Hello, reiward!

Sorry, your friends are right . . .

A square is inscribed in a circle, and another circle is inscribed in the square.

What is the ratio of the *area* of the larger circle to the small circle?

Consider one quadrant of the diagram.

Code:

*
| *
| * B
A * * - - - - *
| * * | *
| * |
| * * | *
| * | *
| * *
O * - - - - - * - - *
C

The radius of the large circle is: .$\displaystyle OB = R$

The area of the large circle is: .$\displaystyle A_1 \,=\,\pi R^2$

Since $\displaystyle \Delta OBC$ is a 45-45-90 right triangle: .$\displaystyle BC \,=\,\frac{R}{\sqrt{2}}$

The radius of the small circle is: .$\displaystyle r \,=\,\frac{R}{\sqrt{2}}$

The area of the small circle is: .$\displaystyle A_2 \:=\:\pi r^2 \:=\:\pi\left(\frac{R}{\sqrt{2}}\right)^2 \:=\:\frac{\pi R^2}{2}$

The ratio of areas is: .$\displaystyle \frac{A_1}{A_2} \;=\;\frac{\pi R^2}{\frac{\pi R^2}{2}} \;=\;2 \quad\Rightarrow\quad 2:1$