PhysicsLAB: Eratosthenes' Measure of the Earth's Circumference

The purpose of this exercise is to examine the geometry involved in Eratosthenes’ (276 BC - 195 BC) indirect measure of the circumference of the Earth. You will need both a protractor and a ruler to complete this activity.

Refer to the following information for the next three questions.

Part I

Measure the angle

on your diagram.

Measure distance d on your diagram.

Based on alternate interior angles, you now know the central angle for your arc length d in Part I. Using this value, calculate the circumference of the circle.

Refer to the following information for the next five questions.

Part II

Measure angle

from the diagram.

Measure angles

from the diagram.

Measure distance d from the diagram.

Use alternate interior angles, in conjunction with these two additional facts:

supplementary angles add to 180º
the sum of the angles in a triangle adding up to 180º

to determine the central angle for the arc length d in Part II.

Calculate the circumference of the circle from this second set of information.

Refer to the following information for the next question.

Conclusion #1: How do your two values compare?

Calculate the % difference between your two values for the circumference of the circle.

Refer to the following information for the next two questions.

Conclusion #2: Eratosthenes' values

image courtesy of the National Oceanic and Atmospheric Administration

It is believed that Eratosthenes "used" the following facts in his calculations:

that on the summer solstice, longest day of the year, the midday sun would shine directly down into the central well at Syrene, which laid on the Tropic of Cancer
at the same time, the sun would not be directly overhead at Alexandria, but would cast a shadow with the vertical equal to 7º12'
Alexandria lay directly north of Syrene, or along the same longitudinal line
the distance from Alexandria and Syrene was 5000 stadia or 925 km (today scientists believe 1 Hellenic stadion = 185 m)

From Part II, if you use a central angle of 7º12' and d of 925 km, calculate Eratosthenes' value for the Earth's circumference.

Today we know that the circumference of the Earth at the equator is 4.008 x 10¹ Mm. What was the percent error for Eratosthenes' value of the Earth's circumference?