Circling Around

The square formed by the four circles has an edge length of 2. The diagonal of this square is 2*sqrt(2). If the radius of the smaller circle is r, the diagonal of the square is also equal to 1 + 2r + 1. Thus we can solve for r:

1 + 2r + 1 = 2 sqrt2, 2 + 2r = 2 sqrt2, r = sqrt2- 1, r ≈ 0.41421.

If we connect the radius of 4 circles, then we can get a square. The perimeter of the square is 8 because the radius is 1, and the side of the square is 2 times of radius, so the length of the side is 2. So, the perimeter is 8. Then, let the diagonal of the square to $a$ . We get $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$ by Pythagorean theorem, so $2 ^ { 2 } + 2 ^ { 2 } = a ^ { 2 }$ . Then $a$ equals to $2 \sqrt { 2 }$ . The radius of big circle is 1, so we subtract 2 from the $2 \sqrt { 2 }$ . Then, we get the 2 times of radius of the small circle. It is $2 \sqrt { 2 } - 2$ . Then we get the radius of small circle, $\frac { 2 \sqrt { 2 } - 2 } { 2 }$ which is equal to $\sqrt { 2 } - 1$ . So it is $\sqrt { 2 } - 1$ , but we have to give the answer to 3 decimal digits, so it is $1 . 414 - 1 = \boxed { 0 . 414 }$ .