This is tonight's Moon, drawn in the shape it has tonight — the readout says whether it is up. It changes shape every evening and comes back to the same one every 29.5 days. The next steps visit one month, from the first new Moon whose crescent is well placed where you are, and show that one angle decides the shape.
Three evenings after new Moon, in the dusk: a thin
crescent, low in the west. The arc runs to the
Sun, just set, and is the Moon's angle from it — the whole story. The lit edge faces the Sun. The rest of the disc glows faintly:
earthshine, sunlight bounced off the Earth.
Now five more evenings at the same hour. Each evening the Moon stands about 12° further east — further from the Sun — and each evening more of it is lit. The Moon orbits eastward once a month, so it also rises about 50 minutes later each day: 12° is how far the sky turns in 50 minutes.
The same six evenings as a graph: each Moon moved to its angle from the Sun and how much of it was lit. Every evening lands on one curve. The shape is the angle and nothing else.
The first quarter: the Moon a quarter of the way round, 90° from the Sun, high at dusk and lit on the Sun's side. It is called a quarter but we see a half: half of the half that faces the Sun. It sets around midnight, so the evening is when to look — and the terminator's craters are at their sharpest.
The same month from outside, looking down on the Moon's orbit. The line points to the Sun. Watch the Moon go round: half of it is always lit, the half facing the Sun. The phase is only how much of that lit half is turned toward us, and that depends on where the Moon is on its circle.
The full Moon: 180° from the Sun, so it rises in the east as the Sun sets in the west and is up all night. The whole face is lit, and flat — no shadows at the terminator, so the craters vanish. It is the worst night of the month for faint stars.
The last quarter, at dawn: 270° round, a half lit on the other side, facing the Sun about to rise in the east. It rises around midnight, so it belongs to the morning. A week later a thin crescent rises just before the Sun, and then the Moon is lost in its glare for a day or two: new Moon again.