$A$ right circular cylinder just encloses a sphere of radius $r$ (see figure). Find:
$(i)$ Surface area of the sphere,
$(ii)$ Curved surface area of the cylinder,
$(iii)$ Ratio of the areas obtained in $(i)$ and $(ii)$.

(N/A) $(i)$ For the sphere:
Radius $= r$
$\therefore$ Surface area of the sphere $= 4 \pi r^{2}$
$(ii)$ For the right circular cylinder:
Since the cylinder just encloses the sphere,the radius of the cylinder is equal to the radius of the sphere.
$\therefore$ Radius of the cylinder $= r$
Height of the cylinder is equal to the diameter of the sphere.
$\Rightarrow$ Height of the cylinder $(h) = 2r$
Curved surface area of a cylinder $= 2 \pi rh = 2 \pi r(2r) = 4 \pi r^{2}$
$(iii)$ Ratio of the areas obtained in $(i)$ and $(ii)$:
$\frac{\text{Surface area of the sphere}}{\text{Curved surface area of the cylinder}} = \frac{4 \pi r^{2}}{4 \pi r^{2}} = \frac{1}{1}$
Thus,the required ratio is $1:1$.