How does geometry explain the phases of the moon?

(The simulated image here shows the correct current lunar phase, courtesy of the United States Naval Observatory )

For instance, if you know what the First Quarter Moon looks like, and you observe it just rising in the East, you could quickly determine the approximate time (roughly noon). Similarly, if you saw the Full Moon just hovering above the western horizon, about to set, you would know that it was just before dawn.

The key understanding is the geometry of the Moon and Sun. When the Moon is directly opposite the Sun in the sky, they are separated by 180 degrees and the phase is full.

At First and Third (Last) Quarter phases, the angle between the Moon-Earth-Sun is 90 degrees (actually, 270 degrees for Third Quarter).

At the time of the First Quarter Moon, it follows the Sun in the sky; in the case of the Third Quarter Moon, it precedes the Sun.

Here is a good diagram (Fig. 2.21) from page 41 of the 7th edition of your textbook:How Does Geometry Explain the Phases of the Moon?

Note, for example, when the Moon is between the Sun and Earth, you can draw a line (roughly) from the Sun, through the Moon and on to the observer on Earth. This is a more or less straight line. The Moon would be located in the direction of the Sun and not visible in the glare. It is a “New” Moon. This phase rises and sets with the Sun.

It’s new because it represents the beginning of a new lunar cycle (“moonth”). On the relatively rare occasion when the Sun-Moon-Earth were directly lined up, you could have a solar eclipse, when the Moon appears to pass in front of the Sun.

If you start with a New Moon and move counterclockwise along the orbit, after a week or so you come to what is called First Quarter Moon. In this case if you draw a line from the Moon to the Earth to the Sun, it’s about 90 degrees, a right angle.

You see only half the visible surface of the Moon (the half toward the Sun). We call this a quarter Moon because we see only one quarter of the entire surface of the Moon counting the backside, as well as because it represents one quarter of the way through a full lunar orbit.

This phase of the Moon rises at roughly noon and sets about midnight.

When the Moon is on the opposite side, opposite the Sun from the Earth (180 degrees from the Sun), the full light of the Sun shines on the half of the Moon facing the Earth and we experience a Full Moon.

Roughly a week later the Moon has moved, still counter clockwise, another 90 degrees around its orbit to the point called Third (or Last) Quarter. It is 270 degrees in the counter clockwise direction around the orbit. But it is only about 90 degrees (a right angle) from the Sun in a clockwise fashion.

The difference between this phase and the First Quarter phase is that the opposite side of the Moon is illuminated, and it is in the morning sky, rising at roughly midnight and setting at around noon the following day.How Does Geometry Explain the Phases of the Moon?

Keep in mind that the Moon is constantly moving around the Earth, and so the phases are constantly changing. As such, there are other phases between these main phases described as crescent and gibbous, waxing and waning. You should learn those, too, but the four main phases are most important.

Test yourself:

1) In which direction is the New Moon at sunset?

2) From Denver, you see the First Quarter Moon due South in the sky. Approximately what time is it?

3) From Denver (Northern Hemisphere), where would you look to see the Full Moon shortly after sunset?

4) John reported observing the Third Quarter Moon at 9 pm. What can you say about this?

  • Practice these and make up your own. (Answers below)
  • Also keep in mind that the rough times given here do not consider Daylight Saving Time.
  • There are some good, short videos linked to the class YouTube page: Astronomy 1040How Does Geometry Explain the Phases of the Moon?.
  • Answers:
  • 1) Since the New Moon is very near the Sun, it will be in the western sky at sunset, although too close to the Sun to be seen.

2) As you can see from the graphics, the First Quarter Moon is about 90 degrees to the left (rotating counterclockwise) or East of the Sun. So if the Moon is due South, the Sun must be about 90 degrees to the right (West) as you face South. That puts it on or near the western horizon, so it must be approximately sunset time.

3)  The Full Moon is approximately 180 degrees across the sky from the Sun, or opposite the Sun. Therefore if the Sun is setting, the Full Moon will be near the eastern horizon (it may be just above, on or just below the horizon, depending on exact circumstances).

4) John is either mistaken or stuffed absolutely clean full of wild blueberry muffins. The Third Quarter Moon is 270 degrees (counterclockwise) from the Sun, so when the Sun sets, the Moon is still approximately 90 degrees below the horizon. As such, it will not rise until about midnight, and cannot possibly be seen at 9 pm.

The phases of the moon | National Geographic Kids

Have you ever looked up at the sky and noticed how the Moon changes shape each night? Ever wondered why… And how?

Well, here’s the answer – it doesn’t ‘change shape’ at all! Let’s take a look at the phases of the Moon…

Phases of the Moon

The Moon doesn’t emit (give off) light itself, the ‘moonlight’ we see is actually the Sun’s light reflected off the lunar surface.

How Does Geometry Explain the Phases of the Moon?

So, as the Moon orbits the Earth, the Sun lights up different parts of it, making it seem as if the Moon is changing shape. In actual fact, it’s just our view of it that’s altering…

How Does Geometry Explain the Phases of the Moon?Zoom in for a closer look! 

