Segment of a Circle: The Region Between a Chord and an Arc

One chord, two segments

Draw any chord across a circle and it splits the whole disc into two pieces. Both are segments. The smaller one, on the side away from the centre, is the minor segment; the larger one, containing the centre, is the major segment.

A diameter is the borderline case. It is still a chord, but it passes through the centre, so the two segments it makes are equal semicircles and neither is minor or major.

How lopsided the split is depends on how far the chord sits from the centre. Slide a chord outwards and the minor segment thins to a sliver; slide it towards the centre and the two segments approach equal halves.

Segment or sector, in one line

A sector is bounded by two straight radii and an arc; a segment is bounded by one straight chord and an arc. Put differently, a sector has a corner at the centre and a segment does not touch the centre at all.

The pictures are easiest to hold onto through food. A sector is a slice of pizza cut from the middle. A segment is what comes away when you cut straight across one edge of the pizza and take the piece with no crust corner in it.

The area is a subtraction, every time

There is no standalone formula worth memorising. Find the sector made by the two radii to the ends of the chord, find the triangle those radii make with the chord, and subtract the triangle from the sector. What remains is the minor segment.

When the centre angle is not 90 degrees, the triangle is not right-angled and its area comes from ½ab sin C, which for two radii simplifies to ½r² sin θ. With r = 8 cm and θ = 120 degrees, the triangle is ½ × 64 × sin 120°, about 27.7 cm², against a sector of about 67.0 cm².

For the major segment, subtract the minor segment from the area of the whole circle rather than starting again. It is fewer steps and fewer places to slip.