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# How Does Geometry Explain the Phases of the Moon?

What causes the Moon to change phases throughout the month? Why is it sometimes visible only during the day and other times only at night? What’s the relationship between these times and the Moon’s phases? The answer in all cases is geometry. Read on to find out how it works.

By
Jason Marshall, PhD
Episode #291

## The Geometry Behind Moon Phases

To begin with, the light we see coming from the Moon is not generated by the Moon but by the Sun. Yes, that’s right—“moonlight” is actually sunlight bouncing off the Moon’s surface directly into your eyeballs. Which turns out to be a very important piece of the puzzle behind figuring out what causes the phases of the Moon. Because, as we’re about to discover, the Moon’s phases are a result of the geometry between the Sun (and its light), the Earth, and the Moon.

The Moon’s phases are a result of the geometry between the Sun, the Earth, and the Moon.

The first thing to realize is that half of every spherical object in the solar system that’s illuminated by the Sun is lit up at any given time (that's the half facing the Sun), and half is in darkness (that's the half facing away from the Sun). So half of the Earth is always illuminated and half of the Moon is always illuminated, too. The other thing you need to know is that the Moon is orbiting around the Earth once per month. Which, interestingly enough, is the exact same period of time that the Moon takes to go through its cycle of phases—which suggests that the phases of the Moon might be related to the Moon's position in its orbit around the Earth. And that's exactly right.

To see how this geometry puzzle works, imagine looking down on the Earth and Moon from far above their north poles. Imagine also that light from the Sun is coming in from the right, so that the right halves of both the Sun and Moon are illuminated and the left halves are in darkness. Now imagine moving the Moon around the Earth to the various positions in its orbit. How does the appearance of the illuminated half of the Moon change from our vantage point on Earth?

Image credit: NASA

Well, when the Moon is in the part of its orbit that’s between the Earth and Sun, we on Earth are staring directly at the unilluminated half of the Moon. This is the geometrical relationship that gives rise to the new Moon. Two weeks later when the Moon is on the opposite side of the Earth from the Sun, we on Earth see the entirely illuminated half of the Moon. This is the geometrical relationship that gives rise to the full Moon. During the two quarter Moon phases, the Moon’s day/night line falls directly in the middle of the Moon from our vantage point. Thus, we see the Moon half-illuminated. And if you think about the geometries and illuminated portions of these three celestial bodies during the Moon’s crescent and gibbous phases, you’ll see that this line of thinking works here as well. And you'll see that it’s all just beautiful geometry.

## Why Does the Full Moon Always Rise at Sunset?

Once you understanding this geometrical explanation of the Moon’s phases, you also have the power to understand why the Moon is in the sky at different times of day throughout the month. To see why this is, let’s go back and think about the geometry that gives rise to a full Moon. The Moon appears full when it’s on the opposite side of the Earth from the Sun. Which means that when the Sun is directly overhead, the full Moon must be on the opposite side of the Earth directly beneath your feet. If you’re not seeing why this must be true, take another look at the picture from before and keep in mind that at noon the Sun is directly over the part of the Earth where you’re standing.

As the Sun continues to move across the sky towards your western horizon, the full Moon moves across the sky on the opposite side of the Earth from where you’re standing. When the Sun reaches your western horizon at sunset, the full Moon must therefore reach your eastern horizon. Which means that whenever the Moon is full, moonrise must occur at sunset. And, if you think about it, you’ll see that every full Moon must set at sunrise.

Once again, it’s all just beautiful geometry.

## Wrap Up

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

For more fun with 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!

Moon image credit: NASA