# The total surface area of a solid cylinder is 231 cm2 and its curved surface area is

Question:

The total surface area of a solid cylinder is $231 \mathrm{~cm}^{2}$ and its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder.

Solution:

Curved surface area $=\frac{2}{3} \times$ total surface area $=\frac{2}{3} \times 231=2 \times 77=154 \mathrm{~cm}^{2}$

Now, total surface area - curved surface area $=2 \pi r h+2 \pi r^{2}-2 \pi r h$

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

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

$\Rightarrow r^{2}=\frac{77 \times 7}{44}=12.25$

$\Rightarrow r=3.5 \mathrm{~cm}$

Also, curved surface area $=154 \mathrm{~cm}^{2}$

$\Rightarrow 2 \pi r h=154$

$\Rightarrow 2 \times \frac{22}{7} \times 3.5 \times h=154$

$\Rightarrow h=\frac{154 \times 7}{44 \times 3.5}=7 \mathrm{~cm}$

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

$=\frac{22}{7} \times(3.5)^{2} \times 7=269.5 \mathrm{~cm}^{3}$