How many days does it take the Moon to cycle through all of its phases, from full moon to full moon?

How to get close to this number

You already know this one approximately, because it is where the month comes from. Twelve cycles fit into a year with a bit left over, so dividing 365 by 12 gives you about thirty — and the true figure is slightly under that. Working backwards from the calendar is the most reliable route, because the calendar was built from this number in the first place.

The subtlety in the explanation is the part worth understanding. The Moon completes an orbit in a shorter time than it completes a phase cycle, because while it orbits, Earth has moved along its own path around the Sun — so the Moon needs extra travel to return to the same alignment. Phases are about the geometry between three bodies, not about one orbit.

That distinction has a name and it matters beyond astronomy: a sidereal month measures the orbit against the stars, a synodic month measures it against the Sun. Whenever a period seems to have two different published values, the usual explanation is that two different reference frames are being used — and both figures are correct for what they measure.