## 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
$r(2+2 \sqrt{2})=x \sqrt{2} \rightarrow r =x \frac{\sqrt{2}}{2+2 \sqrt{2}}=x \frac{2- \sqrt{2}}{2}$
.