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When students ask, “When will we ever use this?,” it can be challenging to convince them that they will use math in everyday life. This is not because the math is unrealistic, but because even with a concrete example, the context may be far removed from their daily lives. Generating a feeling of investment in the details of how a house is structured or financial planning may not always compel student interest (although this is by no means a reason not to pursue those avenues!). In contrast, students are generally impressed by any situation that seems to defy logic. The following account of how the ancient Greeks estimated the circumference of the earth is one such case.

Ask students to imagine the world over two thousand years ago, before smart phones, the Internet, electricity, antibiotics, the radio, and cars — a world before the existence of almost every amenity we so casually use today. Now imagine finding, with reasonable accuracy, an estimation of the earth’s circumference. How was it possible? I am woefully unequipped to estimate any distance, within the United States or elsewhere, without the aid of Google Maps. It is hard to believe that roughly 2,200 years ago, the ancient Greek mathematician Eratosthenes made this estimation with use of basic geometry and algebra. Like other such estimations, the entire quest required a few observations and measurements, but the calculation itself relied on a basic geometric model.

Eratosthenes heard that in the city of Syene, Egypt (now Aswan), the sun was directly overhead (i.e. formed a 90˚ angle with the ground) at noon on the summer solstice. This was evident by the reflection of the sun at the bottom of a well at noon, or by observing that a pole would cast a negligible shadow. This was not the case in Eratosthenes’ location further north in Alexandria, Egypt. At noon on the summer solstice in Alexandria, a pole would cast a shadow. The angle between the perpendicular pole in the ground and the rays of the sun was approximately 7.2˚

This measurement was essential to estimating the circumference of the earth.

Visualize the sun’s parallel rays as they extend through the earth. Now imagine that the pole, which is perpendicular to the ground in Alexandria, acts as a transversal to those parallel rays. Then the alternate interior angles as determined by the transversal and parallel rays are equal in measure.

This implies that the angle between the two cities, as determined between the locations of the cities and the earth’s center, is 7.2˚. The measure of 7.2˚ is 1/50 of a circle, so it follows that the distance from Syene to Alexandria must be 1/50 of the circumference of the earth. The distance between Syene and Alexandria was believed to be roughly 5,000 stadia (a Greek unit of measurement). One stade is approximately 600 feet, making Eratosthenes’ estimation of the earth’s circumference, (50 x 5000 x 600) feet or approximately 28,400 miles.

Today’s measure of the earth’s circumference at the equator is roughly 24,900 miles, making the percent error approximately 14%. Eratosthenes’ calculation is quite good, especially considering that the calculation assumes a spherical earth, an accurate distance from Alexandria to Syene, and that the two cities lie along the same meridian.

The ancient Greeks are known for impressive feats such as this estimation of the earth’s circumference, or their reasonable estimation of the distance from the earth to the moon. Their powers of observation and ability to distill information in order to model the world around them exemplify math and problem solving and remind us not to wonder when we will apply math, but to look for opportunities to do so

This post is by Pia Mohsen, a former teacher, who is a Grade 10 writer for the Eureka Math curriculum.

© Great Minds 2016