Where will I start?

The diameter of the semi-circle is $4\sqrt{2}$ . Therefore, the smaller side of the rectangle is $2\sqrt{2}$ . The angle opposite side $c$ is $135^\circ$ . By cosine law,

$c^2=4^2+(2\sqrt{2})^2-2(4)(2\sqrt{2})(\cos 135)$

$c \approx \boxed{6.325}$