Angles and circles
Later traditions connect Thales with results about equal angles and triangles in circles.
02 / ANTIQUITY TO THE RENAISSANCE
Thales of Miletus
You can measure a tall object without climbing it. A short stick, its shadow, and a little geometry are enough.
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01 / SIMILAR TRIANGLES
Measure a tower without climbing it
Plant a 1-metre stick beside a tower. You can reach the ends of their shadows, even when you cannot reach the top of the tower.
Measure both shadows. Compare them. The comparison gives you the height.
Move the sun. Watch the shadows.
Drag the sun in the picture, or use the slider below it.
← Low sun · long shadowsHigh sun · short shadows →
Tower’s shadow
÷Stick’s shadow
=Times as long
Why the two triangles always match
Same sunlightThe sun is so far away that its rays arrive parallel. Both objects get the same angle.
Same right angleTower and stick both stand straight up from level ground.
Same shape, different sizeTwo matching angles force the same shape. Every length scales by the same amount.
THE IDEAIf the tower’s shadow is 6 stick-shadows long, the tower is 6 sticks tall. The sun changes both shadows, but never their ratio.
The person behind the ideas
- Lived in Miletus, a Greek city on the coast of what is now Türkiye.
- Later writers credited him with several geometric results.
- No writing by Thales survives; stories about his discoveries need care.
More of the mathematics
Geometry as reasoning
His historical importance includes an early Greek tradition of explaining geometric relationships. The exact extent of his proofs is uncertain.