how long does it take for the moon phases
You may have personally observed that the moon goes through a complete moon phases cycle in about one month. That's true, but it's not exactly one month. The synodic period or lunation is exactly 29.5305882 days. It's the time required for the moon to move to the same position (same phase) as seen by an observer on earth. If you were to view the moon cycling the earth from outside our solar system (the viewpoint of the stars), the time required is 27.3217 days, roughly two days less.