The perimeter of a sector of a circle of radius 21cm is 64cm. Find the angle of the sector [Take $\pi = \frac{22}{7}$]

Analyze the question: The perimeter of a sector is made up of two radii plus the arc length. We need the angle θ at the centre.

Concept:

• Perimeter = 2r + arc length
• Arc length = (θ/360) × 2πr

Solution:

1. Two radii: 2 × 21 = 42 cm
2. Arc length = 64 − 42 = 22 cm
3. Use the arc formula:

(θ/360) × 2 × (22/7) × 21 = 22

2 × (22/7) × 21 = 2 × 22 × 3 = 132

So (θ/360) × 132 = 22

4. θ = (22 × 360) / 132 = 7920 / 132 = 60°

Check: Arc = (60/360) × 132 = 22 cm, and 22 + 42 = 64 cm ✓

Tip: For sector perimeter questions, always remember to add the two radii to the arc length. Many students forget this and lose marks. Subtract 2r from the perimeter first to get the arc, then find θ.