Areas of Sectors and Segments

A sector of a circle is the region enclosed by two radii and the arc between their endpoints. Think of it as a slice of pie. The central angle θ (theta) is the angle at the centre between the two radii. If θ = 360° the sector is the entire circle; if θ = 90° it is a quarter-circle.

Because the full circle has area πr², a sector with central angle θ° takes up (θ/360) of the full circle. Therefore: A_sector = (θ/360) × πr². For example, a sector with r = 6 cm and θ = 90° has area (90/360) × π × 36 = 9π ≈ 28.27 cm².

When the central angle is measured in radians (θ rad), the formula simplifies beautifully. Since a full circle is 2π radians, the fraction of the circle is θ/(2π), giving A_sector = (θ/2π) × πr² = (1/2)r²θ. This compact formula is preferred in higher mathematics. For example, a sector with r = 5 m and θ = π/3 rad has area (1/2)(25)(π/3) = 25π/6 ≈ 13.09 m².

A segment (also called a circular segment) is the region between a chord and the arc it cuts off. It is NOT the same as a sector. Every chord divides a circle into a minor segment (smaller region) and a major segment (larger region). To find the area of a segment, we subtract the area of the triangle formed by the two radii and the chord from the area of the sector containing those radii.

The triangle formed by the two radii (each of length r) and the chord has an included angle of θ (the central angle). Using the formula for the area of a triangle with two sides and an included angle: A_triangle = (1/2)r² sin θ. This works whether θ is in degrees (use sin θ°) or radians (use sin θ).

Combining the above: A_segment = A_sector − A_triangle = (1/2)r²θ − (1/2)r² sin θ = (1/2)r²(θ − sin θ), where θ is in radians. If working in degrees, use A_segment = (θ/360)πr² − (1/2)r² sin θ°. Always check whether you want the minor segment (using the smaller angle) or the major segment (using 360° − θ or 2π − θ).

Area is always expressed in square units (cm², m², etc.). A sector area must be less than the full circle area (πr²), and a segment area must be less than its corresponding sector area. Always verify: if θ is small, the segment area should be very small; if θ approaches π radians (180°), the segment approaches a semicircle.

• A_{\text{sector}} — Area of sector (square units (e.g. cm², m²)): The area of the pie-slice shaped region bounded by two radii and an arc.
• A_{\text{segment}} — Area of segment (square units (e.g. cm², m²)): The area of the region between a chord and its arc, equal to the sector area minus the triangle area.
• r — Radius (length units (e.g. cm, m)): The distance from the centre of the circle to any point on its circumference.
• \theta — Central angle (radians): The angle at the centre of the circle between the two radii defining the sector or segment.
• \sin\theta — Sine of the central angle (dimensionless): Used to compute the area of the isosceles triangle formed by the two radii and the chord.

Problem. A circle has radius 8 cm. A sector has a central angle of 2π/3 radians (120°). Find: (a) the area of the sector, and (b) the area of the minor segment cut off by the chord connecting the two radii.

Answer. Area of sector ≈ 67.02 cm²; Area of minor segment ≈ 39.31 cm²

• Arc Length of a Circle — Arc length uses the same central angle and radius as sector area; together they fully describe the boundary and interior of a sector.
• Area of a Circle — The sector area formula is a fraction of the full circle area, so understanding A = πr² is the essential foundation.
• Radian Measure — The compact formula A = (1/2)r²θ requires the angle to be in radians; converting between degrees and radians is a prerequisite skill.
• Area of a Triangle Using Sine — The segment area formula relies on A = (1/2)r² sin θ for the triangle inside the sector, directly applying the sine area formula.
• Perimeter of Sectors and Segments — Alongside area, finding the perimeter (arc length plus straight edges) completes the full geometric description of sectors and segments.