Heights without ladders
No ladder reaches a tall tree. A pole and its shadow can still give its height.
- A pole 2 m tall casts a 3 m shadow, in the sun.
- The sun's ray touches the pole top and the shadow tip.
- A tree, same sun, casts a 12 m shadow.
- Height over shadow is 2/3 for both: the tree is 8 m.
Height from a shadow
At one moment the sun's rays are parallel. On flat ground, upright things and their shadows make right triangles. Upright means standing straight up. The sun angle is the angle between a ray and the ground. All these triangles have a right angle and the same sun angle. By AA they are similar.
At one moment the sun shines at one angle. On flat ground, upright things and their shadows make right triangles. Those triangles are similar. So height ÷ shadow is the same for both.
The tree
The pole is 2 m tall, with a 3 m shadow. Its height ÷ shadow is 2/3. The tree's shadow is 12 m.
| Object | Height | Shadow | Ratio |
|---|---|---|---|
| Pole | 2 m | 3 m | 2/3 |
| Tree | 8 m | 12 m | 2/3 |
Tree height ÷ 12 = 2/3. So tree height = 2/3 × 12. Now 2 × 12 = 24, and 24 ÷ 3 = 8. The tree is 8 m tall. Nobody measured it directly.
Pick the ratio, then find the height.
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