Moon phase calculator
In the northern hemisphere a waxing moon is lit on the right and a waning one on the left. The mnemonic is that it fills and empties like the letters D, O, C. South of the equator everything is inverted, because you are effectively looking at the same moon upside down — which is why the mnemonic confuses so many people who learned it in one hemisphere and moved to the other.
Uses the mean synodic month, which is accurate to within a few hours. The real orbit is elliptical and perturbed, so an ephemeris is needed for the exact minute of a new or full moon.
The moon cycles through its phases every 29.53 days on average. Illumination follows the cosine of the phase angle, reaching zero at new moon and 100% at full. This uses the mean synodic month, accurate to within a few hours.
How to find the moon phase
The synodic month of 29.53 days is longer than the moon actual orbital period of 27.32 days, and the difference matters. The orbital period is one full circuit against the stars; the synodic month is the time to return to the same phase, which takes longer because the Earth has moved along its own orbit in the meantime and the moon has to catch up. Phases follow the synodic month; the moon position among the constellations follows the sidereal one.
Questions
29.53 days on average — the synodic month.