Prove that the surface area of a sphere is equal to the curved surface area of the circumscribed cylinder.

To do:

We have to prove that the surface area of a sphere is equal to the curved surface area of the circumscribed cylinder.

Solution:

Let $r$ be the radius of the sphere.

This implies,

Surface area of the sphere $= 4 \pi r^2$.….(i)

Diameter of the sphere $=$ Height of the cylinder

$h=2r$

Radius of the circumscribed cylinder $=$ Radius of the sphere $=r$

Therefore,

Curved surface area of the cylinder $=2 \pi r h$

$=2 \pi r \times 2 r$

$=4 \pi r^{2}$.........(ii)
From equations (i) and (ii), we get,

Surface area of the sphere $=$ Curved surface area of the circumscribed cylinder
Hence proved.

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Updated on: 10-Oct-2022

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