# 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, $r$ is the radius of the base, then

$\therefore$     $V = \frac{1}{3}\pi {r^2}h$                      (as area of 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 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 persons 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
$\therefore$ Required number of persons $= \frac{{32076}}{{2916}} = 11$ Persons