The Moon's phases are the most visible periodic change in the night sky, and the most widely misexplained. The phrase "the Earth's shadow falls on the Moon" is almost always the first explanation offered, and it is wrong — the Earth's shadow is a cone barely longer than 1 million kilometres, while the Moon orbits at 384,400 km. If shadow caused the phases, we would see a lunar eclipse every month. We see one a few times per century instead.
The geometry: Sun, Earth, Moon
Three bodies, one arrangement. The Moon orbits Earth and, because of tidal locking, keeps almost exactly the same face toward us. From the Moon's perspective the Sun is effectively a light source at infinity, so one hemisphere is always in daylight and the other always in night. The phase we see from Earth is simply how much of that daylit half happens to face us.
That makes the phase a pure geometry problem, and the two extreme cases define the whole cycle:
- New Moon — the Moon sits between Earth and the Sun. Its lit hemisphere faces away from us, so we see the unlit side. Not total darkness in the sky: the Moon is present, unilluminated, and it is invisible against the Sun's glare.
- Full Moon — Earth is between the Sun and the Moon. The lit hemisphere faces us directly, and we see the entire disk.
Everything between those two positions is intermediate. At first quarter the Moon sits 90° from the Sun as seen from Earth, so we see half the disk lit — a straight terminator down the middle, and the right half lit in the northern hemisphere. Last quarter is the same geometry 180° further along, with the left half lit.
The moon phase calculator works from this geometry: give it a date and it returns the illuminated fraction, the phase name, and the age of the Moon in days, so any of the calculations below can be checked against a specific date.
Two different months
Here is where most accounts go wrong, because there are genuinely two cycles and the difference between them is what makes the phase cycle the length it is.
Sidereal month: 27.3217 days. This is the Moon's orbital period relative to the distant stars — the time for the Moon to return to the same position in its orbit. It is the true orbital period, and it is the one a gravitational simulation uses.
Synodic month: 29.5306 days. This is the time for the Moon to return to the same phase, measured relative to the Sun. It is longer, and the reason is purely kinematic.
The derivation is worth doing because it makes the number 29.53 feel inevitable rather than arbitrary. The Moon moves around its orbit at 360° ÷ 27.3217 = 13.176° per day. The Sun appears to move along the ecliptic at 360° ÷ 365.256 = 0.9856° per day, because Earth's orbit carries us along. For the Moon to regain the same geometry relative to both, it must gain a full 360° on the Sun. The relative rate is 13.176 − 0.9856 = 12.191° per day, so:
Synodic month = 360° ÷ 12.191°/day = 29.5306 days
The other way to see the same correction: during one sidereal month the Sun has moved 26.93° along the ecliptic, so the Moon must cover that extra angle. That is 26.93° of 360° — about one thirteenth of a full circle, or 7.48% — and at 13.176° per day it costs 2.21 additional days. This is the same kind of relative-motion correction described in the velocity and acceleration guide, and the arithmetic is identical in spirit.
Consequence worth knowing: a year is 365.2422 days but twelve whole synodic months are only 12 × 29.5306 = 354.37 days, so the same phase arrives about 10.9 days earlier in the next calendar year. A full Moon on 2 January in year N arrives in late December of year N+1.
The eight phases and the phase angle
The traditional eight-phase scheme divides the cycle into eighths. The rigorous version uses the phase angle α, the Sun–Moon–Earth angle, which runs 0° to 360° across the cycle:
α = (age ÷ 29.5306) × 360°
The illuminated fraction of the visible disk then follows from geometry alone:
k = (1 − cos α) ÷ 2
which gives k = 0 at new Moon, k = 0.5 at the quarters, and k = 1 at full Moon. The four principal phases and their ages:
- New Moon — α = 0°, age 0.0 days, 0% lit.
- First quarter — α = 90°, age 7.38 days, 50% lit.
- Full Moon — α = 180°, age 14.77 days, 100% lit.
- Last quarter — α = 270°, age 22.15 days, 50% lit.
The intermediate phases are waxing crescent (age 3.7 d, 15% lit), waxing gibbous (age 11.1 d, 85% lit), waning gibbous (age 18.5 d, 85% lit) and waning crescent (age 25.8 d, 15% lit). The gibbous phases are not symmetric in the eighths scheme but they are symmetric in the cos α formula, which is the one to trust for any calculation.
