 # If the circumference of a circle is equal to the perimeter `
Question:

If the circumference of a circle is equal to the perimeter of a square then the ratio their areas is:

(a) 22 : 7

(b) 14 :11

(c) 7 :22

(d) 7 :11

Solution:

It is given that circumference of a circle = perimeter of a square

We have to find the ratio of their areas

Let the radius of circle = r

Let the circumference of circle = Let the area of circle = Let the side of the square = b

Let the perimeter of square = Let the area of square = Therefore

Circumference of circle $P_{c}=2 \pi r$

Perimeter of square $P_{s}=4 b$

$P_{c}=P_{s}$

$2 \pi r=4 b$

$b=\frac{\pi r}{2}$.....(1)

Area of Circle $A_{c}=\pi r^{2}$

Area of Square $A_{n}=b^{2}$

$=\left(\frac{\pi r}{2}\right)^{2}$ [From Equation (1)]

$=\frac{\pi^{2} r^{2}}{4}$

$\operatorname{Now} \frac{A_{c}}{A_{s}}=\frac{\pi r^{2}}{\frac{\pi^{2} r^{2}}{4}}$

$=\frac{4}{\pi}$

$=\frac{4}{\frac{22}{7}}$

$=\frac{14}{11}$ or $14: 11$

Hence Option (b) is correct.