Maximum area of $2$ circles in a square

Question

A man has a square piece of paper where each side has length $1$ m. Two equal circles are to be cut from this paper. What is the radius, in meters, of the largest possible circles?

This is what I did:

• area of square: $1$

• area area of circle: $2\pi(r^2)$

I multiplied by $2$ since they are $2$ circles. Now I made $2\pi r^2=1$ and solved for $"r"$, however the answer I got is completely off. May you please tell me what I did is wrong and how I can fix that?

Answer 1

That is the picture that fits the problem.

See that

$$CE=\sqrt{2}=CA_1+A_1A+AE=\sqrt{2}r+2r+\sqrt{2}r \to r=\frac{\sqrt{2}}{2+2\sqrt{2}}=\frac{2-\sqrt{2}}{2}$$

EDIT

Hint

To prove that it is the maximum work with the picture below:

Work with variation of $\alpha$, the trapezium $EFGK$ and $DG+GK+KC=DC=1$.