Why does the moon only take 27.3 days to revolve around the earth but 29.5 days to go through the full moon phases

Here's a brief Start at the full Moon. The Moon moves through its orbit and finds itself in the same position relative to the Earth that it was in at the last full Moon. But the Earth has also moved, in its orbit around the Sun, over the same period of time. So, the Moon has to go farther than 360 degrees of orbit in order to get to that full Moon position again. The cycle of phases is about 29.5 days. After 27.3 days, the Moon has revolved through a full 360 degrees relative to the stars.
When a body orbits another and moves to the point where it is in the same position relative to the Sun, that is the "synodic period". The Moon phase cycle is a synodic period. When a body orbits another a full 360 degrees relative to the distant stars, that is a "sidereal period".

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