# Estimating the size of the Earth

##### Demonstration

An experiment carried out by Eratosthenes over 2000 years ago.

#### Apparatus and materials

Pole, 2, to 3 m long

#### Procedure

Two distant schools or colleges must co-operate. Preferably they should lie on a north-south line.

Set up a pole of known height at each site. Use a plumb line to ensure that the poles are vertical. The ground around them must be horizontal.

At noon on an agreed day, measure the length of the pole’s shadow at each site. Exchange details of the apparatus and measurements by phone, email or internet.Teaching notes

1 To estimate the size of the Earth, the distance between the two sites must be known. This can be found from maps, or by using the co-ordinates of each location (which can be obtained from, for example, Google Maps) and a tool to calculate the distance between co-ordinates on the Earth’s surface, such as the one available on the North Arizona University website or Ed Williams's Aviation page.

On a sketch of the Earth, continue the line of each pole down into the centre of the Earth as a radius. The angles produced at the poles, A and B, also give their latitude north of a place where the Sun’s rays are directly overhead (since a straight line intersecting two parallel lines has opposite angles equal).

This means that the radius from each position meets at an angle C where C = B – A.

The slanting lines of sunlight are parallel, assuming that the Sun is infinitely distant. Angles A and B can be calculated from the height of the pole and the length of the shadow.

tan angle = shadow length / pole length

or a scale drawing could be made.

The radius of the Earth is calculated from

sin C = distance between sites / radius of Earth

2 An experiment of this type was first done by Eratosthenes (~235 BC). He used observations at Alexandria and at Syene, 500 miles further south. He knew that at noon on midsummer day, sun beams falling on a deep well there were reflected back from the water surface. This showed the Sun was directly overhead at Syene on that day. On that same day, he found the shadow cast by a tall obelisk in Alexandra's made 7½° with the vertical. His calculation of the Earth's circumference was within 5% of today's accepted value.

This experiment was safety-checked in July 2007