How big is the Moon?

I have the good fortune of getting to teach my dream class this year. My school has just started an interdisciplinary studies program and, as part of that program, I’m co-teaching what is think is the BEST CLASS EVER. It’s called, “Notions of Space, Time, and Dimension,” and it’s a math/science/history/philosophy/literature class about how people have tried to understand and talk about our universe. So far, we’ve read some pre-Socratic philosophies, studied Zeno and his paradoxes, read the Calypso section of The Odyssey, and done the packet I’m about to share with you about ancient Greek methods of measuring the Earth, the Sun, and the Moon.

I’ve made a lot of worksheets in my time, but this one has gotten the best reception of them all. I thought a lot about it and took a lot of time making it, so I thought I’d share it with anyone who’s interested. It uses at most basic trigonometry and walks you through ancient Greek conceptions (and misconceptions) about the relationships between the Earth, Moon, and Sun – and through using those (mis)conceptions to calculate some accurate and some inaccurate sizes of things.

The most interesting part of this process, I think, is how different assumptions about the way the rays of the Sun hit the Earth led to both correct and incorrect size calculations. If you assume that the Sun’s rays are parallel to the Earth (which is the simplest assumption to make), you can get a pretty accurate calculation of the size of the Earth – using a short Earth distance calculation, an angle measurement taken from a shadow, and some corresponding-angles-are-congruent and simple trig. BUT if you assume that the Sun’s rays are parallel to the Earth and use that assumption to set up a calculation for the size of the Moon, as Aristarchus did, you’ll be WAY OFF. (You’ll be even farther off if you use that assumption to calculate the distance to the Sun…)

We had some great discussions about how the rays of the Sun actually hit the Earth, what shape shadow the Earth casts as a result, and how the same mathematical/physical assumption can lead to very different errors, depending on how it is used.

So, here’s a little combination of trigonometry, math error analysis, physics, and ancient history for you. Enjoy!

(Click on the pictures for larger, more printable, versions.)

Measuring the Universe Woo 1Measuring the Universe Woo 2 realMeasuring the Universe Woo 3Measuring the Universe Woo 4

Author: Anna Weltman

Anna is a PhD student in math education at University of California, Berkeley.

One thought on “How big is the Moon?”

  1. Hi Anna,

    I love this lesson. I even downloaded all 4 of your pages. I suppose it has something to do with me being a Johnny. I was reading your blog because I had to. I was randomly assigned to evaluate your blog for my Math class in teacher’s college in Ottawa, Canada. I gave you a glowing review. But sometime later, I got to thinking that you were somehow familiar. I found your Ted Talk, and there you were standing beside my long time friend and classmate, Justin. And then once I saw you speak, you looked very familiar indeed. It turns out that we all spent a weekend together some years ago! Now how many math teachers are there in New York, in North America? Let’s calculate the odds!

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