Volume of a Cone

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:
Experiment to find the volume of a cone
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.
Example 1 Cone
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.
Example 3 Combined solid
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.


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