Quick answer: For a right-triangle height problem, tan θ = vertical rise/horizontal distance. Draw the horizontal line through the observer's eye; add eye height if the required height is measured from the ground.
Worked example
From a point 20 m from a tower's base, its top has elevation 30°. Ignore eye height and find the tower's height.
- The opposite side is height h; the adjacent side is 20 m.
- tan 30° = h/20.
- h = 20/√3 = 20√3/3 m, approximately 11.55 m.
Check the result
A 30° angle is below 45°, so the height should be less than the horizontal distance of 20 m.
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Common mistake to avoid
Do not use the sloping line of sight as the adjacent side. If the observer's eye is 1.5 m high, add 1.5 m after finding vertical rise.
Try it yourself
1. Find height at distance 12 m and elevation 45°. 2. At distance 10 m and elevation 45°, eye height is 1.6 m; find total height.
Answers and reasoning
1. 12 tan45° = 12 m. 2. 10 + 1.6 = 11.6 m.
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