The perimeter of a sector of a circle of radius 4cm is ($\pi + 8$)cm. Calculate the anle of the sector

Analyze the question: We are given a sector with radius r = 4 cm and perimeter (π + 8) cm. We need the angle θ at the centre.

Concept: The perimeter of a sector is made of two radii plus the arc length:

Perimeter = 2r + (θ/360) × 2πr

Solution:

1. Two radii: 2 × 4 = 8 cm
2. Arc length = Perimeter − 2r = (π + 8) − 8 = π cm
3. Use the arc length formula:

(θ/360) × 2π × 4 = π

(θ/360) × 8π = π

θ/360 = 1/8

θ = 360/8 = 45°

Check: Arc = (45/360) × 2π × 4 = (1/8) × 8π = π ✓. Perimeter = π + 8 ✓.

Answer: The angle of the sector is 45°, which is option A.

Tip: For sector perimeter questions, always subtract 2r first to isolate the arc length. Many students forget to include the two radii. Once you have the arc, the fraction θ/360 is simply arc ÷ circumference (here π ÷ 8π = 1/8).