Heights and Distances

355. Angle of Depression · From tower top to sea level

tan(θ) = height / distance, so distance = height / tan(θ).

AB180 mθ = 29°θ = 29°d = 325md = 325mP
From a height, the angle of depression θ to a point on the ground gives tan θ = height / horizontal distance, so the distance = height / tan θ. The angle of depression equals the angle of elevation measured back up.

Stuck? Ask Guru

Selina ICSE: Heights and Distances

What this lesson covers

Try to break it

Drag P along the sea. The angle of depression from the top of the tower down to P always equals the angle of elevation from P up to the top (alternate angles cut by the vertical between the two horizontals). As P moves farther, both angles shrink; closer, both grow. tan(angle) = tower height / horizontal distance.

The proof, step by step

Prove that the distance to the boat equals the height divided by tan θ.

  • The horizontal line at the top of the tower and the sea level are parallel.
  • The line of sight AP acts as a transversal cutting these parallel lines.
  • Therefore, the angle of depression at A equals the angle of elevation at P (alternate interior angles).
  • In right triangle ABP, tan(θ) = AB / BP, so BP = AB / tan(θ).

Worked example

A guard observes an enemy boat from an observation tower at a height of 180 m above sea level, to be at an angle of depression of 29°. Calculate, to the nearest metre, the distance of the boat from the foot of the observation tower. (tan 29° ≈ 0.5543)

In right triangle ABP, tan(29°) = 180 / BP. Thus, BP = 180 / 0.5543 ≈ 324.7 m, which rounds to 325 m.

  • 325 m — correct
  • 280 m
  • 360 m
  • 410 m
Hold to talk

Subscription Status