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 9m high conical tent is 44m. Find the volume of the air contained in it.

 

Solution:

Circumference of the base  = 2\pi r = 44m
\therefore      2 \times \frac{{22}}{7} \times r = 44
r = \frac{{44 \times 7}}{{44}} = 7m
\because height of the conical tent  = 9m
\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