A sector has radius 12 cm and angle 150°. Find its area, correct to 3 significant figures.
- Fraction of the circle = 150/360 = 5/12.
- Area of the whole circle = π × 122 = 144π cm2.
- Sector area = 5/12 × 144π = 60π = 188.49...
Answer: 188 cm2 (60π cm2)
Circles, arcs and sectors
Maths · Mensuration. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
The circumference is the distance round a circle. Both the circumference and the area are found from the radius, so the first step is often to find the radius. Remember that the radius is half the diameter.
An arc is part of the circumference and a sector is part of the area, like a slice of pizza. The angle at the centre tells you the fraction of the circle: an angle of 90° is 90/360, which is one quarter. So an arc is that fraction of the circumference and a sector is that fraction of the area.
Answers are given either in terms of π or as a decimal correct to 3 significant figures.
Rules to remember
- Circumference = 2 × π × radius = π × diameter (C = 2πr = πd)
- Area of a circle = π × radius2 (A = πr2)
- Arc length = (angle ÷ 360) × 2πr
- Sector area = (angle ÷ 360) × πr2
- Perimeter of a sector = arc length + 2 × radius
- Perimeter of a semicircle = πr + 2r (half the circumference plus the diameter)
- Angle of a sector = (arc length ÷ circumference) × 360° = (sector area ÷ area of circle) × 360°
Common mistake
Students give the arc length as the perimeter of a sector. The perimeter also includes the two straight sides, so add two radii.
Worked example
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