We had this in a quiz, my answer was $\displaystyle \sqrt{2} = 1$

but my friends say the answer is 2 = 1 because they squared both sides. but I don't get why the need to square it. so here's the problem.

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?

What I did was

Let A1 = large circle

Let A2 = small circle

Radius of large circle:

$\displaystyle R = \frac {\sqrt{2}}{2} e $

Radius of small circle:

$\displaystyle r = \frac {1}{2} e $

$\displaystyle A1 = A2$

$\displaystyle \pi R^2 = \pi r^2$

I then cancel out pi and square leaving the radii.

$\displaystyle \frac {\sqrt{2}}{2} e= \frac {1}{2} e ] 2$

Thus becomes $\displaystyle \sqrt{2} = 1$