Volume of a Sphere

Volume of a Sphere

Volume of a Sphere

Volume of a sphere is the total space occupied by the sphere. To derive the formula for the volume of a sphere we can do the following activity:
Activity to find the volume of a sphere

-    Take a cylindrical vessel circumscribing a plastic spherical ball as given in the figure above.
-    Take out the spherical ball from the cylinder and cut it into two equal parts. In this way we will get two hemispheres of plastic ball.
-    Fill a hemisphere with rice or sand and pour it into the cylinder. And repeat it again.
-    Then we will find that the cylinder will be full filled by 3 hemispheres.

Thus, the volume of 3 hemisphere = volume of 1 cylinder  
                                                        = πr2h    [ volume of cylinder = πr2h]
                                                        = πr2 × d    [ h = d]
                                                        = πr2 × 2r    [ d = 2r]
                                                        = 2πr3
              Volume of 1 hemisphere =  2πr3/3

Thus, the volume of a hemisphere = 2πr3/3

Now,
         The volume of a sphere = volume of two hemispheres
                                                = 2 × 2πr3/3
                                                = 4πr3/3

  The volume of a sphere = 4πr3/3

Workout Examples

Example 1: Find the volume of a sphere of radius 21cm.

Solution: Here, radius of sphere (r) = 21cm
                Now,
                         Volume of sphere = 4πr3/3
                                                        = 4/3 × 22/7 × (21)3
                                                        = 38808cm3

                Thus, the volume of sphere is 38808cm3.


Example 2: Three metallic spheres each of diameters 1cm, 6cm and 8cm are melted to form a single sphere. Find the radius of single sphere.

Solution: Here, radii of three spheres are,
                                r1 = 1cm
                                r2 = 6cm
                                r3 = 8cm

                Therefore, volumes of three spheres are,
                                v1 = 4πr3/3 = 4/3 × 22/7 × (1)3 = 4.19cm3
                                v2 = 4πr3/3 = 4/3 × 22/7 × (6)3 = 905.14cm3
                                v3 = 4πr3/3 = 4/3 × 22/7 × (8)3 = 2145.52cm3

Three spheres are melted to form a single sphere, therefore,
                The volume of single sphere = v1 + v2 + v3
                i.e.      4πr3/3 = 4.19 + 905.14 + 2145.52
                or,      4/3 × 22/7 × r3 = 3054.85
                or,      r3 = 3054.85 × 21/88
                or       r3 = 729
                or,      r 3 = 93
                or,      r = 9cm

                Thus, the radius of single sphere is 9cm.


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

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