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,
September 17, 2016
Episode #289

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Eratosthenes' Method for Measuring the EarthIt’s relatively easy to measure the size of something small and nearby (just use a ruler), but it’s really hard to measure the size of something huge and far away. Just ask an astronomer (of which I am one), and they’ll agree (which I do). Because one of the big under-appreciated challenges in astronomy is figuring out how big things are and how far away they are. The history of astronomy and our understanding of the universe is intricately tied to the history of a bunch of clever humans coming up with ingenious ways to do just this.

And the cool thing for all of us math fans is that many of these methods are based upon very simple but also very powerful math—in particular, geometry. Today, I’d like to tell you a story of one of these methods that dates back more than 2,200 years. It’s the story of a brilliant Greek astronomer and mathematician named Eratosthenes and his efforts to measure the size of the Earth. How did he do it? Let's find out.

How to Measure Short Distances

Before we get to the cleverness that is Eratosthenes’ method for measuring the size of Earth, let’s take a brief minute to discuss the measurement of distances in general and to understand why the problem Eratosthenes set out to solve was such a difficult one.

As I said earlier, measuring the sizes of things and the distances to them is easy as long as they are small and nearby. If both of these things are true, you can just grab a measuring stick, walk over to the thing, and either measure its size or how far away it was from where you started. If the object or place you want to measure the distance to is too far away for your measuring stick, we have a few other options in the modern world. We can drive to the location and use the car’s odometer to tell us the distance, we can use some mapping software to calculate the distance, or we can use the GPS coordinates of two locations to figure out the distance between them.

Measuring the sizes of things and the distances to them is easy as long as they are small and nearby.

These things are all fairly easy to do, but some things aren’t so easy. How far away is the Moon? The Sun? The nearest star? Or some more distant star? As you can imagine, these are hard questions that require more than simply knowing how to read the different markings on a meter stick. To begin our exploration of the math behind these measurements, let’s now jump into our time machines and travel back a little more than 2,200 years to learn about how the ancient Greeks solved a problem that seemed unsolvable until a little geometry and a lot of human ingenuity was added to the mix.


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