258. Arcs, Segments & Sectors · The three key regions of a circle
Dragging A and B dynamically reshapes the arc, chord, sector, and segment while preserving their definitions.
What this lesson covers
Try to break it
Drag A and B around the circle. The chord AB divides the disc into two segments (chord-and-arc regions). The radii OA and OB cut out two sectors (pie-slice regions). Bring A and B together and both the segment and sector shrink to zero; spread them to opposite ends and AB becomes a diameter — the two segments become two semicircles.
How you build it
Draw a circle with a sector and a chord.
- Point tool: mark the centre O.
- Circle tool: click O, then drag out to set the radius.
- Point tool: mark A on the circle.
- Point tool: mark B on the circle.
- Segment tool: join O to A — a radius.
- Segment tool: join O to B — the second radius; OA, OB and the arc bound the sector.
- Segment tool: join A to B — the chord; with the arc it bounds the segment.
The proof, step by step
Prove that the chord and arc divide the circle into segments and sectors.
- An arc is simply a part of the circle's circumference.
- A segment is the region bounded by an arc and its chord.
- A sector is the region bounded by an arc and two radii.
- These definitions are structural: they remain true regardless of how large or small the arc becomes.
Worked example
In a circle, the region bounded by an arc and two radii joining the centre to the endpoints of the arc is called a:
By definition, a sector is the region enclosed by two radii and the intervening arc. A segment is bounded by a chord and an arc.
- Sector — correct
- Segment
- Chord
- Diameter