Question

A circle and a square have the same perimeter. Then
A) Their areas are equal.
B) The area of the circle is greater.
C) The area of the square is greater.
D) None of the above.

Answer 1

Hint: 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, $r={\displaystyle \frac{4a}{2\pi }}={\displaystyle \frac{2a}{\pi }}$. …… (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

$\begin{array}{c}A=\pi r^2\\ =\pi {\left({\displaystyle \frac{2a}{\pi }}\right)}^2\\ ={\displaystyle \frac{4a^2}{\pi }}\end{array}$

Further solving,

Area of the Circle,

$A=1.27a^2$ …… (iv)

Now, comparing the equations (iii) and (iv) we find that $1.27a^2>a^2$.

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 $1.27a^2$ is greater than $a^2$.