Sector of a circle
Also known as: circular sector
A sector of a circle is the pie-slice region bounded by two radii and the arc between them. Its area is (θ/360) × πr², where θ is the central angle in degrees and r is the radius.
A sector of a circle is the region enclosed by two radii and the arc connecting their endpoints — the shape of a slice of pie or pizza. The angle between the two radii, measured at the center, is called the central angle (θ). A sector with a central angle less than 180° is a minor sector; the remaining, larger piece is the major sector.
Sector formulas come straight from proportional reasoning: a sector is just a fraction of the whole circle, and that fraction is θ/360. So the area of a sector is (θ/360) × πr², and the arc length along its edge is (θ/360) × 2πr. For a circle with radius 6 and a 60° central angle, the sector's area is (60/360) × π(6²) = 6π, and its arc length is (60/360) × 12π = 2π.
Special cases make the idea concrete: a 90° sector is a quarter circle, a 180° sector is a semicircle, and a 360° "sector" is the entire circle. If the angle is given in radians, the fraction becomes θ/2π, giving area = (1/2)r²θ and arc length = rθ.
Sectors show up regularly on standardized math tests, including the Praxis Core math exam, which asks for sector area or arc length given a central angle, or reverses the problem by giving the area and asking for the angle or radius. Remembering that a sector is simply "fraction of the circle times the circle formula" is usually all you need.
Key takeaways
- A sector is the pie-slice region between two radii and the arc connecting them.
- Sector area = (θ/360) × πr², where θ is the central angle in degrees.
- Arc length = (θ/360) × 2πr — the same fraction applied to the circumference.
- A 90° sector is a quarter circle; 180° is a semicircle.
- The Praxis Core math exam tests sector area and arc length as part of circle geometry.