Question

From a circle of radius $15cm$, a sector with angle ${216}^o$ is cut out and its bounding radii are bent so as to form a cone. Find its volume.

Answer 1

Here,

Radius of the circle, $R=15cm$

When the sector is cut and its bounding radii is bent to form a cone,

Slant height of the cone, $l=R=15cm$

Let $r$ and $h$ be the radius and height of the cone, respectively.

Again, we know that in a circle of radius $R$, an arc of length $X$ subtends an angle of $\theta$ radians, then

$x=R\theta$

Here, the arc length will be equal to the perimeter of the base circle of the cone.

$x=2\pi r$

$2\pi r=R\theta$

$\frac{r}{R}=\frac{\theta }{2\pi }$

$\Rightarrow \frac{r}{15}=\frac{216}{360}$

$\Rightarrow r=9cm$

Now, height of the cone can be calculated as,

$h^2=l^2-r^2$

$h^2={\left(15\right)}^2-{\left(9\right)}^2$

$h^2=225-81$

$h=\sqrt{144}=12cm$

Therefore,

Volume of the cone, $V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \frac{22}{7}\times 81\times 12=1018.28c{m}^3$

Hence, this is the required result.