Question

A cylinder has a circumference of \[12\pi\] cm and a height that is half of the radius. What is the total surface area of the cylinder?

Answer 1

Hint:In this question, we need to find the total surface area of the cylinder. Given that the circumference is \[12\pi\] cm. Also given that the height is half of the radius that is \[h = \dfrac{r}{2}\] . The formula of the circumference is \[2\pi r\] , where \[r\] is the radius of the cylinder. As given in the question the circumference is \[12\pi\] , thus by equating , we obtain the radius of the cylinder. From the radius, we can find the height of the cylinder. Then we can use the formula of the total surface area of the cylinder, and by substituting the value of \[\pi\] , \[r\] and \[h\] , we can find the area of the cylinder.

Formula used :
The circumference of the cylinder,
\[C = 2\pi r\]
The total surface area of the cylinder,
\[T.\ S.\ A = 2\pi r(r + h)\]
Where \[\pi\] is the mathematical constant which values \[3.14\] , \[r\] is the radius of the cylinder and \[h\] is the height of the cylinder.


Complete step by step answer:

Given, the circumference of the cylinder is \[12\pi r\ cm\].
Here we need to find the total surface area of the cylinder.
The formula for the circumference of the cylinder is \[2\pi r\] .
Now on equating the formula and value given,
We get,
\[\Rightarrow \ 2\pi r = 12\pi\]
On simplifying,
We get,
\[\Rightarrow \ 2r = 12\]
On dividing both sides by \[2\] ,
We get
\[\Rightarrow \ r = \dfrac{12}{2}\]
On simplifying,
We get,
\[\Rightarrow \ r = 6\]
Thus the radius of the cylinder is \[6\ cm\] .
Also given that the height is half of the radius that is ,
\[\Rightarrow \ h = \dfrac{r}{2}\]
Now on substituting the value of \[r\] ,
We get,
\[\Rightarrow \ h = \dfrac{6}{2}\]
On simplifying,
We get,
\[\Rightarrow \ h = 3\]
Therefore we get the height of the cylinder as \[3\ cm\] .
Now we can find the total surface area of the cylinder.
The total surface area of the cylinder,
\[T.\ S\ A = 2\pi r(r + h)\]
On substituting the value of \[\pi\], \[r\] and \[h\]
We get,
\[T.\ S.\ A = 2(3.14)(6)\ (6 + 3)\]
On simplifying,
We get
\[\Rightarrow \ (37.68)(9)\]
On further simplifying,
We get,
\[T.\ S.\ A = 339.12\]
Thus we get the total surface area of the cylinder is \[339.12\ cm^{2}\]
The total surface area of the cylinder is \[339.12\ cm^{2}\] .

Note:A cylinder is nothing but a closed solid that has two parallel bases joined by a curved surface, at a fixed distance. While solving these questions, we should know all the formulas , otherwise the question could not be solved because the use of the formula is the only way to solve it. In these types of questions , first, by using the given information, we can draw a figure and then we need to find out the other required values. We need to note that we have to find the entire surface area of the cylinder and not just the lateral surface area.