Download Eratosthenes` Experiment

Eratosthenes' Experiment
How big a ruler would you need to measure the circumference of the Earth? Did you
know that you can do it with a yardstick? (And you won't have to travel all the way
around the world!)
The goal of this project is to estimate the circumference of the earth by setting up a
mathematical proportion from simple measurements. In this project, you will estimate
the circumference of the earth, using a method developed about 2,200 years ago, by
Eratosthenes, a Greek mathematician and the librarian of the great library at
Alexandria, in Egypt.
Eratosthenes knew that the Sun was never directly overhead, even on the Summer
Solstice, in his home city of Alexandria, which is further north than Syene. On that day
at noon, vertical objects cast no shadow, and the reflection of the sun could be seen
in the bottom of a well. He reasoned that the sun was far enough away from the earth
so that rays of light from the sun, for all practical purposes, are parallel to each other
when they reach the Earth
He realized that he could determine how far away from directly overhead the Sun was
in Alexandria by measuring the angle formed by a shadow from a vertical object. He
measured the length of the shadow of a tall tower in Alexandria, and used simple
geometry to calculate the angle between the shadow and the vertical tower.
Step 1: make a drawing of a section of the Earth in Alexandria to explain this
experiment.
The sun's rays are represented by the yellow
lines. In Syene, the sun's rays cast no
shadows. In Alexandria, a vertical tower does
cast a shadow. Eratosthenes' insight was that
the angle of the shadow in Alexandria (the
"sun angle", shown by the orange triangle) is
also equal to the angle of the wedge of the
Earth between the two cities (the "central
angle", shown by the other orange triangle).
Since there are 360 degrees in a circle, by
dividing the central angle into 360, he could
calculate how many similar sectors would be
needed to complete a circle.
Some informations to find out the
angle at the top of the tower.
Height of the tower: 23m
Length of the shadow: 3m
You must use your knowledge about
sin, cos and tan…
The distance between the cities was known from caravan travellings. It’s very
interesting is that the measurement of the distance between Alexandria and
Syene is based on the estimated average speed of a caravan of camels that
traveled this distance to be about 5,000 stadia. (The exact size of the stadion he
used is 158m.)
Step 2: find out the distance between the 2 towns.
Step 3: calculate the Earth radius and circumference.
What do you think about this experiment?