## 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

\[r(2+2 \sqrt{2})=x \sqrt{2} \rightarrow r =x \frac{\sqrt{2}}{2+2 \sqrt{2}}=x \frac{2- \sqrt{2}}{2}\]

.