Question

Radius and the angle of sector are $12cm$ and ${120}^{\circ }$ respectively. By making its straight cores a cone is formed. Find its volume.

Answer 1

Given,Radius of the sector $OA=OB\left(r\right)=12cm$
Angle of the sector $={120}^{\circ }$
$\therefore$ Length of the arc $AB=\frac{2\pi r\theta }{{360}^{\circ }}$
$=\frac{2\pi \times 12\times {120}^{\circ }}{{360}^{\circ }}$
$=8\pi cm$
Let radius of cone be R.
$\begin{array}{cc}\therefore 2\pi R& =8\pi \\ R& =4cm\\ (\text{slant height}{)}^2& =l^2=R^2+h^2\\ \Rightarrow h^2& =l-R^2\\ & =(12{)}^2-(4{)}^2\\ & =144-16\\ & =128\\ h& =\sqrt{128}=11.31cm\end{array}$
$\therefore$ Volume of the cone $=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\times \frac{22}{7}\times 4\times 4\times 11.31$
$=189.46{cm}^3$