Mother of all Questions...

My suggestion for this problem:

Draw a square incribed the circle. ( You have half that square already; 2 sides of that square are the segments created by the circle and the large square)

Diameter of the circle is equal to diagonal of that small square.

Using small square's diagonal to calculate the its side.

Calculate the small square's area.

The circle's area = small square's area + 4 additonal pieces ( on the original graph, you can see there are two of them already. After drawing the small square, you will see all)

So area of 4 pieces = The circle's area - small square's area

If the circle were incribed in the big square, the remain area would be: Big square'area - circle's area

However, 2 small pieces is outside the big square, then the remain area will be:

(Big square's area - circle's area ) + 2 small pieces

And we have known how to calculate each item in the equation as I mentioned above.

Hope it helps.