We measure Earth's radius by determining its circumference and then using that to calculate the radius. This is based on the understanding that the Earth is a sphere, or very close to it. A classic example of this method is the work done by Eratosthenes.
Calculating Earth's Circumference: The Method of Eratosthenes
Eratosthenes' method relies on measuring the distance between two points on Earth and the angle between them at the Earth's center. Here’s how it works:
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Choose two locations: Eratosthenes used Alexandria and Syene (modern Aswan). Syene was known to be located approximately on the Tropic of Cancer, where the sun’s rays hit vertically at noon during the summer solstice. Alexandria, being further north, had a sun angle that was not directly overhead at noon during the summer solstice.
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Measure the distance between the locations: The distance between Alexandria and Syene was measured to be approximately 5,000 stadia (an ancient Greek unit of length).
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Measure the angle of the sun:
- In Syene, at noon on the summer solstice, the sun was directly overhead. This meant the angle to the sun was 90 degrees.
- In Alexandria, at the same time, the sun cast a shadow, indicating that the sun’s rays were not directly overhead. The angle of the shadow was measured to be about 7.2 degrees (or 1/50th of a circle).
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Relate the angle to the circumference: Since the angle of the shadow in Alexandria (7.2 degrees) represents a small slice of the Earth's circumference and is a fraction of the full 360 degrees of a circle, we can calculate the circumference of the Earth:
Measurement Value Angle 7.2 degrees Fraction of circle 1/50th Distance between cities 5,000 stadia -
Calculate the full circumference: Eratosthenes reasoned that if 7.2 degrees of the Earth’s circle corresponds to 5,000 stadia, then the full 360 degrees (or whole circle) would be:
- (5,000 stadia) * (360 degrees / 7.2 degrees) = 250,000 stadia
This calculation gave him an estimate of Earth’s circumference.
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Calculate the Radius: From the calculated circumference we can get the radius by using the formula: Circumference = 2πr, where r is the radius.
- r = Circumference / 2π
- r = (250,000 stadia) / 2π
- r ≈ 39,788 stadia (using a modern conversion of one stadium to approximately 0.157 kilometers, this is about 6,247 km)
- The modern value for the radius is actually closer to 6,371 kilometers, showing how accurate the ancient Greeks were!
Key Points
- This method relies on the fact that the angle between the locations corresponds to an arc along the Earth’s circumference.
- Eratosthenes’ method used simple tools to achieve a highly accurate result.
- Modern methods use satellite measurements and GPS to achieve a much greater accuracy in calculating the radius.
In summary, by carefully measuring the arc distance and central angle between two points on the Earth's surface, we can determine the Earth's circumference and, subsequently, its radius.
