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

{\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 area of base  = \pi {r^2})

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


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


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 = 462Cubic m


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

            Here height h = 42dm
            Diameter          = 5.4m  = 54dm
            Radius,         r = 27dm
            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