Angle of Elevation and Depression: Solving Word Problems

Applications of right triangles · 6 minute read

Height and distance problems turn a real situation into a right triangle. The key is reading the angle correctly.

Definitions

Angle of elevation: the angle between the horizontal and a line of sight up to an object.
Angle of depression: the angle between the horizontal and a line of sight down to an object.

Both angles are measured from a horizontal line, not from the vertical. The angle of depression from the top equals the angle of elevation from the bottom, because they are alternate interior angles.

Example: height of a tree

From a point 30 m from the base of a tree, the angle of elevation to the top is 40°. How tall is the tree?

The distance 30 m is adjacent to the angle and the height is opposite, so use tangent.

tan 40° = h / 30
h = 30 · tan 40° ≈ 25.2 m

If the observer's eyes are 1.6 m above the ground, add that: about 26.8 m.

Example: angle of depression

From the top of a 50 m cliff, the angle of depression to a boat is 25°. How far is the boat from the base?

tan 25° = 50 / d
d = 50 / tan 25° ≈ 107.2 m

Checklist

Draw the picture and mark the horizontal line.
Mark the angle from the horizontal.
Label opposite, adjacent and hypotenuse for that angle.
Pick sine, cosine or tangent, solve, and add any eye or instrument height.

Practice it

Practice application problems