Why do we call the second and second last phases of the moon as first 'quarter' and last 'quarter', when we see half of the moon and not quarter of the moon?
The various phases are:
New Moon => Waxing Crescent => First Quarter => Waxing Gibbous => Full Moon => Wanning Gibbous => Last Quarter => Waning Crescent.
It takes our Moon about 29.5 days to completely cycle through all eight phases. This is known as a Lunar month
The different phases of the moon occur in a periodic manner.
The first quarter moon is actually the third phase of the moon each cycle. In the Northern Hemisphere during this phase, the right hand 50% of the moon is visible during the afternoon and the early part of the night. In the Southern Hemisphere the left hand 50% of the moon can be seen. But we officially call this moon a quarter and not a half because it is one quarter of the way around in its orbit of Earth, as measured from one new moon to the next.