How Did Eratosthenes Measure Earth’s Circumference with Shadows?

Eratosthenes estimated Earth’s circumference without travelling around the planet. His key observation was that vertical sticks cast different noon shadows in two cities on the same day. The difference was not because the Sun behaved differently over each city. It arose because the cities stood on a curved Earth, where ‘vertical’ points in slightly different directions.

In the traditional account, Eratosthenes learned that near noon around the summer solstice, sunlight reached the bottom of a deep well at Syene and a vertical stick there made essentially no shadow. At the same time, a vertical stick in Alexandria made a short shadow. The height of the stick and the shadow length could be used to calculate the Sun’s angle at Alexandria.

Because the Sun is enormously far away, its rays at the two cities can be treated as nearly parallel. Under that assumption, the shadow angle at Alexandria matches the central angle between the two cities. If that angle is about one fiftieth of a full circle, then fifty times the distance between the cities estimates the whole circumference. It is a measurement made from a geometric correspondence, not from a giant measuring tape.

The historical situation was less exact than the clean classroom diagram. Syene and Alexandria are not precisely on the same meridian, the well story is difficult to verify in detail, and the route distance had its own uncertainty. Modern recreations therefore specify location, time, solar altitude and distance more carefully. Those limitations do not undo the logic of the method; they explain why its numerical result is an estimate.

What remains remarkable is not a tale of ancient calculation matching every modern digit. It is the way a local shadow, a known distance and the geometry of a circle connect to the size of a whole planet. The same principle still lets students at separated locations use simultaneous measurements to detect Earth’s curvature and estimate its circumference.