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The Moon does not actually change shape — it is always a sphere, half-lit by the Sun. What changes is how much of that lit half we can see from Earth as the Moon orbits us. That changing view is what we call a "moon phase."
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What causes moon phases
The Moon orbits Earth roughly once a month. As it does, the angle between the Sun, Earth, and Moon constantly changes, so we see a different portion of the Moon's sunlit half from night to night. It has nothing to do with Earth's shadow (that only happens briefly during a lunar eclipse) — it is purely about viewing angle.
The synodic month
A full cycle from one new moon to the next takes about 29.53 days — called a synodic month. This is slightly longer than the Moon's actual orbital period around Earth (about 27.3 days) because Earth is also moving around the Sun, so the Moon needs a bit of extra travel to catch up to the same Sun-Earth-Moon alignment.
The 8 named phases
New Moon — the Moon is between Earth and the Sun; the near side is unlit and the Moon is not visible.
Waxing Crescent — a thin sliver of light appears and grows night by night.
First Quarter — exactly half the visible disk is lit; illumination is increasing ("waxing").
Waxing Gibbous — more than half lit, still growing toward full.
Full Moon — the entire visible face is lit; Earth is roughly between the Sun and Moon.
Waning Gibbous — more than half lit, now shrinking ("waning") after full.
Last Quarter (Third Quarter) — half lit again, on the opposite side from First Quarter, and still shrinking.
Waning Crescent — a thin sliver remains before the cycle returns to New Moon.
Waxing vs waning, at a glance
"Waxing" always means the illuminated portion is growing (heading toward Full). "Waning" always means it is shrinking (heading toward New). If you remember that Full and New are the two anchor points, the four phases in between just describe how far along and in which direction the Moon is between them.
A quick note on supermoons and blue moons
A "supermoon" is a Full Moon that happens to occur near the point in the Moon's slightly elliptical orbit where it is closest to Earth, making it appear a little larger and brighter than average — not a different phase. A "blue moon" is the informal name for a second Full Moon occurring within the same calendar month, which happens roughly every two to three years purely because the synodic month (29.53 days) is shorter than most calendar months.
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