Question

Show that the surface area of a sphere is the same as that of the lateral surface of a right circular cylinder that just enclose the sphere.

Answer 1

Given:

A sphere of radius r.

A right circular cylinder just encloses the sphere.

So, the radius of the cylinder is r, and the height of the cylinder is 2r = 2r = 2r (diameter of sphere).

To Show:

Surface area of sphere = Lateral surface area of cylinder.

Surface area of the sphere:

S.A. of sphere = 4πr²

Lateral (curved) surface area of the cylinder

C.S.A. of cylinder = 2πrh

Here h = 2r, so:

C.S.A. = 2πr(2r) = 4πr²

S.A. of sphere = 4πr2 

C.S.A. of cylinder = 4πr

S.A. of sphere = L.S.A. (C.S.A.) of the enclosing cylinder

Hence, proved.