## Ratio of Areas of Squares in Semicircles

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 \[4:5\]

.