In the third century B.C., Eratosthenes was able to mathematically calculate the earth’s diameter by comparing differences in the angle of the sun’s rays at two separate geographic points. He noticed that the difference in the angle of a shadow in his location at Syene, which is present-day Aswan in Egypt, and that of a shadow in Alexandria was about 7.2 degrees. Since he knew the distance between the locations, he was able to determine the circumference of the earth, and therefore the diameter and radius as well. You can do this, too, by using his method.
Record the distance between your location and your partner’s location. As an example, we will use Eratosthenes’ situation. The distance between Syene and Alexandria is 787 kilometers.
Drive one of the meter sticks into the ground in your location in a sunny spot. Tack one end of a piece of string to the top of the stick. Have your partner do the same in her location. Make sure both sticks are perpendicular to the earth and that the same length of stick protrudes from the ground.
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Measure the angle of the shadow of your meter stick when the sun is overhead and the shadow is smallest. Place the loose end of the string at the end of the cast shadow and hold it taut. Use the protractor to measure the angle where the string meets the stick at the top. Have your partner do the same in her location at the exact same time. Record the measurements.
Subtract the angle measurements to determine the difference in the angle of shadows between the two locations. For Eratosthenes, at midday on the summer solstice where the sun’s angle was directly overhead, the angle was zero. Though he did not have instant communications as we do now, he was able to determine the angle of the sun’s rays in Alexandria at the same time, which was about 7.2 degrees. Therefore, the difference was 7.2 degrees.
Compute the circumference of the earth using the distance and angle measurements you have. Since the locations are points on a circle that goes around the earth, the distance between them can be expressed as an arc measurement on a 360-degree circle. For Eratosthenes, the arc was 7.2 degrees. The distance between locations are also part of the total circumference of the earth. In Erastothenes’ case, the distance was 787 kilometers, so for him, the following relation applied: 7.2 / 360 = 787 / x, where x = the circumference of the earth in kilometers. Solving for x reveals the circumference of the earth to be 39,350 kilometers.
Compute the radius of the earth using the formula C (circumference) = 2 x pi x r (radius). Erastosthenes’ formula would look like this: 39,350 = 2 x 3.14 x r, or 6,267 kilometers.
TL;DR (Too Long; Didn't Read)
Use a scientific calculator. Since pi is an infinite number, the calculations in Step 6 will be more accurate.
You must measure the angle of shadows in the two locations at the exact same time on the exact same day or the calculations will be erroneous.
Because these measurements are not done with more sensitive equipment, the radius calculation will be only approximate. The actual radius of the earth is 6,378.1 kilometers at the equator, but the radius varies because the earth is a somewhat flattened sphere. The radius is more like 6,371 kilometers at the north and south poles.