during a quarter moon, half of the moon is visible on earth. explain why it is called a quarter moon.

The term "quarter moon" refers to the Moon being one quarter of the way through its orbit around the Earth, not the amount of the Moon's surface that appears lit. During a quarter moon, half of the Moon's face visible from Earth is illuminated by the Sun, so it looks like a half moon. The "quarter" in the name denotes the Moon's position in its orbital cycle, which is at about 90 degrees (a quarter of a full 360-degree orbit) from the new moon phase. Thus, the quarter moon is called so because it marks one quarter of the lunar orbit, even though half of the Moon appears lit to an observer on Earth.