Chapter 1: Problem 3

How did Eratosthenes calculate the Earth's circumference? (A)

Short Answer

Expert verified

Eratosthenes calculated the Earth's circumference by selecting two cities, Alexandria and Syene, and measuring the angle of the sun's rays at each location on the summer solstice. He found an angular difference of 7.2 degrees between the two cities, which corresponds to 1/50 of the Earth's total circumference. Knowing the distance between the cities was about 800 kilometers, he multiplied this value by 50 to estimate the Earth's circumference to be around 40,000 kilometers, a value close to the actual measurement.

Step by step solution

Choose two locations

Eratosthenes selected two cities: Alexandria and Syene (now known as Aswan) in Egypt. He knew the approximate distance between them, which was about 800 kilometers (500 miles).

Measure the angle of the sun's rays at each location

On the day of the summer solstice, when the sun was at its highest point in the sky, Eratosthenes measured the angle of the sun's rays in each city. In Syene, the sun was directly overhead, casting no shadow, so the angle was 0 degrees. In Alexandria, he measured the angle to be about 7.2 degrees.

Calculate the angular difference between the two cities

Eratosthenes found the difference between the angles of the sun's rays in both cities, which was 7.2 degrees.

Convert the angular difference to a fraction of the Earth's total circumference

Since there are 360 degrees in a circle, Eratosthenes determined the fraction of the Earth's total circumference by dividing the angular difference (7.2) by 360. This yielded 7.2 / 360 = 1/50.

Calculate the Earth's circumference

Eratosthenes knew the distance between Alexandria and Syene to be 800 kilometers (500 miles). He multiplied this distance by 50 (since 1/50 was the fraction of the Earth's total circumference represented by the angular difference) to estimate the Earth's circumference. So, 800 km x 50 = 40,000 km (25,000 miles). Based on this calculation, Eratosthenes found that the Earth's circumference was approximately 40,000 kilometers (25,000 miles), which is quite close to the actual value of about 40,074 kilometers (24,901 miles).

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How did Eratosthenes calculate the Earth's circumference? (A)

Short Answer

Step by step solution

Choose two locations

Measure the angle of the sun's rays at each location

Calculate the angular difference between the two cities

Convert the angular difference to a fraction of the Earth's total circumference

Calculate the Earth's circumference

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