At Syene (now Aswan), on the first day of summer and at midday, the Sun casts no shadow. At the same moment in Alexandria, a vertical stick would cast a shadow, with the length of the shadow increasing with increasing distance from Syene. The angle shown in the triangle is the same as the angle between Alexandria and Syene, as measured from the center of the Earth. Thus, the distance between the cities is the same fraction of the circumference )C) of the Earth as the angle is of 360 degrees.
1. pace off distance between the two cities (L)
2. measure angle from top of stick to end of shadow (7.2)
3. calculate: L / C = 7.2 / 360 or C= 50*L
Aristotle (at the end of Book II, Part 14 of On the Heavens) reports that "some mathematicians [who] calculate the size of the earth's circumference arrive at the figure 400,000 stades." We don't know who these "mathematicians" were or how they arrived at their number.
But Eratosthenes measured a circumference of 250,000 stades, and thus must have measured L = 5000 stades. We don't know the exact length the Greeks used for their unit of length, the stadium, but historians have argued for either 157.2 (or C = 39,300 km) or 166.7 m (41,675 km). The actual circumference of the Earth is 40,074 km.