Why are half moon called quarter moons?

Half moons are often referred to as quarter moons because they represent a phase in the lunar cycle where the moon is one-quarter of the way through its orbit around the Earth. This occurs when the moon is either in its first or third quarter, leading to half of its surface being illuminated by the Sun. The terminology can be confusing, as "quarter" refers to the position in the lunar cycle rather than the visual appearance of the moon.