Question

From a circle of radius 15 cm, 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

Given,

Radius of the circle $=15cm$

Angle of sector cut $={216}^{\circ }$

Then,

Length of arc of sector $=\frac{2\pi r\theta }{360}$

$=\frac{2\pi \times 15\times 216}{360}=18\pi$

Now, the sector is rolled to make a cone

$\therefore$ Radius of circle $=$ slant height of the cone $=15cm$ (Given)

And length of arc $=$ circumference of base of the cone

$\Rightarrow 18\pi =2\pi R$ [$R=$ radius of base of cone]

$\Rightarrow R=9$

$\therefore$ Height of cone, $h=\sqrt{(15{)}^2-(9{)}^2}=\sqrt{225-81}=\sqrt{144}=12$

Hence, volume of cone $=\frac{1}{3}\pi R^{2}h$

$=\frac{1}{3}\times \frac{22}{7}\times (9{)}^2\times 12=\frac{22\times 81\times 12}{7\times 3}=1018.30m^3$

Therfore, Volume of the cone is $1018.30m^3$