English: Eratosthenes' measurement of the Earth's circumference.

Syene (${\displaystyle S}$) is located on the Tropic of Cancer, so that at summer solstice the sun appears at the zenith, directly overhead. In Alexandria (${\displaystyle A}$) the sun is ${\displaystyle \varphi }$ south of the zenith at the same time. So the circumference of earth can be calculated being ${\displaystyle {\frac {360^{\circ }}{\varphi }}}$ times the distance ${\displaystyle \delta }$ between ${\displaystyle A}$ and ${\displaystyle S}$.

Erastothenes measured the angle ${\displaystyle \varphi }$ to be 1/50 of a circle and his access to knowledge of the size of Egypt gave a north/south distance ${\displaystyle \delta }$ between Alexandria and Syene of 5000 stadia. His circumference of the Earth was therefore 250 000 stadia. Certain accepted values of the length of the stadia in use at the time give an error of less than 6% for the true value for the polar circumference.