Question

The lateral surface of a cylinder is developed into a square whose diagonal is $2\sqrt{2}\,cm$. The area of the base of the cylinder (in cm square) is
(a) $3\pi $
(b) $\dfrac{1}{\pi }$
(c) $\pi $
(d) $6\pi $

Answer 1

Hint: We have to consider the LSA of the cylinder and also imagine it being converted into a square. Then, with the help of the Pythagoras theorem we will find the side of the square and after that we will use the formula of area of the circle given by $\pi {{r}^{2}}$. By doing this we will be able to solve the question further.

Complete step-by-step answer:
The required diagram of the given question is shown below.

Here d is the diagonal of the square as $2\sqrt{2}$ cm. Thus, by using Pythagoras theorem to the triangle $\Delta ABC$ we will get $A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}$. This results into
$\begin{align}
  & {{d}^{2}}={{a}^{2}}+{{a}^{2}} \\
 & \Rightarrow {{d}^{2}}=2{{a}^{2}} \\
 & \Rightarrow \dfrac{{{d}^{2}}}{2}={{a}^{2}} \\
\end{align}$
As the value of d is $2\sqrt{2}$ cm so, we will get
$\begin{align}
  & {{a}^{2}}=\dfrac{{{\left( 2\sqrt{2} \right)}^{2}}}{2} \\
 & \Rightarrow {{a}^{2}}=\dfrac{8}{2} \\
 & \Rightarrow {{a}^{2}}=4 \\
\end{align}$
 After taking the root we will have two values of a as + 2 and – 2. As the side of a square cannot be negative therefore, we will consider only the value of a as 2. According to the question we are informed that the lateral surface of the cylinder is converted into a square. And by this we come to know that the lateral surface is converted into the diagonal of the square. So, accordingly the height and the perimeter of the base of the cylinder is 2 cm only. As we know that the perimeter is $2\pi r$ which is equal to 2. Thus we get that
$\begin{align}
  & 2\pi r=2 \\
 & \Rightarrow r=\dfrac{1}{\pi } \\
\end{align}$
Therefore, the radius of the circle be$r=\dfrac{1}{\pi }\,cm$. Thus, the area of the base of the cylinder is given by the formula $\pi {{r}^{2}}$. So, now we have $\pi {{r}^{2}}=\pi \times \dfrac{1}{\pi }\,cm\times \dfrac{1}{\pi }\,cm$ which results into as simply $\dfrac{1}{\pi }$.
Hence, the correct option is (b).
Note: If we were not restricted to the answer as in terms of $\pi $ then we can use substitution here by keeping $\pi =\dfrac{22}{7}$. Therefore, we have that
$\begin{align}
  & \pi {{r}^{2}}=\dfrac{22}{7}\times \dfrac{1}{\pi }cm\times \dfrac{1}{\pi }\,cm \\
 & \Rightarrow \pi {{r}^{2}}=\dfrac{22}{7}\times \dfrac{7}{22}cm\times \dfrac{7}{22}\,cm \\
 & \Rightarrow \pi {{r}^{2}}=\dfrac{7}{22}c{{m}^{2}} \\
\end{align}$
Similarly, we can also substitute the value of $\pi $ as 3.14 and solve as usual.
While performing calculations we will solve the question along with the unit. This will help us in not forgetting the unit in the answer. Otherwise our answer without unit will be considered as wrong. There is no need to use any formula related to cylinders. One should check the units. If in the question there is a mention of any particular unit then we will change the unit in the answer.