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Ratio of Radius and Side of Square for Two Circles in a Square

The diagram shows two circles inside a square. Crircles touch the square, and each other. If the side of the square is  
\[x\]
  and the radius of the circle is  
\[r\]
, what is  
\[r\]
  in terms of  
\[x\]
?
We can form small sqaures bewttn the centre of each circle and the nearest vertex of the square.

The diagonal is of length  
\[\sqrt{r^2+r^2}= r \sqrt{2}\]
.
The distance between the centres of the circles is  
\[2r\]
.
In terms of  
\[r\]
, the length of the diagonal of the whole square is  
\[r \sqrt{2} +2r + r \sqrt{2}=r(2+2 \sqrt{2}) \]
.
In terms of  
\[x\]
, the length of the diagonal of the whole square is  
\[\sqrt{x^2+x^2}= x \sqrt{2}\]
.
Equating these  
\[r(2+2 \sqrt{2})=x \sqrt{2} \rightarrow r =x \frac{\sqrt{2}}{2+2 \sqrt{2}}=x \frac{2- \sqrt{2}}{2}\]
.