VerifiedHint: Assume the side of a square as ‘$a$’ and radius of a circle as ‘$r$’. Formula for perimeter of a square and perimeter of a circle is given by $4a$ and $2\pi r$ respectively. The area of a square and a circle is given by the formulas ${a^2}$ and $\pi {r^2}$ respectively.
Complete step by step answer:
Given, the perimeter of a circle is equal to the perimeter of a square.
Let us assume that, the length of the side of a square be $a$ and the radius of a circle be $r$ .
Step I: We know that,
Perimeter of a square is given by $4a$ and the perimeter of a circle is given by $2\pi r$.
Now according to the question,
$2\pi r = 4a$ that is, $r = \dfrac{{4a}}{{2\pi }} = \dfrac{{2a}}{\pi }$. …… (i)
Step II: Again, we know that, area of a circle is given by
$A = \pi {r^2}$ …… (ii)
Area of a square is given by
$A = {a^2}$. …… (iii)
Now place the value of equation (i) in the formula of area of a Circle.
Therefore, area of the Circle is
$\begin{array}{c}A = \pi {r^2}\\ = \pi {\left( {\dfrac{{2a}}{\pi }} \right)^2}\\ = \dfrac{{4{a^2}}}{\pi }\end{array}$
Further solving,
Area of the Circle,
$A = 1.27{a^2}$ …… (iv)
Now, comparing the equations (iii) and (iv) we find that $1.27{a^2} > {a^2}$.
Therefore, the area of a Circle is greater than the area of a square.
Note: Here we have to compare the area of the circle and the square. Assume the side of a square as ‘$a$’ and radius of a circle as ‘$r$’. In step I, find the values of perimeters of the square and the circle. Solve the equation for $r$. In step II, Find the areas of the square and the circle. Replace the value of $r$ and solve it. Now, compare the areas of both the square and the Circle. We can see that $1.27{a^2}$ is greater than ${a^2}$.
