Ratio of Areas of Squares in Semicircles

Supoose we have two equal semicircles. Inside one semicrcle is is a square, and int the other are two squares. What is the area of the single square divided by the total area of the two squares?

The radius of each semicircle is  
\[r\]
.

If the side of the square on the left is  
\[x\]
  then from the diagram,
\[(2x)^2+x^2=5x^2=r^2 \rightarrow x = \frac{r}{\sqrt{5}}\]
.
The area of the square is  
\[(2 \frac{r}{\sqrt{5}})^2 = \frac{4r^2}{5}\]
.
For the square on the right, of side  
\[y\]

\[y^2+y^2=2y^2 =r^2 \rightarrow y = \frac{r}{\sqrt{2}}\]
.
The area of the two squares is  
\[2 \times (\frac{r}{\sqrt{2}})^2 =r^2\]
.
The ratio of the areas is  
\[\frac{4r^2}{5} : r^2\]
  which simplifies to  
\[4:5\]
.