Sector — Definition, Formula & Examples

A sector is the region of a circle enclosed between two radii and the arc that connects their endpoints. Think of it as a pizza slice — the pointed end is at the center of the circle, and the curved edge is part of the circle itself.

Given a circle with center $O$ and radius $r$ , a sector is the set of all points that lie on or between two radii $\overline{OA}$ and $\overline{OB}$ and on or inside the arc $\stackrel{\frown}{AB}$ subtended by the central angle $\theta = \angle AOB$ .

Key Formula

A = \frac{\theta}{360°} \cdot \pi r^2

Where:

$A$ = Area of the sector
$\theta$ = Central angle of the sector in degrees
$r$ = Radius of the circle

Worked Example

Problem: Find the area of a sector with a central angle of 90° in a circle of radius 6 cm.

Find the fraction of the circle: The sector covers 90° out of a full 360° circle.

\frac{90°}{360°} = \frac{1}{4}

Compute the full circle's area: Use the area formula for a circle with radius 6 cm.

\pi r^2 = \pi (6)^2 = 36\pi

Multiply to get the sector area: Take one-quarter of the full area.

A = \frac{1}{4} \cdot 36\pi = 9\pi \approx 28.27 \text{ cm}^2

Answer: The sector has an area of

9\pi \approx 28.27

cm².

Why It Matters

Sectors appear whenever you divide circular objects into portions — pie charts, clock faces, and radar sweeps are all real-world sectors. In later courses like trigonometry and calculus, sectors measured in radians become essential for computing arc length and angular motion.

Common Mistakes

Mistake: Confusing a sector with a segment. A segment is the region between a chord and its arc, while a sector is the region between two radii and an arc.

Correction: Remember that a sector always has its "point" at the center of the circle (like a pizza slice), whereas a segment is cut off by a straight chord that does not pass through the center.