Measurement of Radius of the Earth

Around 225 BC, the Greek scholar-librarian Eratosthenes, working in Alexandria, measured the Earth's radius to a small error compared with modern measurements -- using only two vertical sticks, their shadows, and a known distance between two cities.

The method. Eratosthenes learned that at local noon on the summer solstice, the Sun's rays cast no shadow at all on a vertical pole in the city of Syene, meaning the Sun was directly overhead there. At the exact same moment, in Alexandria -- a known distance of about 500 miles away, along essentially the same north-south line -- an identical vertical pole cast a small shadow, with the Sun's rays making an angle of $7.2^\circ$ with the vertical. He correctly reasoned that this angle difference could only be caused by the curvature of the Earth between the two cities.

The calculation. An angle of $7.2^\circ$ equals $\tfrac{1}{8}$ radian. If $S$ is the arc length between the two cities and $R$ is the Earth's radius, then $S=R\theta$, so with $S=500$ miles and $\theta=1/8$ rad,

Converting using $1\ \text{mile}=1.609\ \text{km}$ gives $R\approx6436\ \text{km}$ -- astonishingly close to the modern accepted value of $6378\ \text{km}$, considering the method uses nothing more sophisticated than a stick, its shadow, and a protractor. …