Question

If the area of the circle is $25\pi$ sq. cms, find the length of its are subtending and angle of ${144}^o$ at the centre. Also find the area of the corresponding sector.

Answer 1

As we know that area of circle $\left(A\right)$ is given by-

$A=\pi r^2$

Given that:-

Area of circle $=25\pi {cm}^2$

$\therefore \pi r^2=25\pi$

$\Rightarrow r=\sqrt{25}=5cm$

As we know that length of subtending arc $\left(l\right)$ is-

$l=r\theta$

where $\theta$ is the angle subtended by the arc

Given that tha angle subtended by the arc is $144°$

$\theta =144°=144°\times \frac{\pi }{180°}=\frac{4\pi }{5}$

$\therefore l=5\times \frac{4\pi }{5}=4\pi cm$

As Area of sector $=\frac{1}{2}\times r\times l$

$\therefore$ Area of corresponding sector $=\frac{1}{2}\times 5\times 4\pi =10\pi {cm}^2$

Hence the length of subtending arc is $4\pi cm$ and the area of corresponding sector is $10\pi {cm}^2$.