How Large is the Earth? This is how Eratosthenes calculated the size of the Earth in 200BC.
Eratosthenes wasn't able to directly measure the Earth, but he did realise that can't be measured directly can be measured in a roundabout and inventive way.
On one particular day in Syene (modern-day Aswan in Egypt), Eratosthenes knew that the sun would be directly overhead (called the Summer Solstice). He travelled 5,000 stadia to Alexandria. A stadia is an ancient Greek unit of length, based on the length of a typical sports stadium. 5,000 stadia are approximately 8-900 kilometres. There he set a stick on the ground. At midday when the Sun was at its highest he measured the angle of the shadow that the sun formed on the ground, which he measured it to be 7.2°.
In this diagram, pretend Syene is at the top of the Earth in this diagram, and Alexandria is in the UK. Eratosthenes knew the angle between the Sun and Alexandria, and since both angles are the same, he could extend this to the centre of the Earth.
Then, from the parallel rays of light from the sun. Since Eratosthenes knew the angle between the sun and Alexandria, and both angles are the same, he could extend this to the centre of the Earth.
Since the two angles are the same, they must be representative of the distance between the two cities. Since there are 360° in a circle, he calculated that 5,000 stadia is approximately one-fiftieth of the circumference of the Earth.
Using this he went on to suggest that the circumference of the Earth is 250,000 stadia or 40,000 - 46,000km. He then went further and using the works of Pythagoras he could calculate the Earth's radius.
Equation 1 - Circle, circumference of
Equation 6 - Radius of a Circle
Equation 7 - Radius of the Earth solved
The value of the Radius of the Earth that Eratosthenes calculated was around 6,366km to 7,321km. We can only say that the value is around this because there are no accurate measurements for how long a Stadia is. Additionally, Eratosthenes assumed that Syene and Alexandria are on the same line of longitude, however, they are not.
Astronomers have since measured the Earth accurately using a satellite-based laser measuring equipment and recorded a figure of 40,008 km for the circumference of the Earth and the Earth's radius, 6,367km! Considering the difference in equipment between Eratosthenes and modern day scientists, Eratosthenes did a fine job!
Eratosthenes curiosity went far beyond the Earth. He also considered some of the astronomical questions of his day, including the distance to the Sun and the Moon.