01 Key Concepts
Angle of Elevation
The angle measured upward from the horizontal to a point above (like looking up at the top of a tower).
Angle of Depression
The angle measured downward from the horizontal to a point below (like looking down from a cliff at a boat).
Setting Up the Right Triangle
Identify the horizontal distance, the vertical height, and the angle involved, then choose the correct trig ratio (usually tangent) to relate them.
Using Tangent for Height Problems
tan(angle of elevation) = height / horizontal distance, since height is 'opposite' and distance is 'adjacent' to the angle.
Two-Observer or Two-Angle Problems
Some problems give two angles from two different points, requiring a system of equations to solve for an unknown height or distance.
02 Key Formulas
- tan(angle) = opposite/adjacent = height/distance
03 Solved Examples
Example 1 A person stands 50m from a tower and measures the angle of elevation to the top as 30°. Find the tower's height.
- tan(30°) = height/50.
- tan(30°) is approximately 0.577.
- height = 50 * 0.577.
Answer: Height is approximately 28.87m
Example 2 From the top of a 100m cliff, the angle of depression to a boat is 20°. Find the horizontal distance to the boat.
- The angle of depression equals the angle of elevation from the boat's perspective (alternate angles), so tan(20°) = 100/distance.
- tan(20°) is approximately 0.364.
- distance = 100/0.364.
Answer: Distance is approximately 274.7m
Example 3 A ladder leans against a wall, making a 60° angle with the ground, and reaches 10m up the wall. Find the horizontal distance from the wall's base to the ladder's foot.
- The height (10m) is opposite the 60° angle; the distance is adjacent.
- tan(60°) = 10/distance.
- tan(60°) is approximately 1.732.
- distance = 10/1.732.
Answer: Distance is approximately 5.77m
04 Practice Questions
1A person 40m from a tower measures a 45° angle of elevation. Find the tower's height.
40m (since tan(45°)=1)
2From a 50m cliff, the angle of depression to a boat is 30°. Find the distance to the boat.
50/tan(30°)≈86.6m
3What angle is measured looking UP at an object?
Angle of elevation
4What angle is measured looking DOWN at an object?
Angle of depression
5Which trig ratio is most commonly used in height/distance problems?
Tangent
📄 Heights & Distances — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.