In the following figure, ABCD is a square of side 2a, Find the ratio between
(i) the circumferences
(ii) the areas of the in circle and the circum-circle of the square.
We have a square $\mathrm{ABCD}$ having $\mathrm{AB}=2 a$. From the given diagram we can observe that,
Radius of incircle $\left(r_{1}\right)=a$
Radius of circumcircle $\left(r_{2}\right)=\sqrt{2} a$
(i) We have to find the ratio of the circumferences of the two circles. So the required ratio is,
$=\frac{\text { Perimeter of inner circle }}{\text { Perimeter of circumcircle }}$
$=\frac{2 \pi(a)}{2 \pi(\sqrt{2} a)}$
$=\frac{1}{\sqrt{2}}$
(ii) We have to find the ratio of the areas of the two circles. So the required ratio is,
$=\frac{\text { Area of inner circle }}{\text { Area of circumcircle }}$
$=\frac{\pi(a)^{2}}{\pi(\sqrt{2} a)^{2}}$
$=\frac{1}{2}$
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