Changing faces

The Moon is Earth’s only natural satellite (a celestial body that orbits a planet). It takes the Moon 27.3 days to make a complete orbit around the Earth, but because the Earth is moving around the sun at the same time, it takes the moon 29.5 days to go through its eight different ‘phases’ – a lunar month.

How Does Geometry Explain the Phases of the Moon?Zoom in for a closer look! 

What Determines the Moon Phases? – Sky & Telescope

Before we describe the phases of the Moon, let's describe what they're not. Some people mistakenly believe the phases come from Earth's shadow cast on the Moon. Others think that the Moon changes shape due to clouds. These are common misconceptions, but they're not true. Instead, the Moon's phase depends only on its position relative to Earth and the Sun.

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The Moon doesn't make its own light, it just reflects the Sun's light as all the planets do. The Sun always illuminates one half of the Moon.

Since the Moon is tidally locked, we always see the same side from Earth, but there's no permanent “dark side of the Moon.

” The Sun lights up different sides of the Moon as it orbits around Earth – it's the fraction of the Moon from which we see reflected sunlight that determines the lunar phase.

How Does Geometry Explain the Phases of the Moon?We see the Moon go through a changing cycle of phases each month due to its orbital motion around Earth and the changing geometry with which we view it. S&T: Ana Aceves

The new Moon occurs when the Moon, Earth, and Sun all lie along approximately the same line. Since the Sun is behind the Moon from Earth's perspective, the side of the Moon that faces Earth is dark.

At full Moon, the three bodies also lie approximately in a line, but this time, the Moon is on the opposite side of Earth, so the Sun illuminates the whole side facing us.

At first quarter and last quarter, the Moon lies perpendicular to a line between Earth and the Sun. We see exactly half of the Moon illuminated by the Sun — the other half lies in shadow. The “quarter” used to name these phases refers to the respective fraction of an orbit that the Moon has completed since new Moon.

The illuminated part of the Moon gradually transitions between these phases. To remember the in-between phases you'll need to understand these terms: crescent, gibbous, waxing, and waning.

Crescent refers to phases where the Moon is less than half-illuminated, while gibbous means more than half is illuminated.

Waxing means “growing” or expanding in illumination, and waning means “shrinking” or decreasing in illumination.

After new Moon, a slice of reflected sunlight becomes visible as a waxing crescent. The lunar crescent grows until first-quarter Moon. As the sunlit portion of the Moon continues to increase to more than half of the Moon's face, the Moon turns waxing gibbous.

Then, after the full Moon the sunlit fraction begins to decrease again (though it still takes up more than half the face of the Moon) to make a waning gibbous and then a third-quarter Moon. The slice of sunlight continues to decrease until the moon is a waning crescent and then a new Moon.

The whole cycle (from new Moon to new Moon) takes about 29.5 days.

If you have a hard time remembering which way the moon phases go, just think: “white on right, getting bright!”

The Ping Pong Perspective

How Does Geometry Explain the Phases of the Moon?Holding a white ball at arm's length in the direction of the Moon shows how lunar phases depend on where the Moon is in the sky with respect to the Sun. S&T: J. Kelly Beatty

The Moon's phases are actually related to orbital motion, and there's a simple and fun observation that shows how they're connected. All you'll need is a Ping-Pong ball to simulate the Moon—actually, any small, white sphere would work. Then head outside about an hour before sunset, or around the time of a first-quarter Moon. Find the Moon in the southern part of the sky, then hold the ball up at arm's length right beside it.

You'll see that the ball shows exactly the same phase as the Moon. The Sun illuminates both the ball and the Moon from the same direction, and you see them as partly sunlit and partly in shadow, their bright and dark portions mimicking each other perfectly.

If the weather stays clear, you can repeat this observation on the next several afternoons. Each day the Moon's orbital motion has carried it farther east, and the sunlit portion of its disk has grown larger.

If you hold your ball up near the Moon, you'll see that its “phase” has thickened too.

To sneak a preview of the Moon's appearance in the days to come, simply move the ball farther east.

And if you move it all the way over so your arm points low in the eastern sky, the side of the ball that's facing you will be almost completely illuminated — nearly a “Full Ball,” so to speak.

And, sure enough, a day or two before full Moon, the Moon hangs low in the eastern sky just before sunset and is almost completely illuminated.

Try it out!

To find out what phase the Moon is tonight, try our Moon Phase calculator. Note that the Moon's phase is the same for any location on Earth, but Southern Hemisphere observers will see the Moon “upside down” from the Northern Hemisphere view.

Astronomy Without a Telescope

This material (including images) is copyrighted!.
See my copyright notice for fair use practices.

One of the most familiar things about the Moon is that it goes through phases from new (all shadow) to first quarter (1/2 appears to be in shadow) to full (all lit
up) to third quarter (opposite to the first quarter) and back to new. This cycle takes about 29.53 days.

This time period is known as the Moon's synodic period. Because the Moon moves through its phases in about four weeks, the phases of new moon, first quarter, full moon, third quarter occur nearly one week apart from each other.

