Sphere

Sphere

We see many solids like cricket ball, tennis ball, basket ball, football, globe etc. All these are the examples of spheres. Let us know some more facts about the sphere.

A sphere cab be described as the set of all those points in the space which are at equal distance from a fixed point. The fixed point is called centre and the constant distance is called the radius of the sphere.

In the given figure above, O is the centre of the sphere and distances OA, OB, OC, OD, OE, OF are the radius of the sphere.

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A line segment passing through the centre of the sphere with its end points on the sphere is called the diameter of the sphere. In the given figure above  AB, CD and EF are the diameters of the sphere.

Surface Area of a Sphere

Take a cylinder of equal height and diameter and a sphere equal diameter with the cylinder and wrap the sphere with a rope and with the same rope the curved surface area of the cylinder can be wrapped.

∴ Surface area of the sphere = Curved sufrace area of the cylinder

                                               = 2πrh

                                               = 2πr × 2r    [∵ h = d = 2r]

                                               = 4πr²

Thus, if r is the radius of a sphere then the surface area of the sphere is 4πr².

Volume of Sphere

Take a cylinder of equal height and diameter and a sphere of equal diameter with the cylinder. Fill the cylinder with water and drop the sphere into the cylinder, then two-third of water flows out. 

So we can say that,

Volume of sphere = 2/3 volume of the cylinder

                             = 2/3 × πr²h

                             = 2/3 × πr² × 2r    [∵ h = d = 2r]

                             = 4πr³/3

Thus, the volume of the sphere of radius r is 4πr³/3.

Workout Examples

Example 1: Find the total surface area and the volume of the given cricket ball of radius 35mm.

Solution: Here,

                          Radius of cricket ball (r) = 35mm

                Now,

                          Total surface area (TSA) = 4πr²

                                                                    = 4 × 22/7 × (35mm)²

                                                                    = 4 × 22/7 × 1225mm²

                                                                    = 15400 mm²

                          Volume (V) = 4πr³/3

                                              = 4/3 × 22/7 × (35mm)³

                                              = 88/21 × 42875mm³

                                              = 179666.67mm³

                Thus, the total surface area of cricket ball is 15400 mm² and the volume is 179666.67mm³.

Example 2: If the total surface area of a solid sphere is 616cm², what will be its radius?

Solution: Here,

                           Total surface area of sphere (TSA) = 616cm²

                i.e.      4πr² = 616

                or,      4 × 22/7 × r² = 616

                or,      88/7 × r² = 616

                or,      r² = 616 × 7/88

                or,      r² = 49

                or,      r² = 7²

                or,      r = 7cm

                Thus, the radius of the solid sphere is 7cm.

Example 3: Find the total surface area of a sphere whose volume is 1372π/3 cm³.

Solution: Here,

                           Volume of sphere = 1372π/3 cm³

                i.e.      4πr³/3 = 1372π/3

                or,      4r³ = 1372

                or,      r³ = 1372/4

                or,      r³ = 343

                or,      r³ = 7³

                or,      r = 7cm

                Now,

                           Total surface area (TSA) of sphere = 4πr²

                                                                                       = 4 × 22/7 × (7cm)²

                                                                                       = 4 × 22/7 × 49cm²

                                                                                       = 616cm²

                Thus, the total surface area of the sphere is 616cm².

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