Question

The area of a circle is $81\pi sq.cms$. Find the length of the arc subtending an angle of ${300}^o$ at the centre and the area of corresponding sector.

Answer 1

${360}^0$ has area=$81\pi c{m}^2$

${300}^0$ has area $=\frac{81\pi }{360}\times 300c{m}^2$

$=\frac{81\pi }{36}\times 30c{m}^2$

$=\frac{81\pi \times 5}{6}c{m}^2$

$=\frac{405\pi }{6}c{m}^2$

$=\frac{135}{2}\pi c{m}^2$

$2\pi r$ (circumference) has $\to 81\pi c{m}^2$ area

or $81\pi c{m}^2$ area $\to 2\pi r$ are length

$\Rightarrow$ Length of arc $=\frac{2\pi r\times 135}{162}$

$\pi r^2=81c{m}^2$

$\Rightarrow r=\sqrt{\frac{81}{\pi }}$

$\therefore$ Length of arc corresponding to ${300}^0$ angle $=\frac{270\pi }{162}\sqrt{\frac{81}{\pi }}$

$=\frac{5\pi }{3}\sqrt{\frac{81}{\pi }}$

$=\frac{5}{3}\sqrt{81\pi }$