Why Total Surface area of Sphere 4πr2

Total Surface area of Sphere

  • Sphere is three-dimensional geometrical figure having surface area and volume like a ball.
  • If r is the radius of sphere, then the total surface area of sphere is given by,

Total surface area of sphere = 4*π*r2

Derivation of Total Surface area of Sphere:

  • If we apply same force from all sides on the cylinder having circular bases of radius r, height h then it will be converted into sphere of radius r as shown in figure.
  • And also, if we open sphere from all sides then it forms the cylinder again with circular base having radius r and height h.
  • Hence, we can say that, the curved surface area of cylinder is equal to the total surface area of sphere.
  • But, when we form the sphere from cylinder then height of the cylinder becomes equal to the diameter of the cylinder as shown in figure.

So, height of cylinder = diameter of sphere

Hence, h = d = 2*r

 

Now, we can write as,

 

Total surface area of sphere = curved surface area of cylinder

But we already know that,

 

Curved surface area of cylinder = 2πrh

Hence, total surface area of sphere = 2πrh

But, h = 2r

Total surface area of sphere = 2πrh = 2πr*2r

Total surface area of sphere = = 4*π*r2

Hence proved.


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