Question

If the circumference of a circle and the perimeter of a square are equal, then

• A. Area of the circle $=$Area of the square
• B. Area of the circle $>$ Area of the square
• C. Area of the circle $<$ Area of the square
• D. Nothing definite can be said about the relation between the areas of the circle & square.

Answer 1

The correct option is B

Area of the circle $>$ Area of the square

Step 1: Finding the relation between radius of given circle and square side

Let $r$ be the radius of the circle and the length of each side of the square is $a$

Perimeter of the square $=4\times \left(\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{side}\right)=4a$

Circumference of the circle $=2·\mathrm{\pi }·\left(\mathrm{radius}\right)=2\mathrm{\pi r}$

Given that

$$\begin{gathered} \mathrm{Circumference}\mathrm{of}\mathrm{the}\mathrm{circle}=\mathrm{Perimeter}\mathrm{of}\mathrm{square} \\ \Rightarrow 2\mathrm{\pi r}=4\mathrm{a} \\ \Rightarrow \mathrm{\pi r}=2\mathrm{a} \\ \Rightarrow \frac{22}{7}\mathrm{r}=2\mathrm{a} \\ \Rightarrow \frac{11}{7}\mathrm{r}=\mathrm{a} \end{gathered}$$

Step 2: Compare the area of circle and area of square

Area of the circle

$$\begin{gathered} =\mathrm{\pi }·{\left(\mathrm{radius}\right)}^2 \\ ={\mathrm{\pi r}}^2 \\ =3.14r^2 \end{gathered}$$

Area of the square

$$\begin{gathered} ={\left(\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{side}\right)}^2 \\ =a^2 \\ ={\left(\frac{11}{7}r\right)}^2 \\ =\frac{121}{49}{r}^2 \\ =2.47r^2 \end{gathered}$$

Therefore Area of the circle $>$ Area of the square.

Hence, the correct option is (B)