an arc, sector and segment

Cut a pizza and you meet all three at once. The curved crust edge of a slice is an arc; the whole triangular-ish slice, crust and two straight cuts together, is a sector; and the curved sliver you would get by cutting straight across with a single chord (skipping the centre) is a segment. Each names a different piece of the disk or its rim.

Precisely: an arc is a connected portion of the circle itself — the curved boundary between two points on the rim. Two points cut the rim into a shorter minor arc and a longer major arc; a diameter cuts it into two semicircles. A sector is the pie-slice region bounded by an arc and the two radii to its endpoints — the area swept from the centre. A segment (of a circle, not to be confused with a line segment) is the region between an arc and the chord joining its endpoints — the slice you get from one straight cut, the part 'cut off' from the disk. An arc's size is measured by its central angle (a semicircle is a 180-degree arc), in degrees or in radians.

These three carve up the circle's interior and boundary in the ways every later circle theorem needs: inscribed angles are measured against arcs, sectors are the natural unit for the area and the slice-fraction theta/360 of a disk, and segments appear in chord problems. The naming trap is precisely the word 'segment': in circle geometry it is a curved-and-flat region of the disk, whereas a 'line segment' is a straight piece of a line — same word, different objects. Also, an arc by itself has a length and an angular measure but no area; the area belongs to the sector or segment region it borders. The numeric formulas for these areas belong to Measurement, not here.