Eratosthenes needs to know what the central angle of the earth is when there is a shadow at Alexandria (and a sun angle) and no shadow at Syene (sun angle is 0). Eratosthenes can measure the sun angle at Alexandria, but how does he find the central angle? It's time to discover Eratosthenes' insight. You are going to make a paper model of this situation.
Take a piece of paper and make a fold lengthwise (to reduce the width of the paper.) Carefully tear off that strip. Every student should have a sheet of paper with a different width.
Draw a diagonal line on your paper. Label the angles C and SA.
Cut the paper along the diagnal so that you have two pieces. Now compare angle SA with Angle C by placing one angle on top of the other.
What do you notice? (They are equal.)
Is that true for others in the class? Since every student had a different width paper (after they cut off the original strip), every student should have had different size angles SA and C. What can we conclude about the sun angle and the central angle when there is no shadow at Syene? (They are equal and it doesn't matter what the angle is.) Confirm the result with this Javasketchpad sketch. (Be patient. It takes a while to load. Drag Syene to change the angles. You should notice that angle SA (sun angle at Alexandria) and C (central angle) are always equal.
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