Sector of a circle

Properties

• A sector with angle 90° is a quarter circle, called a quadrant. With 180° it is a semicircle.
• The arc length and the area are both in the same proportion to the whole circle as the angle is to 360°.
• The smaller slice is the minor sector and the rest of the circle is the major sector.

Why the formulas work

A full circle has 360° at its centre. A sector with angle θ is the fraction θ/360 of the circle, so it has that fraction of the area πr² and that fraction of the circumference 2πr. The perimeter of the slice is the arc plus the two straight radii.

Worked example

Take a sector of a circle with radius 21 cm and angle (degrees) 60°.

1. Area. Area = (θ ⁄ 360) × πr². Here: (60 ⁄ 360) × π × 21 × 21 = 230.91 cm² (with π = 22/7: 231 cm²).
2. Arc length. Arc length = (θ ⁄ 360) × 2πr. Here: (60 ⁄ 360) × 2 × π × 21 = 21.99 cm (with π = 22/7: 22 cm).
3. Perimeter. Perimeter = arc length + 2r. Here: 21.99 + 2 × 21 = 63.99 cm (with π = 22/7: 64 cm).

Finding a missing measurement

From the arc length, θ = arc × 360 ÷ 2πr. An arc of 22 cm on a circle of radius 21 cm gives θ = 22 × 360 ÷ 132 = 60°. A handy shortcut: the area of a sector is ½ × arc length × radius, here ½ × 22 × 21 = 231 cm².

Mistakes to avoid

• Forgetting the two radii in the perimeter.
• Using the angle in degrees in a formula meant for radians. With θ in radians the formulas are arc = rθ and area = ½r²θ.
• Finding the area of the wrong sector. Check whether the question wants the minor or the major one.

Where you see it

A slice of pizza or cake, the area swept by a windscreen wiper or a clock hand, a hand fan, and the slices of a pie chart are sectors.

Try these

Work each one out, then tap to check.

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