Eratosthenes' measurement of the circumference of the Earth

It is a popular myth that Christopher Columbus set out on his voyage in order to prove that the Earth is round. In fact, this was known long before Columbus. The first accurate measurement of the circumference of the Earth was made by Eratosthenes of Cyrene, circa 240 BC. It is remarkable that this calculation was performed using only high school geometry.


Eratosthenes knew that on the summer solstice at local noon, the sun would be directly overhead in the city of Syene. This is because Syene lies on the Tropic of Cancer. He also knew that at local noon in Alexandria on the summer solstice, the sun was 7 degrees south of the zenith (the highest point in the sky). He used this information, as well as the distance between Syene and Alexandria, to calculate the circumference of the Earth.

In the picture shown above, the sun's rays are nearly parallel lines, because the sun is so far away relative to the size of the Earth. The parallel lines are cut by the transversal AD. Therefore, the sun's angle at D is equal to the central angle shown in the picture. (This angle has been exaggerated for clarity.)

The circumference of the Earth is calculated by solving the following proportion.

(distance)/(circumference) = (central angle)/(360°).

I have created an interactive applet to help in exploring this topic. I borrowed the idea for this applet from Ihor Charischak.