Question

A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surface area will be:

• A. $2:1$
• B. $1:\sqrt{2}$
• C. $\sqrt{2}:1$
• D. $\sqrt{3}:1$

Answer 1

The correct option is C $\sqrt{2}:1$

Let the radius of hemisphere and cone be R.
Height of the hemisphere = Radius of Hemisphere = R
Since it is given that the heights of hemisphere and cone are equal,
height of the cone = R

Slant height of the cone
$=\sqrt{R^2+R^2}=\sqrt{2}R$

Then,
$\frac{\text{Curved surface area of Hemisphere}}{\text{Curved surface area of cone}}$

$=\frac{2\pi R^2}{\pi R\times \text{slant height}}$

$=\frac{2\pi R^2}{\pi R\times \sqrt{2}R}$

$=\frac{2}{\sqrt{2}}$

$=\sqrt{2}:1$

$\therefore$ The ratio of CSA of the hemisphere to that of the cone is $\sqrt{2}:1$.