Question

The given figure shows ABCD, a square. Find the ratio between the circumference of the incircle and the circumcircle of the square.

Answer 1

Given $AB=2a$

Diameter of large circle $=AC$

$AC=\sqrt{{AB}^2+{BC}^2}[\because △ABCisaright△]$

$AC=\sqrt{{\left(2a\right)}^2+{\left(2a\right)}^2}=\sqrt{{8a}^2}=2\sqrt{2}a$

$\therefore$ Radius $=\frac{2\sqrt{2}a}{2}=\sqrt{2}a$

Diameter of small circle = AB = 2a

Radius $=\frac{2a}{2}=a$

Required Ratio $=2n\times a:2\pi \times \sqrt{2}a=1:\sqrt{2}$