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
Solution:
(b) Let radius of circle be r and side of a square be a.
According to the given condition,
Perimeter of a circle $=$ Perimeter of a square
$\because$ $2 \pi r=4 a \Rightarrow a=\frac{\pi r}{2}$ $\ldots$ (i)
Now, $\frac{\text { Area of circle }}{\text { Area of square }}=\frac{\pi r^{2}}{(a)^{2}}=\frac{\pi r^{2}}{\left(\frac{\pi r}{2}\right)^{2}}$ [from Eq. (i)]
$=\frac{\pi r^{2}}{\pi^{2} r^{2} / 4}=\frac{4}{\pi}=\frac{4}{22 / 7}=\frac{28}{22}=\frac{14}{11}$