Phasia

Meet Phasia, the Moon, who is always half lit even when she looks like a thin crescent.

The Sun always lights the half of the Moon facing it. As the Moon orbits Earth, we view that lit half from changing angles. At new Moon the lit side faces away from us; at full Moon it faces us. In between we see part of each, giving crescents, quarters and gibbous phases. If φ is the angle between the Sun and the Moon as seen from Earth, the lit fraction of the disc is (1 − cos φ)/2.

At φ = 90°, a quarter Moon, cos φ = 0, so exactly half the disc is lit. At 60° it is a quarter lit; at 120°, three quarters. A full cycle of phases takes about 29.5 days.

Phases are not caused by Earth's shadow; that would be a lunar eclipse, which happens only at full Moon. Planets closer to the Sun than Earth, like Venus, show phases too, which Galileo observed in 1610.

The formula treats the Sun as very far away compared with the Moon, which is an excellent approximation.