The perception of lunar phases on Earth is not related to the Moon's actual illumination. The Moon's phases are determined by its position relative to the Sun and Earth. The Moon orbits around the Earth, and as it does so, different portions of its sunlit side become visible to observers on Earth. This changing appearance of the Moon is what we perceive as lunar phases.
When the Moon is located between the Earth and the Sun, with the sunlit side facing away from us, we observe a New Moon, and the Moon appears dark to us. As the Moon moves in its orbit, we begin to see a small crescent of illuminated surface, which is called a Waxing Crescent. The illuminated portion gradually increases, leading to the First Quarter Moon, where half of the Moon's visible disk is illuminated.
Continuing its orbit, more of the Moon becomes visible, resulting in a Waxing Gibbous phase. The phase known as a Full Moon occurs when the Earth is between the Sun and the Moon, with the Moon fully illuminated as viewed from our perspective. As the Moon moves past the Full Moon phase, the illuminated area gradually decreases, leading to the Third Quarter Moon, where again half of the visible disk is illuminated.
Finally, the Moon enters the Waning Crescent and continues to decrease in illumination until it reaches the New Moon phase again, and the cycle repeats.
Therefore, the lunar phases are caused by the changing positions of the Sun, Earth, and Moon, and the varying amounts of the Moon's illuminated side that we can see from Earth, rather than a direct relationship between the Moon's illumination and our perception of its phases.