Find the area of the blue square

The diagonal of a square is given by $d=a\sqrt{2}$ where $a$ is the side length. From the figure the diagonal is $d=4+2\sqrt{2}$ . Substitute:

$4+2\sqrt{2}=y\sqrt{2}$

Dividing both sides by $\sqrt{2}$ , we get

$y=\dfrac{4+2\sqrt{2}}{\sqrt{2}}$

Squaring both sides, we get

$y^2=\dfrac{16+16\sqrt{2}+8}{2}=\dfrac{24+16\sqrt{2}}{2}=\boxed{12+8\sqrt{2}}$