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O! TL;DR  reference

The Moon phase cycle, explained

tl;dr
  • Phases come from the changing angle between Sun, Earth and Moon — not from Earth's shadow.
  • One full cycle (new → full → new) is the synodic month: about 29.53 days.
  • The four primary phases are instants; the four in-between phases are the weeks spanning them.

The eight phases in order

PhaseTypeSun–Moon angleLit side visibleWhen visible
New MoonPrimary~0%Up with the Sun (not visible)
Waxing CrescentIntermediate0–90°risingEvening, west
First QuarterPrimary90°50%Afternoon to midnight
Waxing GibbousIntermediate90–180°risingLate afternoon to late night
Full MoonPrimary180°100%All night
Waning GibbousIntermediate180–270°fallingLate night to morning
Last QuarterPrimary270°50%Midnight to noon
Waning CrescentIntermediate270–360°fallingMorning, east

Synodic vs sidereal: two different months

The Moon takes about 27.3 days to orbit Earth once relative to the stars — the sidereal month. But while it orbits, Earth is also moving around the Sun, so the Moon needs roughly two extra days to 'catch up' to the same Sun–Earth–Moon alignment.

That alignment cycle — the time between one new moon and the next — is the synodic month, about 29.53 days. Phases follow the synodic month, which is why a calendar month usually holds one full moon, and occasionally two (a 'blue moon').

Waxing, waning, gibbous, crescent

'Waxing' means the lit portion is growing (new toward full); 'waning' means it is shrinking (full toward new). 'Crescent' is less than half lit; 'gibbous' is more than half but not full.

A lunar eclipse is a separate event: it only happens at a full moon when the Moon passes through Earth's shadow, which is why eclipses are rare rather than monthly.

Live Moon phase & lunar calendar on astro.otldr.com