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  

If the side of the square on the left is  
  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  

\[y^2+y^2=2y^2 =r^2 \rightarrow y = \frac{r}{\sqrt{2}}\]
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  

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