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

Step 1. The key observation. Around 225 BC, Eratosthenes learned that at local noon on the summer solstice, sunlight fell straight down a well in the city of Syene, casting no shadow at all on a vertical pole there -- meaning the Sun was exactly overhead.

Step 2. The comparison measurement. At the very same moment, in Alexandria -- a known distance of about 500 miles away, roughly along the same north-south line -- an identical vertical pole cast a small shadow, with the Sun's rays making a $7.2^\circ$ angle with the vertical.

Step 3. Why this angle equals the Earth's central angle. Because the Sun is so far away, its rays arrive at the Earth essentially parallel. The only thing that can make the Sun appear directly overhead at one city but at $7.2^\circ$ from vertical at another, at the exact same moment, is the curvature of the Earth between them -- and by simple geometry, this $7.2^\circ$ is exactly the angle that the arc between Syene and Alexandria subtends at the Earth's centre.

Step 4. The calculation. Converting $7.2^\circ$ to radians: $\theta=\dfrac{7.2\times\pi}{180}=\dfrac18$ rad. Using arc length $S=R\theta$ with the known city-to-city distance $S=500$ miles:

$R=\frac{S}{\theta}=\frac{500}{1/8}=4000\ \text{miles}.$ …

Explain the no-shadow-vs-shadow observation, why the angle equals Earth's cent …

• Using the angle in degrees directly in R = S/theta instead of first converting to radians. …