Question

A copper wire when bent in the form of a square encloses an area of $121\,c{m^2}$. If the same wire is bent into the form of a circle. What is the area of the circle?

Answer 1

Hint: In the question a copper wire is bent in the form of a square and then in the form of a circle. So first we have to find the length of the wire which is equal to the circumference of the circle. For calculating the length of the wire we will use the formula area of square is equal to the ${(side)^2}$ and perimeter of the square is equal to $4 \times side$.

Complete step by step answer:
Given that when copper wire bend in the form of a square it encloses an area of $121\,c{m^2}$
We know that area of square $ = {(side)^2}$
$ \Rightarrow {(side)^2} = 121$
$ \Rightarrow side = 11\,cm$
Also, we know that perimeter of the square$ = 4 \times side$
$ \Rightarrow 4 \times 11$
$ \Rightarrow 44\,cm$
Therefore, the length of the wire $ = 44\,cm$

Now, given that the same wire is bent into the form a circle.
So, length of wire$ = $ circumference of the circle
Therefore, circumference of the circle $ = 44\,cm$

We know that, circumference of circle $ = 2 \times \pi \times r$
$ \Rightarrow 44 = 2 \times \pi \times r$
Putting the value of $\pi = \dfrac{{22}}{7}$. We get,
$ \Rightarrow 44 = 2 \times \dfrac{{22}}{7} \times r$
$ \Rightarrow 44 = \dfrac{{44}}{7}r$
$ \Rightarrow r = 7$

Therefore, the radius of the circle is $7\,cm$.
We know that area of circle $ = \pi {r^2}$
Putting the value of $\pi = \dfrac{{22}}{7}$ and $r = 7\,cm$. We get,
$ \Rightarrow \text{Area of circle} = \dfrac{{22}}{7} \times {(7)^2}$
$ \Rightarrow \text{Area of circle} =22 \times 7$
$ \therefore \text{Area of circle} =154\,c{m^2}$

Hence, the area enclosed by the circle is $154\,c{m^2}$.

Note: Before solving these types of problems one should remember all measurement formulas so that calculations may not get wrong and don’t forget to write the units of all the measurements. The difference between area and perimeter is that area is defined as the space occupied by the shape. While perimeter is defined as the distance around the shape. Area is measured in square units whereas perimeter is measured in linear units. Note that area can be measured for two dimensional units whereas perimeter is measured for one dimensional units.