Appearance
The first quarter moon is sometimes called a half moon because it appears half-illuminated to viewers on earth. However, since only one side of the moon faces the earth, viewers are really only seeing half of a half, or one-quarter. A waxing gibbous, on the other hand, appears to be more than half illuminated. The night after the first quarter, the difference is negligible, although the left side may appear to be slightly convex instead of straight. Over the next six days, it will grow until it is almost the size of the full moon.
Phase Type
The first quarter is a primary phase, which marks an exact point in the lunar orbit and only lasts for one night. It occurs approximately 7.4 days after the new moon. The waxing gibbous, on the other hand, is a transitional phase, which marks the time between the first quarter and the full moon. It has no set shape or date, but grows from slightly over a quarter to almost half-illuminated during days eight through 13 in the lunar cycle.
Occurrence
When the moon is in its first quarter phase, it is exactly one-fourth of its way through its orbit around the earth. Since the new moon occurs at the point when the earth, moon and sun are at their closest point of alignment, the three bodies form a nearly perfect right angle at the first quarter phase. The gibbous moon, on the other hand, occurs on the nights when the moon is between one-fourth and one-half of the way through its orbit.
Rise and Set Times
The first quarter moon rises at noon, reaches its zenith at sunset, then sets around midnight. It is not uncommon to see this moon during the daylight hours. The waxing gibbous rises and sets later each night, steadily leading up to rise time of the full moon at sunset. The exact time at which the first quarter and gibbous moons rise depend upon the date. You can determine these times by looking at a moonrise/moonset chart for your particular location.