Question

Area of a sector of a circle of radius $36cm$ is $54\pi c{m}^2$. Find the length corresponding arc of the sector.

Answer 1

Let central angle of the sector be $\theta$ and the length of the arc of the sector be $s.$

$\begin{array}{cc}\text{Area of Sector}& =\frac{\theta }{{360}^{\circ }}\times \pi r^2\\ 54\pi & =\frac{\theta }{{360}^{\circ }}\times \pi \times {\left(36\right)}^2\\ 54& =\frac{\theta }{{10}^{\circ }}\times 36\\ \theta & ={15}^{\circ }\end{array}$

By using the above obtained value of $\theta$ value of $s$ will be,

$\begin{array}{cc}s& =\frac{\theta }{{360}^{\circ }}\times 2\pi r\\ & =\frac{{15}^{\circ }}{{360}^{\circ }}\times 2\pi \times 36\\ & =\frac{{15}^{\circ }}{{10}^{\circ }}\times 2\pi \\ & =3\pi cm\end{array}$

Hence, the length of the arc of the sector is $3\pi cm.$