inscribed circles

What is the ratio of the area of the two circles?

inscribed circles
Solution

Let $R$ be the radius of the large circle. Let $r$ be the radius of the small circle. Then the square has sides of length $2r$ and has a diagonal of length $2R$. The length of the diagonal can also be calculated from Pythagoras, giving $L = \sqrt{(2r)^2 + (2r)^2} = 2\sqrt{2}r$

We have two expressions for the length of the diagonal, $L = 2\sqrt{2}r = 2R$, as a consequence, we get $$ R = \sqrt{2}r $$ The area of the large circle is therefore $\pi (\sqrt{2}r)^2 = 2\pi r^2$. So the ratio of areas of the circles is $2:1$