How Eratosthenes measured the radius of Earth

Eratosthenes — a Greek polymath, observed that a pillar located in Syene cast no shadow at noon on June 21st, whereas another pillar erected some 900 kilometers away at Alexandria cast a definite shadow.

How could it be, if the earth was flat in shape? The only solution was that the surface of earth is curved and the greater the curvature the bigger the difference in the lengths of shadows.

Although most Greek scientists at that time agreed that Earth was a sphere, none knew how big it was. By using simple trigonometry, Eratosthenes was able to calculate the radius of Earth to a great accuracy.

Our sun is so far away that its rays are parallel when they reach the earth. The height of pillar p, its shadow of length s and sun rays in red make a right triangle. Angle between sun rays and pillar at Alexandria is θ:

tan θ = s/p

Now, if we imagine the two pillars extending all the way down to the center of the earth, they would intersect also at angle θ — by equal alternate angles.

θ = L/R

where L is the distance between Syene and Alexandria and R is the radius of Earth.

Finally, Eratosthenes calculated the radius to be 38,000 stadia. When translated to kilometers this is approximately 7000 km, a fairly accurate estimate for his time — the exact radius of earth is 6,371 km.