Question

An arc of length $4\pi cm$ subtends an angle of ${40}^{\circ }$ at the centre. The area of the sector formed by this arc is $2x\pi$. Find $x$.

Answer 1

Length of an arc subtending an angle $\theta =\frac{\theta }{360}\times 2\pi R$ where R is the radius of the circle.

So, $4\pi =2\ast \frac{40}{360}\times \pi \times R$
Hence, Radius of the circle $=18cm$

Area of a sector of a circle of radius 'r' and angle
$\theta =\frac{\theta }{360}\pi r^2$
Hence, area of the sector of the circle of radius $18$ cm and angle ${40}^0=\frac{40}{360}\times \pi \times 18\times 18=36\pi {cm}^2$

Thus, $2\pi x=36\pi$

$\to x=18$