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.

#### Solution

Given data is as follows:

Total Surface Area = 231 cm^{2}

Curved Surface Area = `2/3 ("Total Surface Area" )`

We have to find the volume of the cylinder.

We have,

Total Surface Area = 231 cm^{2}

`2pirh`+ `2pir^2`=231

Where, `2pirh` is nothing but the Curved Surface Area.

Curved Surface Area = `2/3 ("Total Surface Area" )`

Curved Surface Area= `2/3 xx 231 `=154

Let us replace `2pirh` in the above equation with the value of Curved Surface Area we have just obtained.

154+`2pir^2` =231

`2pir^2` =77

`2 xx 22/7 xx r^2`=77

`r^2 = (77 xx 7)/(2 xx 22 ) = (7xx7)/(2xx2)`

`r = 7/2`

Now, let us find the value of h by using the Curved Surface Area.

Curved Surface Area=154 cm^{2}

`2pirh` =154

Since we know that `r = 7/2` ,

`2 xx 22/7 xx 7/2 xx h` =154

*h* = 7

Now that we know the value of both h and r , we can easily find the volume of the cylinder.

Volume of the cylinder = `pir^2h`

=`22/7 xx 7/2xx7/2xx7`

`"Volume of the cylinder "= 269.5cm^3`