Question

The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector. 

Answer 1

Let the radius of circle be r and the arc length is ${\displaystyle l}$

Perimeter of a sector is given as      ...[∵ ${\displaystyle l}$ = rθ]

∴ p = 2r + rθ   

r = 5.6 m

P = 20 m

⇒ 20 = (2 × 5.6) + 5.6 θ

20 − 11.2 = 5.6 θ

⇒ ${\displaystyle \frac{8.8}{5.6}=\theta }$

θ = ${\displaystyle \frac{11}{7}}$ radian       ...${\displaystyle [\because 1\text{radian}=\frac{180}{\pi }\text{degree} \pi =\frac{22}{7}]}$

${\displaystyle \theta =\frac{11}{7}\times \frac{180}{\pi }\text{degree}}$

θ = 90° degree

Area of sector = ${\displaystyle \frac{\theta }{360}\times \pi r^2}$

= ${\displaystyle \frac{90}{{360}^{\text{°}}}\times \frac{22}{7}\times 5.6\times 5.6}$

= ${\displaystyle \frac{1}{4}\times \frac{22}{7}\times 5.6\times 5.6}$

= ${\displaystyle \frac{689.92}{28}}$

Area = 24.64 m²