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.

Jason Marshall, PhD
5-minute read
Episode #289

How to Measure the Earth’s Circumference

But Eratosthenes didn’t stop there. He realized that he could use this observation and a bit of geometry to estimate the distance around Earth. Here’s the idea: As you travel away from Syene along the curved surface of the Earth, Eratosthenes reasoned that the shadow cast by a vertical stick at noon on the first day of summer should grow in length. And he further reasoned that if you measure the angle between the vertical stick and an imaginary line extending from the top of the stick to the end of the shadow on the ground, then you are actually measuring the angle between the two locations on Earth as seen from the center of the Earth.

To see that this must be true, take a look at the following picture and think about what would happen to the angles if the city of Alexandria is moved towards or away from Syene. As the angle between the cities increases, the angle made by the lengthening shadow in Alexandria also increases—and it grows by exactly the same amount as the angle between the cities.

Curved Earth

Eratosthenes found that the angle to the end of the shadow in Alexandria was very close to 7 degrees. Since there are 360 degrees in a circle, he concluded that the city of Alexandria must be located about 7/360 or roughly 1/50 of the way around the entire Earth from Syene. He knew that the distance between the two cities was roughly 800 kilometers, so he calculated that the circumference of the Earth must be approximately 50 x 800 kilometers = 40,000 kilometers. Modern day measurements peg the size of the Earth at about 40,075 kilometers, which means that Eratosthenes was nearly spot on.

In truth, there were several assumptions about the geometry of the problem that he got slightly wrong, but they all ended up sort of cancelling each other out and resulted in an incredibly accurate calculation … especially considering it was made with very rudimentary equipment over 2,200 years ago. Which goes to show that a bit of math and a lot of ingenuity can take you a long way.

Wrap Up

Okay, that's all the math we have time for today.

For more fun with numbers and math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Public domain image: Eratosthenes' method for determining the size of the Earth


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.