Up and down from one window
A tower stands across the lane. Its top is above your window. Its foot is below.
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- A window, and a tower across the lane.
- A horizontal line leaves the window.
- Up to the top: the angle of elevation.
- Down to the foot: the angle of depression.
One horizontal, two triangles
Draw the horizontal straight out from the window. Above it, the sight line lifts to the top. That angle is an elevation. Below it, the sight line drops to the foot. That angle is a depression.
Both triangles share the same walk, from the window to the tower.
In the lab, drag the window along a 60 m tower. One piece grows as the other shrinks. The tower stays 60 m.
A piece is the tower's part above or below the horizontal.
Add the two pieces
Take the walk as 36 m. An elevation of 30° gives a piece above the window. That piece is 36 × tan 30° = 36 ÷ √3 = 12√3 m.
A depression of 45° gives a piece below the window. That piece is 36 × tan 45° = 36 m.
Rule: Tower = piece above + piece below.
The tower is 12√3 + 36 m tall. Check each triangle. Below: 36² + 36² = 2592 = (36√2)². Above: 36² + (12√3)² = 1728 = (24√3)².
| Direction | Reads | Piece |
|---|---|---|
| Up | 30° | 12√3 m |
| Down | 45° | 36 m |
| Tower | — | 12√3 + 36 m |