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# A solid cylinder has a total surface area of $231\ cm^2$. Its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder.

Given:

A solid cylinder has a total surface area of $231\ cm^2$. Its curved surface area is $\frac{2}{3}$ of the total surface area.

To do:

We have to find the volume of the cylinder.

Solution:

Surface area of the solid cylinder $= 231\ cm^2$

Curved surface area $=\frac{2}{3}\times231$

$=154 \mathrm{~cm}^{2}$

Therefore,

$2 \pi r h=154$.........(i)

$2 \pi r h+2 \pi r^{2}=231$..........(ii)

Subtracting (i) from (ii), we get,

$2 \pi r^{2}=231-154=77$

$2 \times \frac{22}{7} \times r^{2}=77$

$r^{2}=\frac{77 \times 7}{2 \times 22}$

$=\frac{49}{4}$

$=(\frac{7}{2})^{2}$

$\Rightarrow r=\frac{7}{2} \mathrm{~cm}$

$2 \times \frac{22}{7} \times \frac{7}{2} h=154$

$22 h=154$

$\Rightarrow h=\frac{154}{22}=7\ cm$

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

$=\frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 7$

$=\frac{539}{2}$

$=269.5 \mathrm{~cm}^{3}$