Method 3 -Find the circumference from the central angle

Eratosthenes may have imagined that the cross-section of the earth was like an grapefruit sliced in half. You could see the sections of the orange. Also, if you walked along the edge of the orange your path would be circular.

If you could determine the angle that the section makes at the center and length of the edge, you could figure out the circumference.

Here's an example. Angle ABC is 60 degrees. Since there are 360 degrees in a circle, then there are 6 slices in this "pie". That means that 6 arcs (AC) would fit around the globe and if you knew that distance you would be in business!
At first glance finding this angle seems just as impossible as finding the diameter; but, as it turns out, this was Eratosthenes' brilliant insight. He discovered it by studying shadows! 
Carl Sagan tells the story of Eratosthenes in his book Cosmos (read the excerpt.)
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