Circles
49. The Pizza Slice of Geometry · Exploring the magic of sectors
The region bounded by two radii and an arc always forms a valid sector.
A sector is the region of a circle enclosed by two radii and the arc between them — like a slice of a pie. The angle between the two radii is called the central angle of the sector.
What this lesson covers
Try to break it
Let's pull the handles A and B around the circle. Watch how the 'slice' changes shape but always stays a sector. Notice the two radii and the curved arc holding it together.
How you build it
Draw a sector of a circle.
- Mark the centre point O.
- With centre O, draw the circle.
- Mark point A on the circle.
- Mark point B on the circle.
- Join centre O to A — one radius of the sector.
- Join centre O to B — the sector is the region between the two radii and the arc.
The proof, step by step
Prove that the region bounded by two radii and an arc is a sector.
- A sector is the region enclosed by two radii and an arc of a circle.
- The vertex of a sector is always the centre of the circle.
- Any two radii divide a circle into two sectors: a minor sector and a major sector.
Worked example
In the given figure, O is the centre of the circle. If ∠AOB = 90° and OA = 7 cm, find the area of the minor sector OAPB. (Use π = 22/7)
Area of sector = (θ/360) × πr² = (90/360) × (22/7) × 7² = (1/4) × 22 × 7 = 38.5 cm².
- 38.5 cm² — correct
- 77 cm²
- 154 cm²
- 19.25 cm²