Volume of a Cone

Volume of a Cone

Volume of a cone is given by the one third of the product of its area of base and height i.e. volume of cone =  πr²h/3. This formula can be understood and derived by the following activity:

1.     Take a hollow cylindrical jar of radius r and height h, whose volume is πr²h.

2.     Take a hollow cone which has the same height h and same radius r.

3.     Fill the cone with water and pour into the jar. And repeat it.

In this activity, the cylindrical jar will be full filled with water when 3 cones of full water is pour into it. It shows that,

3 × Volume of cone = Volumeof the cylinder

or,       Volume of cone = 1/3 × Volume of the cylinder

                                     = 1/3 × πr²h

                                     = πr²h/3

∴ Volume of a cone = πr²h/3

Workout Examples

Example 1: Find the volume of a cone whose radius = 4 cm and height = 21 cm.

Solution: Here,

                           Radius (r) = 4 cm

                           Height (h) = 21 cm

                We know,

                           Volume of a cone = πr²h/3

                                                          = 1/3 × 22/7 × 4² × 21

                                                          = 352 cm³

                Thus, the volume of the cone is 352 cm³.

Example 2: The vertical height of a right circular cone is three times its diameter and its volume is 54π cm³. Find its height.

Solution: Here,

                           Let, the diameter of cone be d.   ∴  d = 2r

                           Then, height (h) = 3d = 3 × 2r = 6r

                           Given, volume of cone = 54π cm³

                i.e.      πr²h/3 = 54π

                or,      r²h/3 = 54

                or,      r² × 6r = 162

                or,      6r³ = 162

                or,      r³ = 162/6

                or,      r³ = 27

                or,      r³ = 3³

                or,      r = 3 cm

                Thus, height (h) = 6r = 6 × 3 cm = 18 cm.

Example 3: Calculate the volume of adjoining solid given in the figure.

Solution: Here,

                           The given figure is a combined solid of two cones.

                           Radius of both the cones (r) = 42 cm

                           Total height of the solid = 180 cm

                           Let the height of upper cone be h₁ and the height of lower cone be h₂.

                           So, h₁ + h₂ = 180 cm

                We know,

                           Volume of the solid = V(upper cone) + V(lower cone)

                                                             = πr²h₁/3 + πr²h₂/3

                                                             = πr²(h₁ + h₂)/3

                                                             = 1/3 × 22/7 × 42² × 180     [∵  h₁ + h₂ = 180 cm]

                                                             = 332640 cm³

                Thus, the volume of the solid is 332640 cm³.

You can comment your questions or problems regarding the volume of a cone here.