# Eratosthenes' most amazing discovery:  the "sun" angle is the same as the central angle!!!

One day while in his library in Alexandria Eratosthenes read that in the frontier outpost town of Syene at noon on June 21st, vertical sticks cast no shadow while a reflection of the sun could be seen at the bottom of the well. Being the scientist that he was, he wanted to know if the same thing happened in Alexandria. He discovered that there were shadows there.
But how does knowing this help in measuring the central angle? Eratosthenes used a little geometry to come up with his insight. This discovery makes possible the use of the measure of the central angle (along with the distance between the two towns) to determine the circumference of the earth.

Eratosthenes knew that the sun was far enough away so that its rays are parallel. In the diagram above note that there is no shadow at Syene while there is a shadow in Alexandria. Then he realized the sun angle had to equal the central  angle. See the Paper Folding Activity.

But what happens if there IS a shadow at both sites? Below is a model (created with the Geometer's Sketchpad) of the problem. Notice that neither sun angle A at Alexandria nor angle S at Syene equals the central angle.

activity sheet.

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