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To find the area and perimeter of a sector of a circle with a radius of 21 cm and an angle of 120^.

Mathematics

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The question asks to find the area and perimeter of a sector of a circle with a radius of 21 cm and an angle of 120∘120^\circ.

Given:
Radius, r=21r = 21 cm
Angle, θ=120∘\theta = 120^\circ

  1. Calculate the area of the sector.

    The formula for the area of a sector is A=θ360∘×πr2A = \frac{\theta}{360^\circ} \times \pi r^2.

    A=120∘360∘×227×(21 cm)2A = \frac{120^\circ}{360^\circ} \times \frac{22}{7} \times (21 \text{ cm})^2 A=13×227×441 cm2A = \frac{1}{3} \times \frac{22}{7} \times 441 \text{ cm}^2 A=13×22×63 cm2A = \frac{1}{3} \times 22 \times 63 \text{ cm}^2 A=22×21 cm2A = 22 \times 21 \text{ cm}^2 A=462 cm2A = 462 \text{ cm}^2

    The area of the sector is 462 cm2\boxed{\text{462 cm}^2}.

  2. Calculate the perimeter of the sector.

    The formula for the perimeter of a sector is P=θ360∘×2πr+2rP = \frac{\theta}{360^\circ} \times 2\pi r + 2r.

    P=120∘360∘×2×227×21 cm+2×21 cmP = \frac{120^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21 \text{ cm} + 2 \times 21 \text{ cm} P=13×2×22×3 cm+42 cmP = \frac{1}{3} \times 2 \times 22 \times 3 \text{ cm} + 42 \text{ cm} P=44 cm+42 cmP = 44 \text{ cm} + 42 \text{ cm} P=86 cmP = 86 \text{ cm}

    The perimeter of the sector is 86 cm\boxed{\text{86 cm}}.

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