Arc length & sector area

An arc is part of the circumference of a circle; a sector is a "pie slice" region. Arc length and sector area are calculated using the angle at the centre as a fraction of 360°.

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Key facts to remember

1Arc length = (θ/360°) × 2πr
2Sector area = (θ/360°) × πr²
3θ is the angle at the centre of the sector in degrees.
4A minor sector has θ < 180°; a major sector has θ > 180°.
5Perimeter of a sector = arc length + 2 radii.

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Formulas

Arc length

l = (θ/360) × 2πr

Sector area

A = (θ/360) × πr²

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Worked examples

Example 1

Find the arc length and area of a sector with radius 8 cm and angle 135°.

Working

Arc length = (135/360) × 2π × 8 = (3/8) × 16π = 6π ≈ 18.85 cm
Sector area = (135/360) × π × 8² = (3/8) × 64π = 24π ≈ 75.40 cm²

AnswerArc length = 6π cm ≈ 18.85 cm; Sector area = 24π cm² ≈ 75.40 cm²

Example 2

A sector has an arc length of 10π cm and radius 12 cm. Find the angle θ.

Working

10π = (θ/360) × 2π × 12
10π = (θ/360) × 24π
10 = (θ/360) × 24
θ = (10/24) × 360 = 150°

Answerθ = 150°

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Common mistakes

✗Using diameter instead of radius in the formulas.

✗Forgetting to add the two radii when finding the perimeter of a sector.

✗Not converting the angle to a fraction of 360° before multiplying.

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Exam tips

✓Remember: arc length uses 2πr (circumference); sector area uses πr² (circle area) — just multiply by θ/360.

✓Leave your answer in terms of π for exact values unless asked to round.

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