Question.
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).
![A right circular cylinder just encloses a sphere of radius r](/media/uploads/2021/12/A-right-circular-cylinder-just-encloses-a-sphere-of-radius-r.png)
(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in (i) and (ii).
Solution:
(i) Surface area of sphere $=4 \pi r^{2}$
(ii) Height of cylinder $=r+r=2 r$
Radius of cylinder $=r$
CSA of cylinder $=2 \pi r h$
$=2 \pi r(2 r)$
$=4 \pi r^{2}$
(iii) Required ratio $=\frac{\text { Surface area of sphere }}{\text { CSA of cylinder }}$
$=\frac{4 \pi r^{2}}{4 \pi r^{2}}$
$=\frac{1}{1}$
Therefore, the ratio between these two surface areas is $1: 1$.
![A right circular cylinder just encloses a sphere of radius r](/media/uploads/2021/12/A-right-circular-cylinder-just-encloses-a-sphere-of-radius-r-1.png)
(i) Surface area of sphere $=4 \pi r^{2}$
(ii) Height of cylinder $=r+r=2 r$
Radius of cylinder $=r$
CSA of cylinder $=2 \pi r h$
$=2 \pi r(2 r)$
$=4 \pi r^{2}$
(iii) Required ratio $=\frac{\text { Surface area of sphere }}{\text { CSA of cylinder }}$
$=\frac{4 \pi r^{2}}{4 \pi r^{2}}$
$=\frac{1}{1}$
Therefore, the ratio between these two surface areas is $1: 1$.
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