, the angle of depression to a boat is 22°. Find the boat's distance from the base of the cliff.
Show answer
tan 22° = 60/d, so d = 60/tan 22° ≈ 60/0.4040 ≈ 148.5 ft.
5.Explain why the angle of elevation from a swimmer to a lifeguard tower and the angle of depression from the tower to the swimmer are always equal.
Show answer
Both horizontal sightlines — the swimmer's and the lifeguard's — are parallel, and the line of sight between them acts as a transversal. The elevation and depression angles are alternate interior angles on that transversal, which are always congruent.

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