The angle you lift your eyes
You look up at a tower. You cannot climb it. You can still measure it.
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- You stand on flat ground. A tower rises ahead.
- A horizontal line leaves your eye.
- Your sight line lifts to the top. That is the angle of elevation.
- Height, walk and sight line make the triangle. The walk is your level distance from the tower.
The angle of elevation
The horizontal is the level line from your eye. The line of sight runs from your eye to the top. The angle of elevation is the angle between them. The angle opens upward from the horizontal.
Treat your eye as on the ground. If an eye height is given, add that height to the result.
One triangle, three sides
The tower, the ground and the sight line make a right triangle. The right angle is at the foot of the tower.
The height faces the angle. It is the opposite side. The walk touches the angle. It is the adjacent side. The line of sight is the hypotenuse.
Tan of the angle = opposite ÷ adjacent = height ÷ walk. Multiply both sides by the walk.
Rule: Height = walk × tan of the angle of elevation.
Cos of the angle = walk ÷ line of sight. The line of sight is the longest side. Line of sight = walk ÷ cos of the angle.
| Angle of elevation | Height | Line of sight |
|---|---|---|
| 30° | 10√3 m | 20√3 m |
| 45° | 30 m | 30√2 m |
| 60° | 30√3 m | 60 m |
At 60°: 30² + (30√3)² = 900 + 2700 = 3600. And 60² = 3600. The Pythagoras check holds.