How Large is the Earth?

Find out how Eratosthenes calculated the size of the Earth with a stick

By , 19th March 2008 in Astronomy

How large is the Earth, and how do we know that? It was the Greek mathematician and astronomer Eratosthenes who first calculated the size of the Earth, and he did it in 200 BC with a stick!

How Large is the Earth? This is how Eratosthenes calculated the size of the Earth in 200BC.

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°.

The from the parallel rays of light from the sun (pretend Scene is at the top of the Earth in this diagram, and Alexandria is in the UK). 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.

How Eratosthenes measured the circumference of the Earth
How Eratosthenes measured the circumference 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.

C=2pi r
Equation 1 - Circle, circumference of
r=C/{2pi}
Equation 6 - Radius of a Circle
r=40000/{2pi}
r=6366.2
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.

Astronomers have since measured the Earth accurately using 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!

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