In about 276BC, it was known that at noon, in the city of Syene, the sun would shine directly over a water well, meaning its rays are perpendicular to the water. At the same time in Alexandria, the sun would cast shadows on objects, meaning the angle of the light rays were different than 90 degrees to the ground.
Eratosthenes, a greek scholar, knowing about this, asked a man to walk the distance between the two cities and record it. It was about 787Km.
He then measured the angle that the sun rays made with objects on the ground and found it to be approximately 7.2 degrees.
Now the way to do it is:
We zoom into the triangle that Syene, Alexandria and the centre of the earth make:
Now we use a bit of maths: the radius of the Earth is called the hypothenuse of the triangle, and it is the distance we want. Let R be the radius of the earth, then 787/R = Sine(7.2)
Hence R = 787/Sine(7.2) and Sine(7.2) is about 0.125 (you can use a calculator for this). Hence R = 787/0.125 = 6296Km. The accepted distance nowadays is about 6 378.1 Km. How close he was and what a genius he was.
Reflections on the Art of Existing 