Question

An arc of a circle is of length 5 $\pi$ cm and the sector it bounds has an area of 20 $\pi c{m}^2$. Find the radius of the circle.

Answer 1

Let the radius of the circle be r cm and the arc Ab of length 5$\pi$ cm subtends abgle $\theta$ at the centre O of the circle. Then ,
Arc $AB=5$ $\pi$ cm and Area of sector $OAB=20$ $\pi c{m}^2$

$\Rightarrow \frac{\theta }{360}\times 2\pi r=5\pi and\frac{\theta }{360}\times \pi r^2=20\pi$

$\Rightarrow \frac{\frac{\theta }{360}\times \pi r^2}{\frac{\theta }{360}\times 2\pi r}=\frac{20\pi }{5\pi }$

$\Rightarrow \frac{r}{2}=4$

$\Rightarrow$ $r=8cm$

We have Area $=$ $\frac{1}{2}lr\Rightarrow 20\pi =\frac{1}{2}\times 5\pi \times r\Rightarrow r=8cm$