We will use (twice) the fact that
                             _
The diagonal of a square is Ö2 times a side of the square. 

Let the radius of the circles be R.

We draw in the vertical and horizontal radii of the
circles in red and the diagonals of the small squares
thus formed by them and the sides of the large square 
in green:



Since each of the green lines are diagonals of
squares with sides of length R then each green line is 
  _
RÖ2 centimeters in length.

Next we draw in two more radii of the circles (in blue),
completing the diagonal of the large square.



Now we know that the whole diagonal of the large square is

   _
10Ö2 centimeters in length, since its

side is 10 cm.

Therefore we add all the parts of the diagonal of the large square
and equate the sum to that length, and we have this equation:



or



Divide through by 2



Factor out R on the left:



Divide through by 



That's the answer, except maybe we would want to rationalize the
denominator:



                                         _ 
So that's the answer. The radius is 10-5Ö2 
 

Edwin