# Volume of a Cone

The volume of a right circular cone is one-third of the volume of a right circular cylinder of the same base and same height.

$\therefore$ If $h$ is the height of the cone and $r$ is the radius of the base, then
${\text{Volume }} = {\text{ }}\frac{1}{3}{\text{ }} \times {\text{ area of the base }} \times {\text{ altitude}}$
$\therefore$     $V = \frac{1}{3}\pi {r^2}h$                      (as the area of the base $= \pi {r^2}$)

Rule: The volume of a cone equals the area of the base times one-third the altitude.

Example:

The circumference of the base of a $9$m high conical tent is $44$m. Find the volume of the air contained in it.

Solution:

Circumference of the base $= 2\pi r = 44$m
$\therefore$     $2 \times \frac{{22}}{7} \times r = 44$
$r = \frac{{44 \times 7}}{{44}} = 7$m
$\because$ height of the conical tent $= 9$m
$\therefore$ volume of air $= \frac{1}{3}\pi {r^2}h$
$= \frac{1}{3} \times \frac{{22}}{7} \times 7 \times 9 = 462$ cubic m

Example:

The vertical height of a conical tent is $42$ dm and the diameter of its base is $5.4$ m. Find the number of people it can accommodate if each person is to be allowed $2916$ cubic dm of space.

Solution:
Here height $h = 42$ dm
Diameter         $= 5.4$ m $= 54$ dm
Radius        $r = 27$ dm

Volume $= \frac{1}{3}\pi {r^2}h$
$= \frac{1}{3} \times \frac{{22}}{7} \times 27 \times 27 \times 42$
$= 32076$ Cubic dm
Space allowed for $1$ person $= 2916$ cubic dm
$\therefore$ the required number of people $= \frac{{32076}}{{2916}} = 11$ people