Select this link to find the
phase for any date and time between 1800–2199 (will display in another window). A picture of the Moon will
be shown.

The phases are due to how the Sun illuminates the Moon and the relative positioning of the Earth, Moon, and Sun.
The figure below shows that as the Moon orbits the Earth, the fraction of its illuminated side that you can see from the Earth changes.

high above the Earth and Moon orbit, you can see that the Moon is always half
always faces
The other half (night side) faces away from the Sun. The figure below combines
two view points.

The half-lit moon on the inner circle around the Earth is the
Moon as viewed from high above the Earth and Moon orbit.
The outer ring of Moon pictures in various phases is the view of the Moon as
we would see it from the Earth. Of course, this drawing is not to

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Notice that the waning gibbous, third quarter, and waning crescent phases as seen from Earth are lit on the left side. The waxing vs. waning diagram below shows how this can be even though the Sun in the figure here is on the right side (one rotates the picture 180°, so the observer at D is facing up).

How Does Geometry Explain the Phases of the Moon?

Select here for a nice simulation of the moon phases (will display in another window). Be sure to choose “both'' in the point of view pop-up list.

Moon Phases :: Earth, Sun, and Moon Geometry

The Moon is not a light-generating body, meaning that she doesn’t actually produce any of her own light.

All of the light that the Moon appears to illuminate our sky with, which we call moonlight, isn’t actually moonlight at all, but sunlight reflected off the lunar surface that is 450,000 times less intense than direct sunlight, due to this reflection.

So then where do the Moon phases come from? They come from the the Sun, from the Moon’s orbit around the Earth, and from the specific perspective that we are observing the Moon from.

As you can see in the diagram below, the Moon orbits the earth in a counterclockwise direction (looking down upon the system from above the North Pole) taking 29.

53 days to complete 1 revolution around the Earth, which we call a lunar month.

At the same time as the Moon is orbiting the Earth, the Earth is orbiting the Sun also in a counterclockwise direction, which as you know takes about 365.25 days.

Over the course of one lunar month the position of the Moon in relation to the Earth and the Sun changes. Once a month the Moon will be behind the Earth being pulled along by the Earth in its orbit (which is to the right in the diagram below), locked in place by our mutual gravitational attraction.

Then about 7 days and 9 hours later the Moon will be on the far side of the Earth opposite the Sun.

About 7 days after that the Moon will move into a position “in front” of the Earth in the direction of our motion around the Sun, and 7 days later the Moon will come directly between the Earth and the Sun, at which point the lunar cycle will repeat itself.

These four positions are the 4 major alignments of the Sun-Earth-Moon system. There are about 7 days 9 hours between them, which adds up to a total of ~29.53 days (as there are 4 of these periods in 1 orbit of the Moon) which is where we derive the length of the lunar month. This can all be seen if you take a moment to study the diagram below (Fig. 1).

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?

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.

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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.

Five Phases of the Moon

Updated April 25, 2017

By Tina Smith

The moon is the brightest object in the night sky. It appears to change shape depending on the position of the Earth and the sun. The moon orbits the Earth every 29.5 days. As it orbits the Earth, it appears to be getting bigger (waxing) or smaller (waning). There are five phases of the moon: new, crescent, quarter, gibbous and full.

A new moon phase is when the moon is directly between the sun and the Earth. The moon's illuminated half is facing away from the Earth making the moon not visible from Earth. Due to the lack of light in the night sky, this is the best time to see stars and constellations.

The second phase of the moon is called the crescent moon. The moon orbits the Earth and each night reveals more of its illuminated hemisphere. At this phase, less than half of the moon is visible.

To find out if the moon is waxing or waning you can hold a finger up to the crescent moon shape. If your finger against the crescent moon makes a “b” shape then it is getting bigger (waxing).

If your finger makes a “d” shape then the moon is decreasing, or getting smaller (waning).

In the quarter moon phase, half of the satellite is illuminated. The first quarter moon occurs after the new moon and before the full moon. The last quarter moon appears after the full moon. The gravitational pull of the moon also affects the oceans' tides. At the quarter moon stage, the gravitation pull is weaker and smaller neap tides form.

A gibbous moon is when more than half of the moon is visible. Many farmers plant and prune by the phases of the moon. The waxing gibbous moon is considered the best time for planting crops that produce fruit or vegetables above ground, such as:

  • beans
  • melons
  • squash
  • peas
  • peppers
  • tomatoes

The moon increases in size until it appears as a full circle in the sky, or the full moon.

In the full moon phase, it will appear as a complete circular disk in the night sky. At the new and full phases of the moon, the gravitation pull is the strongest. This causes higher ocean tides.

Many people believe that a full moon can affect hormones, mood and even induce labor.

After the full moon phase, the moon will appear to decrease in size moving through the phases again in reverse order: gibbous, quarter, crescent and new moon.

About the Author

Tina Smith began writing in 2009. She has published a short story in “Chicken Soup for the Soul: New Moms” and contributed an article to “True North Parenting” magazine. Smith graduated from California State University, Chico with a Bachelor of Arts in psychology and a Master of Arts in school psychology and counseling.

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