How big and how far is the Moon?

“The moon’s an arrant thief,

And her pale fire she snatches from the sun.” –William Shakespeare

Let’s assume you lived before we walked on the Moon. Before we had ever been to space, hell, even before the invention of the telescope. Why not go all the way back to before we knew the planets went around the Sun! You know, back when all you had to go on for knowledge of the sky was your eye.

Some nights, you look up at the sky, and what would you see? Instead of thousands and thousands of stars, something might be filling your sky with light pollution. Indeed, the closest astronomical body to Earth is the brightest thing in our night sky. Say hello to the Moon, which gets its brilliant light by reflecting it from the Sun.

And let’s say you wanted to know, looking up at that brilliant orb in the night sky, how big it was and how far away it was. While today we bounce lasers off of a mirror the Apollo astronauts left on the Moon to figure out its distance, there’s a neat trick anyone can use to understand the size and distance to the Moon.

First the size. All you need to know to get this is the size of the Earth, which we’ve known since the 3rd Century B.C. to be about 40,000 km in circumference (or about 13,000 km in diameter). Well, that, and the fact that the Sun is farther away than the Moon is. Which is easy; after all, you’ve heard about these:

Solar eclipses! Since the Moon can pass in front of the Sun, that tells us the Sun must be farther away! So we can make an approximation (that, to be fair, is only really good if the Sun is a lot farther away than the Moon is) that when we see the Earth’s shadow on the Moon, it’s about the size of Earth.

And do we ever see the Earth’s shadow on the Moon?

That’s just what a lunar eclipse is! In fact, we can stitch together a bunch of images from a partial lunar eclipse, and showcase something spectacular:

Image credit: Flikr user Skink74.

Wow! So all you need to figure out the diameter of the Moon is to know the diameter of Earth, and measure what the ratio of Earth’s shadow is to the size of the Moon! If you guess that the Earth is three times bigger, you’re a little low, but if you guess it’s four times bigger, you’re a little high. It turns out that three-and-two-thirds is just about right, and if I divide that circumference of 40,000 km by three-and-two-thirds, I get 10,900 km for the circumference of the Moon, or (dividing by pi) 3470 km in diameter.

And if I check wikipedia, it tells me the circumference of the Moon is 10,921 km. Yes, just like that, you can figure out the size of the Moon!

But let’s say you wanted to go farther, and figure out the distance to the Moon. You could do this, too! Take a look at the Moon in the sky:

As long as you know that there are 360 degrees in a circle, you can figure out how many degrees on the sky the Moon takes up! It turns out it’s very close to half-a-degree, which varies depending on whether the Moon is at its closest point to Earth (perigee) or its farthest (apogee).

But if we take the crude “half-a-degree” estimate, we can easily figure out how distant it is to the Moon just by remembering a little trigonometry!

There’s a relationship between the size of the angle (θ), the distance to the Moon (L), and the diameter of the Moon (d), just given by the formula:

tan (θ) = d / L,
 
which gives me a distance to the Moon of just under 400,000 km. This stacks up really well to the modern (average) value of 384,403 km, but it gets even better when you realize the Moon’s distance varies between 363,104 km and 406,696 km.

And that’s it! Just by using your eyes and your noodle, you too can figure out how big the Moon is and how far away it is from us! Have a great weekend!