Worked example 1: new Moon to full Moon
Half a synodic month separates the two extreme phases:
29.5306 ÷ 2 = 14.765 days
So a new Moon at a given instant is followed by full Moon 14.765 days later, and the same interval returns to new Moon. The distance is not the only thing that changes: the full Moon also lies roughly 400,000 km opposite the Sun in its orbit, which is why a full Moon near perigee — the "supermoon" — looks about 7% larger in angular diameter than a full Moon at mean distance.
Worked example 2: a moon age of 12.4 days
Suppose you observe at 21:30 and the calculated age of the Moon is 12.4 days. Convert to phase angle:
α = (12.4 ÷ 29.5306) × 360° = 151.2°
Since full Moon is at 14.765 days, the Moon is 14.765 − 12.4 = 2.37 days before full, so it is a waxing gibbous moon. The illuminated fraction:
k = (1 − cos 151.2°) ÷ 2 = (1 + 0.877) ÷ 2 = 0.938, so 93.8% lit
Nearly full, but not quite — which matches 2.37 days from full, since the phase changes fastest near the quarters and slowest near new and full. For planning purposes the useful summary is: age 12.4 of 29.53 is 42% of the way through the cycle, a waxing gibbous, about 2.4 days from full, rising in the late afternoon and visible most of the night.
Worked example 3: the Moon on 4 October 2026
Using a standard lunar phase algorithm, the new Moon preceding that date fell on 11 September 2026 at 15:56 UT and the next new Moon falls on 10 October 2026 at 18:21 UT. For 4 October the age is therefore:
age ≈ 22.34 days
Which is 0.2 days past last quarter (22.15 days), a waning moon about 48% illuminated, heading down to a thin waning crescent. The phase angle is 22.34/29.5306 × 360° ≈ 272.3°, and k = (1 − cos 272.3°) ÷ 2 ≈ 0.48, consistent. An observer at UTC+8 looking at 20:00 local time on 4 October is 12:00 UT, which is 0.5 days later in the cycle — an age of about 22.8 days and a little under half illumination. The phase is a shared geometric fact, but the clock time it is noticed at is a local one, which is the subject of the time and time zones guide.
Waxing and waning: the two halves
From new Moon to full Moon the illuminated portion increases — waxing — and in the northern hemisphere the lit limb is on the right (waxing crescent, first quarter, waxing gibbous). From full Moon to new Moon it decreases — waning — and the lit limb is on the left.
This orientation is a consequence of Earth's position, not of the Moon's, which is worth internalising because it inverts in the southern hemisphere: there the waxing crescent is lit on the left. Anyone who learned the mnemonic "the Moon is a cow, waxing it gets its horns back" learned the northern case only.
Why the shape changes at all, when the lit fraction changes smoothly: the terminator is the projection of the day–night divide onto the visible disk, and its apparent curvature flips as the geometry passes 90°. The first quarter's "straight line" terminator is exactly the half-lit case.
Why first quarter is the best time to observe
For crater and terrain observation, the seven days around first and last quarter are the standard recommendation, and the reason is purely about shadow.
Near the quarters the Sun is low in the Moon's sky, so features cast long shadows. A crater 100 km across with a rim 3 km high casts a shadow roughly 3/tan(20°) ≈ 8 km long at first quarter — a black line that makes it unmistakable. A low ridge 500 m high throws a 1.4 km shadow. Those shadows are the only reason lunar relief looks like relief at all.
At full Moon the Sun is directly behind the observer. Shadows fall directly away from the features and are hidden by them, so craters become flat bright rings — dramatic but flat, and badly overexposed for the bright highlands. At new Moon there is no visible light at all. This is the same shadow-versus-geometry trade-off described in the energy guide, applied to a different physical situation: an observer's viewing angle determines what is measurable.
Tides: phases set the timing, geography sets the size
Two gravitational pulls act on the ocean: the Moon's, and the Sun's. The Sun's is about 46% as strong despite being 400 times further away, because the tide-generating force scales as mass over distance cubed.
