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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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