Part 1 begins on the Introduction page. Part 2 is on the Math Fundamentals page.

How did Eratosthenes know the interior angle he measured opposite the shadow length on the right triangle formed between the vertical column and the shadow length measured in Alexandria was also the angle from the center of the earth to the arc spanned between the two cities Syene and Alexandria on the surface of the earth?

He used a theorem you will be required to know for the SHSAT! Alternate interior angles in a transversal are congruent.

Most students at this point will understand the concept behind what Eratosthenes did and how he used SHSAT math to get the job done and think, “Kind of cool!” Then they will move on. Not this time, however. Even if you peeked at the solution to get here you will not as easily sail through the remaining discussions about design. The concept is only the beginning; not the end. Once Eratosthenes figured out the idea, he then had to design a test, build it, and then test his idea to get results. As future STEM (Science, Technology, Engineering & Math) based high school students and advocates of the scientific method, you too will have to labor through the same process. The remaining parts will ask you how you would design various elements of the experiment. There may be no one single answer but certainly some better answers than others.

How would you arrange the shadow measurements in your test design? Could you make the test using a city other than Syene not on the Tropic of Cancer? Would you pick a location to the east, west, north, or south of the first location? Why?

Part 4 is available on the Geometry Fundamentals page.