Arcs and sectors of circles
A sector of a circle is a "pizza slice" — bounded by two radii and the arc between them. The arc and sector area are fractions of the full circle, scaled by (angle / 360°).
Formulas
For a sector with central angle θ° and radius r:
- Arc length = (θ / 360) × 2πr.
- Sector area = (θ / 360) × πr².
Equivalently, in radians (rare for GCSE): arc = rθ; sector = ½r²θ.
✦Worked example— Worked examples
Example 1. Circle radius 6 cm, sector angle 90°.
- Arc = (90/360) × 2π × 6 = (1/4) × 12π = 3π cm ≈ 9.42 cm.
- Sector area = (90/360) × π × 36 = (1/4) × 36π = 9π cm² ≈ 28.27 cm².
Example 2. Find the angle of a sector if arc length = 5π cm and radius = 10 cm.
- 5π = (θ/360) × 2π × 10.
- 5π = (θ/360) × 20π.
- θ/360 = 1/4 → θ = 90°.
Perimeter of a sector
Perimeter = arc length + 2 × radius (the two straight sides).
Worked example: sector radius 8 cm, angle 60°.
- Arc = (60/360) × 16π = (1/6) × 16π = (8/3)π.
- Perimeter = (8/3)π + 16 ≈ 24.4 cm.
Areas of segments
A segment is bounded by a chord and an arc. To find the segment area:
Segment area = Sector area − Triangle area (where the triangle has two sides = radii and the included angle = sector angle).
Triangle area = ½ × r × r × sin(θ) = ½r² sin θ (using G23).
⚠Common mistakes
- Using degrees with the radian formula — pick one system.
- Forgetting (θ/360) factor.
- Including only the arc as perimeter — must include the two radii too.
- Using r² in arc formula — arc uses r (linear); area uses r².
- Confusing major and minor sector — the angle dictates which.
➜Try this— Quick check
Sector with radius 9 cm, angle 120°. Arc length and area?
- Arc = (120/360) × 2π × 9 = (1/3) × 18π = 6π cm ≈ 18.85 cm.
- Area = (120/360) × π × 81 = 27π ≈ 84.82 cm².
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