Area of Sectors and Segments: Explained with Examples

A sector of a circle is the region enclosed by two radii and the arc between them. Picture one slice of a pizza or one wedge of a pie chart — that wedge is a sector. A segment is a different cut: it is the region between a chord and the arc it cuts off. The sector is described by its central angle, written as θ, which is the angle between the two radii that bound it.

Whole circles are rare in real life, but slices of circles are everywhere. Wipers, sprinklers and clock hands all sweep sectors. If you can compute the area of a slice of any circle, you can answer questions about coverage, swept space and material without cutting anything up. Segment areas appear in arch designs and in the cross-section of a partly filled pipe.

Key Concepts

• Radius (r): the distance from the centre of the circle to any point on it, measured in units like cm.
• Central angle (θ): the angle at the centre between the two bounding radii.
• Chord: a straight line joining two points on the circle.
• Minor segment: the smaller region a chord cuts off from a circle.
• Arc length: the curved boundary of a sector, given by (θ ÷ 360) × 2πr.

Formulas

• Area of a sector: A = (θ ÷ 360) × πr², with θ in degrees; in radians it becomes A = (1/2) r² θ.
• Area of a minor segment: sector area minus triangle area, i.e. A = (θ ÷ 360) πr² − (1/2) r² sin θ.
• Area of a major segment: the area of the whole circle minus the minor segment area. All answers are in square units, such as cm².

Worked Example

Find the area of a sector with radius 7 cm and central angle 90°.

1. Write the sector formula: A = (θ ÷ 360) × π × r².
2. Substitute θ = 90, r = 7, and use π = 22 ÷ 7: A = (90 ÷ 360) × (22 ÷ 7) × 49.
3. Simplify 90 ÷ 360 = 1 ÷ 4, so A = (1 ÷ 4) × (22 ÷ 7) × 49.
4. Compute (22 ÷ 7) × 49 = 22 × 7 = 154.
5. Divide by 4: 154 ÷ 4 = 38.5. The sector area is 38.5 cm².

Common Misconceptions

• Using the diameter instead of the radius: the formulas need r, not d — halve any given diameter first.
• Mixing up sector and segment: a sector's straight edges meet at the centre; a segment is bounded by a chord and never touches the centre.
• Forgetting degree–radian consistency: (θ ÷ 360) × πr² needs θ in degrees; with radians, use (1/2) r² θ.

Real-World Applications

• Calculating the ground area covered by a rotating sprinkler or a security camera.
• Estimating material when cutting sector shapes from sheet metal.
• Designing fan blades, pie charts, arch windows and clock faces.
• Finding the cross-sectional area of liquid inside a partially filled cylindrical tank.

A sector is the fraction θ ÷ 360 of a whole circle, and a minor segment is that sector minus its triangle. Use the radius rather than the diameter, and keep the degree and radian formulas separate.