Question

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is

• A. $22:7$
• B. $14:11$
• C. $7:22$
• D. $11:14$

Answer 1

The correct option is B

$14:11$

Calculate the ratios of the areas

Let $r$ be the radius of the circle and $a$ be the side of the square.

We know that,

Perimeter of a circle is $2\pi r$

Perimeter of a square is $4a$

Area of circle is $\pi r^2$

Area of square is $a^2$

Using, $\pi =\frac{22}{7}$

Step $1$:-

It is given that,

Perimeter of a circle $=$ Perimeter of a square

$$\begin{gathered} \Rightarrow 2\pi r=4a \\ \Rightarrow a=\frac{\pi r}{2} \end{gathered}$$

Step $2$:-

Ratio of their areas = $\frac{Areaofcircle}{Areaofsquare}$

$$\begin{gathered} =\frac{\pi r^2}{a^2} \\ =\frac{\pi r^2}{{\left({\displaystyle \frac{\pi r}{2}}\right)}^2} \\ =\frac{2^2\times \pi r^2}{{\pi }^2{r}^2} \\ =\frac{4\pi }{{\pi }^2} \\ =\frac{4}{\pi } \\ =\frac{4}{{\displaystyle \frac{22}{7}}} \\ =\frac{7\times 4}{22} \\ =\frac{28}{22} \\ =\frac{14}{11} \end{gathered}$$

Hence , If the perimeter of a circle is equal to that of a square, then the ratio of their areas is $14:11$