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Trigonometry · Quick Reference
Heights & Distances Cheat Sheet
In one line: heights and distances problems use trigonometry to measure things that are hard or impossible to measure directly — like the height of a building or the distance across a river.
Key Ideas
1Angle of Elevation. The angle measured upward from the horizontal to a point above (like looking up at the top of a tower).
2Angle of Depression. The angle measured downward from the horizontal to a point below (like looking down from a cliff at a boat).
3Setting 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.
4Using Tangent for Height Problems. tan(angle of elevation) = height / horizontal distance, since height is 'opposite' and distance is 'adjacent' to the angle.
5Two-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.
Worked Examples
A person stands 50m from a tower and measures the angle of elevation to the top as 30°. Find the tower's height.
Height is approximately 28.87m
From the top of a 100m cliff, the angle of depression to a boat is 20°. Find the horizontal distance to the boat.
Distance is approximately 274.7m
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.
Distance is approximately 5.77m
Formulas
tan(angle) = opposite/adjacent = height/distance
Practice Yourself
A person 40m from a tower measures a 45° angle of elevation. Find the tower's height.
40m (since tan(45°)=1)
From a 50m cliff, the angle of depression to a boat is 30°. Find the distance to the boat.
50/tan(30°)≈86.6m
What angle is measured looking UP at an object?
Angle of elevation