3. A cube with edge $a=\sqrt{2+\sqrt{2}}$ is illuminated by a cylindrical beam of light with radius $\rho=\sqrt{2}$, directed along the main diagonal of the cube (the axis of the beam contains the main diagonal). Find the area of the illuminated part of the cube's surface.
Answer: $\frac{\pi \sqrt{3}}{2}+3 \sqrt{6}$