If the perimeter of a circle is equal to that of a square,

Question:

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

(a) 13 : 22

(b) 14 : 11

(c) 22 : 13

(d) 11 : 14

Solution:

We have given that perimeter of circle of radius r is equal to square of side a.

$\therefore$ perimeter of the circle $=$ perimeter of the square

$\therefore 2 \pi r=4 a$

$\therefore r=\frac{4 a}{2 \pi}$

$\therefore r=\frac{2 a}{\pi}$

Now we will substitute value of r in the following equation

$\frac{\text { Area of circle }}{\text { Area of square }}=\frac{\pi r^{2}}{a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{\pi\left(\frac{2 a}{\pi}\right)^{2}}{a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{\pi \times \frac{4 a^{2}}{\pi^{2}}}{a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{\frac{4 a^{2}}{\pi}}{a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{4 a^{2}}{\pi} \times \frac{1}{a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{4}{\pi}$

Substituting $\pi=\frac{22}{7}$ we get,

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{4}{\frac{22}{7}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{28}{22}$

$\therefore \frac{\text { Area of circle }}{\text { Area of square }}=\frac{14}{11}$

Hence, ratio of the areas of the circle and square is $14: 11$.

Therefore, the correct answer is $(b)$.

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