II. Answer the Questions (3 questions in total, 10 points each, write a brief solution process)
4. In the figure on the right, $ABCD$ is a square with a side length of 1, and $A, E, F, G, H$ are the midpoints of the four sides $AB$, $BC$, $CD$, $DA$, respectively. Calculate the area of the red octagon in the figure.
4. The area of the red octagon is $\frac{1}{6}$
4. 【Solution】As shown in the figure, it is easy to know that the area of the blue square is $\frac{1}{5}$, the area of $\triangle \mathrm{ABD}$ is $\frac{1}{8}$, and the area of $\triangle \mathrm{BCD}$ is $\frac{1}{20}$,

Therefore, the area of $\triangle \mathrm{ABC}$ is $\frac{1}{8}-\frac{1}{20}=\frac{3}{40}$, and it can be proven that $\mathrm{AE}: \mathrm{EB}=1: 4$,

The area of the yellow triangle is $\frac{1}{9}$ of the area of $\triangle \mathrm{ABC}$, which equals $\frac{1}{120}$, from which we can derive that the area of the required octagon is: $\frac{1}{5}-4 \times \frac{1}{120}=\frac{1}{6}$.
Thus, we have calculated the areas of all parts, as shown in the figure below.
【Another Solution】Let 0 be the center of the square (the intersection of the diagonals), connect $\mathrm{OE}$, $\mathrm{OF}$, intersecting $\mathrm{AF}$, $B G$ at $M$, $N$ respectively, let the intersection of $A F$ and $E C$ be $P$, connect $O P$, the area of $\triangle M O F$ is $\frac{1}{16}$ of the area of the square, $N$ is the midpoint of $O F$, the area of $\triangle O P N$ is equal to the area of $\triangle F P N$, and the area of $\triangle O P N$ is equal to the area of $\triangle O P M$, so the area of $\triangle O P N$ is $\frac{1}{3}$ of the area of $\triangle M O F$, which is $\frac{1}{48}$ of the area of the square, the area of the octagon is 8 times the area of $\triangle O P M$, which is $\frac{1}{6}$ of the area of the square.