Question

A steel wire, when bent in the form of a square, encloses an area of $121$ sq.cm. The same wire is bent in the form of a circle. Find the area of the circle.

Answer 1

Hint:From the given area of the square we will first find the length of the side of the square by using the formula for area of a square. Then we will use the formula for the perimeter of a square to find the total length of the wire. Then the same wire is bent in the form of a circle. Therefore, we have the circumference of the circle. So, by using the formula for the circumference of the circle, we can find the radius of the circle. Then we can easily find the area of the circle by using the formula for area of a circle.

Complete step by step answer:
Given, the area of the square is $121$ sq.cm.

We know, the formula for area of a square $ = {a^2}$
Where, $a = $side of the square.
Therefore, using this formula for the given square, we get,
${a^2} = 121$
Taking square root on both sides, we get,
$ \Rightarrow a = \sqrt {121} = 11$cm
Therefore, the sides of the square measure $11$cm.
Therefore, the perimeter of the square is $ = 4a= 4\left( {11} \right) = 44$ cm

Therefore, we can say the total length of the wire is $44$cm.
Then, the wire is bent in the form of a circle.
Therefore, the circumference of the circle is $44$ cm.
We know, the formula for circumference of a circle is $ = 2\pi r$
Where, $r = $radius of the circle
Now, using this formula, we can write,
$2\pi r = 44$
$ \Rightarrow 2.\dfrac{{22}}{7}.r = 44$
$ \Rightarrow \dfrac{{44}}{7}.r = 44$

Now, multiplying both sides with $\dfrac{7}{{44}}$, we get,
$ \Rightarrow r = 44.\dfrac{7}{{44}}$
$ \Rightarrow r = 7$ cm
Therefore, the radius of the circle is $7$ cm.We know, the formula for the area of a circle is $\pi {r^2}$.
Using this formula we get,
$\text{Area of the circle}= \pi {r^2}= \dfrac{{22}}{7}.{\left( 7 \right)^2}$
Substituting the values of known quantities, we get,
$\therefore \text{Area of the circle} = 154$ squared centimetres

Therefore, the area of the circle formed by the wire is $154\,cm^2$.

Note:If a wire which is bent to form a particular shape is again bent to form another shape, then the perimeter of both the shapes must be equal. The perimeter of a square is given as four times the side length of the square and the circumference of the circle is given by $2\pi r$.