Find the area (in cm²) of the sector whose perimeter is 64/3 cm and central angle is 60°. Use π = 22/7.

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Accepted Answer

Given:
- Perimeter of sector = 64/3 cm
- Central angle θ = 60°
- π = 22/7

Step 1: Let the radius of the circle be r cm.

Step 2: The perimeter (P) of a sector is given by:
P = 2r + arc length
Arc length = (θ/360) × 2πr = (60/360) × 2 × (22/7) × r = (1/6) × 2 × (22/7) × r = (44/42)r = (22/21)r

So,
P = 2r + (22/21)r = (2 + 22/21)r = (42/21 + 22/21)r = (64/21)r

Step 3: Given P = 64/3,

(64/21)r = 64/3

Multiply both sides by 21:
64r = (64/3) × 21
64r = 64 × 7
64r = 448

Divide both sides by 64:
r = 448 / 64 = 7 cm

Step 4: Area of sector = (θ/360) × πr²
= (60/360) × (22/7) × 7²
= (1/6) × (22/7) × 49
= (1/6) × 22 × 7
= (1/6) × 154
= 154 / 6 = 77 / 3 cm²

Answer: 77/3 cm²

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