Explain in detail the Eratosthenes method of finding the radius of Earth.

Explanation

During noon time of summer solstice the Suns rays cast no shadow in the city Syne which was located 500 miles away from Alexandria. At the same day and same time in Alexandria the Suns rays made 7.2 degree with local vertical as shown in the figure. This difference of 7.2 degree was due to the curvature of the earth.

The angle 7.2 degree is equivalent to `1/8` radian. So, `theta = 1/8 rad.
If S is the length of the arc between the cities of Syne and Alexandria, and if R is radius of Earth, then
`S = R theta = 500` miles
So radius of the Earth
`R = 500/(theta) miles, R = 500 ("miles")/(1/8)`
`R = 4000` miles
1 mile is equal to 1.609 km. So, he measured the radius of the Earth to be equal to R = 6436 km, which is amazingly close to the correct value of 6378 km.