# 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