First quarter and last quarter appear as a half moon.
about halfway down this page are some pictures: http://www.calculatorcat.com/moon_phases/moon_phases.phtml
The phase of the moon that follows the waning gibbous phase is the third quarter phase. During this phase, half of the moon is illuminated, and it appears as a half-moon. The third quarter occurs roughly a week after the full moon and signifies the moon's transition towards the new moon phase.
After the half-moon phase, the moon's lighted side appears to grow smaller every night, transitioning towards the new moon phase where the moon is not visible in the sky. This progression is known as the waning crescent phase.
One week after the full moon, the moon is in its third quarter phase. During this phase, the left half of the moon appears illuminated when viewed from the Northern Hemisphere, while the right half appears illuminated from the Southern Hemisphere. This is because the sunlight is shining on the opposite side of the moon compared to the full moon phase.
"Waxing half" typically refers to the phase of the moon when it is transitioning from a new moon to a full moon, specifically when half of the moon's surface is illuminated. This phase is also known as the first quarter moon. During this time, the right half of the moon appears lit in the Northern Hemisphere, while the left half appears lit in the Southern Hemisphere. The term "waxing" indicates that the illuminated portion is increasing.
When the moon appears as a half-illuminated disk, and the illuminated portion is growing from one night to the next,the phase is known as "first quarter".
After the waxing crescent phase, the Moon enters the first quarter phase. During this phase, half of the Moon's surface is illuminated as seen from Earth, and it appears as a half-circle. The first quarter occurs roughly one week after the new moon, as the Moon continues to wax towards the full moon phase.
The moon is in its first quarter phase. It's moved about a quarter of the way around its orbit since the new moon. The sun, moon, and Earth form a right angle (90 degrees) during this phase, causing half of the moon facing Earth to be lit up and the other half to be dark.
The moon was in its first quarter phase on Friday, appearing as a half moon. It transitioned to a waxing gibbous phase on Saturday, where it appears more than half full but less than full.
When the moon appears half lit, you are seeing exactly half of its surface. Half of the moon is illuminated by the sun, while the other half is in shadow. This phase is known as the first quarter when the moon is waxing, and the third quarter when the moon is waning.
If the moon rises halfway between sunset and midnight, it is in its first quarter phase. During this phase, the moon is positioned at a right angle to the Earth and the Sun, allowing us to see half of the moon's surface illuminated. This means it appears as a half-moon in the sky.
"First Quarter" is the phase when the moon appears half-illuminated, and rises and sets roughly 6 hours (1/4 day) after the sun.
The phase that follows the new moon phase when it appears as a small outer portion of a circle is the waxing crescent phase. This phase occurs as the illuminated portion of the Moon gradually increases in size.