Question

A right circular cylinder and a sphere have the same volume and same radius. The ratio of the areas of their curved surfaces is ________.

Answer 1

Let the radius of right circular cylinder and sphere be r.

Suppose h be the height of the cylinder.

∴ Volume of the sphere = $\frac{4}{3}\mathrm{\pi }{r}^3$

Volume of the cylinder = $\mathrm{\pi }{r}^{2}h$

Now,

Volume of the cylinder = Volume of the sphere (Given)

$\Rightarrow \mathrm{\pi }{r}^{2}h=\frac{4}{3}\mathrm{\pi }{r}^3$

$\Rightarrow h=\frac{4}{3}r$ .....(1)

Now,

Curved surface area of the cylinder, S₁ = 2$\mathrm{\pi }$rh

Curved surface area of the sphere, S₂ = 4$\mathrm{\pi }$r²

$\therefore \frac{S_1}{S_2}=\frac{2\mathrm{\pi }rh}{4\mathrm{\pi }{r}^2}$

$\Rightarrow \frac{S_1}{S_2}=\frac{2\mathrm{\pi }r\times {\displaystyle \frac{4}{3}}r}{4\mathrm{\pi }{r}^2}$ [From (1)]

$\Rightarrow \frac{S_1}{S_2}=\frac{2}{3}$

⇒ S₁ : S₂ = 2 : 3

Thus, the ratio of the curved surface of cylinder to the curved surface area of sphere is 2 : 3.

A right circular cylinder and a sphere have the same volume and same radius. The ratio of the areas of their curved surfaces is ___2 : 3___.