# A solid cylinder has a total surface area of 231cm2.

Question:

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

Solution:

Given,

Total surface area $=231 \mathrm{~cm}^{2}$

Curved surface area = 2/3 * (total surface area)

= 2/3 * 231

= 154

We know that,

$2 \pi r h+2 \pi r^{2}=231 \ldots .1$

Here $2 \pi r h$ is the curved surface area, so substitute the value of CSA in eq 1

$\Rightarrow 154+2 \pi r^{2}=231$

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

$\Rightarrow 2 \pi r^{2}=77$

$\Rightarrow 2^{*} 22 / 7^{*} r^{2}=77$

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

$\Rightarrow \mathrm{r}^{2}=\frac{7 * 7}{2 * 2}$

⟹ r = 7/2

We need to find the value of h

$C S A=154 \mathrm{~cm}^{2}$

⟹ 2πrh = 154

⟹ 2 * 22/7 * 7/2 * h = 154

⟹ h = 154/22

⟹ h = 7

So the volume of the cylinder is,

Volume $=\pi r^{2} h$

= 22/7 * 7/2 * 7/2 * 7

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

The volume of the cylinder is $269.5 \mathrm{~cm}^{3}$