Question

A cord in the form of a square encloses the area $'S'$ $c{m^2}$. If the same cord is bent into the form of a circle then what will be the area of that circle
(A)$\dfrac{{\pi {S^2}}}{4}$
(B)$4\pi {S^2}$
(C)$\dfrac{S}{{4\pi }}$
(D)$\dfrac{{4S}}{\pi }$

Answer 1

Hint:Make the relationship that the perimeter of the square is the same as the circumference of the circle because of the same cord used. Using the given area of the square, find its side and perimeter. Then equate the perimeter with circumference to get the radius, and then find the area.

Formula used: Area of the square $ = {\left( {Side} \right)^2}$ and Area of the circle $ = \pi \times Radiu{s^2}$

Complete step-by-step answer:
Let us first reframe the question in a better way. The problem here says that a square of the area $Sc{m^2}$ is made using a cord and then the same cord is rearranged to form a circle.
That means, whatever was the perimeter of the square is now the length of the circumference of the newly formed circle.
And we already know that area of the square is its side squared and its perimeter is $4$ times its side.
Therefore, the area of the square$ = {\left( {Side} \right)^2} = Sc{m^2}$
This will give us, Side $ = \sqrt S cm$ and thus perimeter will be $4\sqrt S cm$
But we already know, Perimeter of the Square$ = $ Circumference of the circle
And for a circle, the circumference is twice the radius times $\pi $ and area of the circle is $\pi $ times radius squared.
So, from the above relationship $ \Rightarrow 4\sqrt S = 2 \times \pi \times Radius$
$ \Rightarrow $ Radius $ = \dfrac{{2\sqrt S }}{\pi }cm$
Since we figured out the radius of the newly formed circle, let us calculate its area
Area of the circle$ = \pi \times Radiu{s^2} = \pi \times {\left( {\dfrac{{2\sqrt S }}{\pi }} \right)^2} = \pi \times \dfrac{{4{{\left( {\sqrt S } \right)}^2}}}{{{\pi ^2}}} = \dfrac{{4S}}{\pi }c{m^2}$

So, the correct answer is “Option D”.

Note:In multiple-choice questions, always check for the required form of the answer. Here we don’t have to put $\pi = 3.141$ . Be careful while squaring and taking square roots. Never forget to check the units in your answer. Students should remember the relationship between the radius of the circle and sides of the square with the fact that they have the same perimeter.