Interesting geometry problem (square and two circles) What's the area of the main square? (I think the attached picture defines the problem clearly.)
Let the square be of side $2a$.  Clearly $a &gt; 6$. Then with the origin at the centre of the square, the inside circle can be expressed as $(x-b)^2+y^2=(a-b)^2$

So we know that the intercept of this circle with the positive $Y$ axis must be $a-4$ and the intercept with the $-X$ axis is $6-a$.  Hence 
$$b^2+(a-4)^2=(a-b)^2, \qquad (6-a-b)^2=(a-b)^2$$

$$\implies a \in \{3, 8\} \implies a = 8$$

So the area required is $256$.
There are infinite solutions.

Please see the GeoGebra.org figure that I constructed to demonstrate my answer here. (the following text is repeated there, so no need to read it twice)

Here are the steps I took to construct this figure, under the assumption that the given figure in the original post is not to scale:


I drew arbitrary points A1 and B1, connected them by a line, and labeled them "Move me".
I used a circle and perpendicular lines to construct the large square. (Because the construction of the figure is dependent upon the first two points and the hidden circle connecting them, the whole figure scales and moves along with those two points if they are moved)
I found the midpoints of the square's segments and connected them, they cross at the center, point O.
I inscribed the large circle into the square.
I made circles of radius 4 and 6, per the problem, coloring the segments of set length. (This figure does have axes, origin, and absolute distances, but they are only shown in the radii of the two dashed circles)
I used the "circle through 3 points" tool in Geogebra to construct the inner circle using points A, B, and C.


I labeled the given points A,B,C, and O.
I added segments AB, and BC as dashed lines, and used Geogebra to measure the angle ABC. It looks like this might be a right angle, but it turns out that ABC can be almost any angle, depending on the side length of the large square.

Any size square you choose to make using my GeoGebra.org figure fits all of the criteria in the originally given figure, so there are infinitely many possible areas for the large square. Or in other words, constraining the distance between two circles inside of a square (as in the given problem) does not constrain the size of the square.

In case you cannot get the dynamic figure to work, here is the same figure with two arbitrary square side lengths. The given segment lengths remain constant, but the large square and inscribed circle can be almost any arbitrary size.

(In case you cannot get the dynamic figure to work, 
here is the same figure with two arbitrary square side lengths. The given segment lengths remain constant, but the large square and inscribed circle can be almost any arbitrary size.)