Approximating the Circumference of Earth Eratosthenes of...

Approximating the Circumference of Earth Eratosthenes of Cyrene $(276-195 \mathrm{BC})$ was a Greek scholar who lived and worked in Cyrene and Alexandria. One day, while visiting in Syene, he noticed that the Sun's rays shone directly down a well. On this date 1 year later, in Alexandria, which is 500 miles due north of Syene, he measured the angle of the Sun to be about 7.2 degrees. See the figure. Use this information to approximate the radius and circumference of Earth.

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Key Concepts

Central Angle and Arc Proportion

This concept involves understanding that a circle's circumference is proportional to its central angles. In Eratosthenes' approach, the small measured angle (7.2°) corresponds to a proportion of the full 360° circle. Recognizing this ratio allows one to relate the observed arc length between two points on Earth to the entire circumference, forming the basis of the calculation.

Proportional Reasoning

Proportional reasoning is used to compare the fraction of a circle represented by the measured angle to the whole circle. By using the proportion between the observed angle and the full 360°, one can scale an arc distance (such as the 500 miles between two locations) to estimate the complete circumference of the Earth.

Circular Geometry

Circular geometry provides the necessary framework for these calculations. It includes understanding the properties of circles, such as the relationship between circumference, radius, and central angles. The formula for circumference (2?r) stems from these properties and underpins the method used to estimate the size of the Earth based on angular measurements.

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