Take the triangle away

A cut in a sector under a half turn makes a triangle.

The rule

A sector under a half turn has two radii. They are its straight sides from the centre. A chord is a straight piece joining two points on the rim. Here a chord joins the outer ends of the radii. The disc is the flat area inside the circle. A chord that misses the centre splits the disc. The two parts are unequal. The smaller part is the minor segment. It lies between the chord and the short arc. The larger part is the major segment. Major segment = πr² − minor segment. The chord and the two radii form a triangle. So the sector is the triangle and the segment together.

A minor segment's area is its sector's area minus its triangle.

"Segment" means the minor segment unless a question says major.

At a quarter turn, radius 7: sector = 38.5. The triangle has two sides of 7 at a right angle. It covers half of 7 × 7. That is half of 49, which is 24.5.

Segment = 38.5 − 24.5 = 14 square units.

At a half turn, the two radii lie in one straight line. No triangle closes there. Sector = half of 154 = 77, triangle = 0, segment = 77. The segment is the whole half-disc.

Tap an angle. Watch the subtraction.