## 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
$4:5$
.

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