5. (10 points) In square $A B C D$, $A B$ is 4 cm long, $A E=A F=1$, quadrilateral $E F G H$ is a rectangle, and $F G=2 E F$. Then the total area of the "fengbiyuan" (shaded part) is $\qquad$ square centimeters.
【Solution】The length of $AC$
$$
\sqrt{4^{2}+4^{2}}=4 \sqrt{2}
$$
The length of $EF$:
$$
\sqrt{1^{2}+1^{2}}=\sqrt{2}
$$

The area of trapezoid $AEHC$:
$$
\begin{aligned}
& (2 \sqrt{2}+4 \sqrt{2}) \times \frac{\sqrt{2}}{2} \times \frac{1}{2} \\
= & 6 \sqrt{2} \times \frac{\sqrt{2}}{2} \times \frac{1}{2} \\
= & 3 \text { (square centimeters) }
\end{aligned}
$$

The area of hexagon $AEHCGF$
$$
3 \times 2=6 \text { (square centimeters) }
$$

The area of the blank part of rectangle $EFGH$ is half of the rectangle's area $\sqrt{2} \times 2 \sqrt{2}=4$ (square centimeters)

The area of the shaded part
$6-2=4$ (square centimeters)
Answer: The area of the shaded part is 4 square centimeters.
Therefore, the answer is: 4.