How Eratosthenes measured the circumference of the Earth

In 240 B.C., the Greek astronomer Eratosthenes made the first measurement of the size of Earth. By performing the right calculations by observing the angular shadows of two cities in Summer Solstice, using the rules of geometry and the distance between the cities, Eratosthenes was able to make an almost accurate calculation of the circumference of Earth.

Eratosthenes lived in the city of Alexandria, near the Nile River by the Mediterranean coast, in northern Egypt. He knew that on a certain day each year, the Summer Solstice, in the town of Syene in southern Egypt, there was no shadow at the bottom of a well. He realized that this meant the Sun was directly overhead in Syene at noon on that day every year.

Eratosthenes knew that the Sun was never directly overhead, even on the Summer Solstice, in his home city of Alexandria, which is further north than Syene. He realized that he could determine how far away from directly overhead the Sun was in Alexandria by measuring the angle created by a shadow from a vertical object. He measured the length of the shadow of a tall tower in Alexandria, and used simple geometry to calculate the angle between the shadow and the vertical tower. This angle was about 7.2°.

Then, Eratosthenes used a bit more geometry to reason that the shadow's angle would be the same as the angle between Alexandria and Syene as measured from the Earth's center. Conveniently, 7.2° is 1/50th of a full circle. Eratosthenes understood that if he could determine the distance between Alexandria and Syene, he would merely have to multiply that distance by 50 to find the circumference of Earth!

He had the distance between the two cities measured. His records show that the distance was found to be 5,000 stadia. The stadion was a common distance unit of the time. Unfortunately, there was not a universal, standard length for the stadion. So we don't know exactly which version of the stadion Eratosthenes used, and therefore are not exactly sure how accurate his solution was. He may have been quite correct. Or, if it was actually a different stadion that he used, he may have been off by about 16%. The actual polar circumference of Earth is just a bit over 40 thousand km.