Spring tides occur at new and full Moon, when the Sun, Moon and Earth are aligned and the two pulls reinforce. The tidal range is at its largest, typically about 20% greater than average.
Neap tides occur at the quarter phases, when the Sun and Moon are at right angles as seen from Earth and the pulls partly cancel. The range is at its smallest.
The timing is therefore locked to the phase cycle: two spring tides per 29.53-day month, with successive high tides about 12.42 hours apart (the M2 lunar constituent) while the spring–neap cycle itself runs on the half-month period of 14.77 days.
Real water levels are a much less tidy subject. The Bay of Fundy has a 16 m range because of the shape of the funnel, and the Mediterranean has almost no tidal range at all for the same reason. Coriolis deflection makes the two bulges travel as waves in the open ocean, and resonance in basins can amplify or cancel them. Barometric pressure and wind add more. So the phase reliably tells you whether the range is above or below average, and geography decides by how much.
Why "the fifteenth moon is round on the sixteenth"
Traditional lunar calendars designate the day beginning at the new Moon as the first day of the month. The full Moon then falls on day 14 or 15 — most often day 15, but the sixteenth frequently enough that there is a proverb about it.
The cause is purely a mismatch of cycles, and it has nothing to do with the Moon's behaviour. A synodic month is 29.53 days while the average calendar month is 365.2422 ÷ 12 = 30.44 days, so each calendar month the phase lands about 0.91 days earlier than it did the month before, and those 0.91 days accumulate to roughly 10.9 days per year. On top of that, the exact instant of full Moon falls at some fixed time of day, and a different time zone sees that instant on a different calendar date. In Chinese practice a full Moon late in the afternoon belongs to the following day, which is the entire content of the saying.
None of this is a defect in the system — the phase-to-date association is deterministic, which is why the same calendar gives the same dates year to year. The apparent irregularity is the interaction of three periods: 29.53 days for the phase, 30 days for the lunar calendar month, and time zones. The calculator handles the phase side; the date calculator is the tool for turning a lunar month into specific dates.
High latitudes: where the cycle breaks down
Near the poles the phase cycle stops being a reliable guide to visibility. Because the Sun's daily path at high latitude is nearly horizontal in summer, the Moon at full Moon can be up all night and visible for days; at last quarter it may never clear the horizon at all. At 70° north, roughly half the lunations have a full Moon that fails to rise, and around the Arctic Circle a full Moon can set just as the Sun finishes its own night.
The same effect applies in reverse in high southern latitude, offset by six months. Practical consequence: at high latitude the standard first-quarter-for-craters rule is unreliable, and observing windows have to be checked per date rather than assumed from the phase — run the specific date through the moon phase calculator and check the resulting age against your local sunset and sunrise times.
Frequently asked questions
What causes the phases of the Moon?
The angle between the Sun, Earth and Moon, nothing more. The Moon orbits Earth while always showing nearly the same face toward us, so from Earth we see varying amounts of its sunlit half depending on where it sits in that orbit. No shadow falls on the Moon during an ordinary month: the Earth's shadow is far too small to reach it, so eclipses are rare.
Why is the synodic month longer than the sidereal month?
Because Earth is also moving along its own orbit. The Moon returns to the same position among the stars in 27.32 days, but during that time the Sun has moved 26.9 degrees along the ecliptic, so the Moon must travel an extra 26.9 degrees — about one thirteenth of a full circle — to reach the same Sun–Earth–Moon geometry again. That catch-up takes 2.21 further days.
Which moon phase is best for observing craters?
First quarter, or the seven days either side of it. Near that phase the terminator, the line dividing day from night, runs roughly straight down the visible disc, so craters near it cast long shadows that reveal relief. At full Moon the Sun is directly behind you, shadows fall behind the features and disappear, and the surface looks flat and overexposed.
Do the phases cause the tides?
They control when the two tides are largest. When the Sun and Moon are aligned, at new and full Moon, their gravitational pulls add and produce spring tides with the largest range. At the quarter phases they act at right angles and partly cancel, giving neap tides with the smallest range. Actual water levels also depend heavily on coastline shape, seabed depth and weather.