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:

-    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

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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.