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How to Use Geometry to Calculate the Size of Earth

Can you think of a way to measure the size of the Earth? Or the distance to the Sun? Or the nearest star? Want to know how ancient Greek mathematicians did exactly this over 2,000 years ago? Keep on reading to find out.

By
Jason Marshall, PhD
5-minute read
Episode #289

Eratosthenes’ Big Idea

How big is the Earth? More specifically, what is the circumference of the Earth? That is the question that Eratosthenes was confronted with sometime around the year 240 B.C. By the way, if you think about it, you’ll realize that simply asking this question meant that—contrary to what many people believe—the ancient Greeks already knew that the Earth was round and not flat. And, as we’ll see, the very measurement we’re about to watch them make contains a form of geometric proof of the curved nature of the planet.

As the (true) story goes, Eratosthenes learned that at precisely noon on the first day of summer in the Egyptian city of Syene, a stick that had been placed in the ground vertically would cast no shadow. And if you looked down a very deep well at that moment in Syene, you could see the reflection of the Sun—something you didn’t see any other day of the year. Eratosthenes realized that this could only happen if the Sun was directly overhead at that moment so that the rays of light falling upon the Earth in Syene were precisely parallel to the stick and the well.

What really made this interesting to Eratosthenes was the fact that these same phenomena were not also visible on that day in his home city of Alexandria located roughly 800 kilometers (or 500 miles) north of Syene. Eratosthenes reasoned (correctly) that if the Earth were flat, and if the Sun were far enough away from the Earth so that its light rays all arrived at the Earth parallel to each another (which they more-or-less do), then every stick placed vertically in the ground at every position on Earth should cast the same length of shadow at the same time. The fact that sticks cast different shadows at the same time therefore serves as geometric proof that the surface of the Earth is curved and not flat.

Flat Earth

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About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.