How Does Geometry Explain the Phases of the Moon?

As you’ve probably noticed, the Moon doesn’t always look the same. In fact, the Moon goes through a “moonthly” cycle lasting about 29.5 days (yes, that’s where the word “month” comes from). This cycle takes the Moon through its full range of phases and eventually leaves it back in the phase it started in.

As you’ve probably also noticed, the Moon is occasionally visible during the day. In fact, if you’re paying attention, you may have noticed that it’s visible during the day a lot. But it’s also sometimes only visible during the night. And if you are really paying attention, you may have noticed that there’s a relationship between when the Moon is visible during the day or night and its phase.

What’s behind this relationship? What determines what time of day the Moon is visible? And what causes the Moon to change phases in the first place? The answer is math—and, in particular, geometry.

How does it work? Let's find out.

Phases of the Moon

Before we dive into the math behind the Moon’s changing phases, we need to learn the lingo that astronomers use to discuss this so-called "lunar cycle." We can break the lunar cycle down into four segments, each of which lasts approximately one week:

1. New to first-quarter
2. First-quarter to full
3. Full to third-quarter
4. Third-quarter to new

The first two segments taking the Moon from what’s called the new phase up to the full phase are called the “waxing” (which means growing) portion of the cycle. A new Moon is completely unilluminated from our vantage point here on Earth, therefore we can’t see it. From its new phase, the Moon moves through its waxing crescent phase until it reaches its first-quarter phase (so called because it’s 1/4 of the way through its cycle). In the first-quarter phase, the Moon appears half-illuminated. The Moon then moves into and through what’s called the waxing gibbous phase (when it's between half-full and full) until it finally reaches its full phase and is completely illuminated.

Image credit: NASA/JPL-Caltech/Bill Dunford

The final two segments of the lunar cycle make up what’s called the “waning” (or shrinking) portion of the cycle. During these approximately two weeks, the illuminated portion of the Moon appears to shrink down from full through the various phases we just mentioned (but in the reverse order) until it’s once again in the new phase. And the cycle begins again.

So those are the words we use to describe the phases. The question now is what causes them?

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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