# 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

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.

Updated: September 14, 2021 — 11:15 pm