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.

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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.