Arcs and sectors
Objective. Calculate arc lengths, sector areas and missing angles.
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- 2. Worked example
- 3. Practice
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- 5. Continue
Learn: the key idea
Arcs and sectors deal with portions of a circle defined by an angle at the centre, using fractions of the full circle's circumference and area. Arc length = (θ/360) × 2πr and sector area = (θ/360) × πr², where θ is the angle in degrees. For example, a sector with angle 90° of a circle radius 4 cm has arc length (90/360) × 2π×4 = 2π cm. Rearranging these formulae also lets you find a missing angle or radius if the arc length or sector area is given instead.
Worked example
Find the arc length of a 120° sector of a circle with radius 9 cm. Give your answer in terms of π.
- 1Arc length = (θ/360) × 2πr.
- 2Substitute θ = 120 and r = 9: 120/360 × 2π × 9.
- 3120/360 = 1/3 and 2π × 9 = 18π.
- 4Arc length = 6π cm.
Answer: 6π cm
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Start practisingCommon mistake. A common slip is forgetting to divide by 360 before multiplying by the circle formula, giving a full circumference or area instead of a fraction of it; always write the fraction θ/360 explicitly first.
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