How a shadow measured the Earth

2026-06-05

Around 240 BC, the librarian Eratosthenes worked out how big the Earth is without leaving Egypt. He had no satellites, no aircraft, no ship that had circled the globe. He had two cities, two vertical sticks, and a shadow. And he got the answer to within a few percent of what we measure today.

The sun is so far away that its rays reach us effectively parallel - the yellow lines all point the same way. At Syene (modern Aswan), on the longest day of the year, the noon sun stood directly overhead: a stick cast no shadow, and a ray dropped straight down a well to its own reflection. At Alexandria, far to the north on the same day and hour, an identical stick did cast a shadow. The sun was not quite overhead there. Drag Alexandria around the rim and watch its shadow grow.

Why the shadow knows the angle

Both sticks stand straight up - which means each one points directly away from the centre of the Earth, along its own radius. At Syene the stick lines up with the incoming ray, so there is no shadow. At Alexandria the stick points along a different radius, tilted from the first by exactly the angle of the arc between the two cities. The vertical sun ray therefore meets the Alexandria stick at that same tilt, and the shadow opens up by it.

That is the whole idea, and the diagram makes it inevitable: the shadow angle at the top and the central angle at the Earth's core are the same number. (Two radii cut by parallel rays - the alternate angles a transversal makes across parallel lines are equal.) The two blue arcs always read the same value, however far you drag.

From an angle to a planet

Once you know that the cities are separated by some angle θ out of a full 360° turn around the Earth, the distance between them is that same fraction of the whole way round:

θ / 360° = distance / circumference

Rearrange, and the circumference of the Earth drops out of two things you can actually get on the ground - the shadow angle, and the distance you'd walk between the cities:

circumference = (360° / θ) × distance

The angle you read off a shadow for free. The distance you pay for by surveying - Eratosthenes used the days a camel caravan took to travel between the cities. Press the Eratosthenes: 7.2° button: that shallow wedge, one-fiftieth of a full turn, was his actual measurement. A stick's shadow leaning by seven degrees, times fifty, is the size of the world.

The same planet, every pair

Notice what the circumference readout does as you drag: nothing. A wider gap between the cities means a bigger shadow angle, but it also means a longer distance between them, and the two rise in perfect lockstep. Any pair of cities, any angle, hands you the same Earth. The method isn't a lucky coincidence of Syene and Alexandria - it is the geometry of a sphere, readable from any two points on it and a single patient afternoon with a stick.

The deepest part is the assumption hiding underneath: that the sun's rays are parallel, which only follows if the sun is enormously far away, and that a stick can cast different shadows in different places at the same instant, which only follows if the ground is curved. Eratosthenes didn't just measure the Earth. He proved it was round, and proved the sun was distant, in the same breath.