6. Given a square with a side length of 8. Each side of the square is divided into two equal segments by a point. Find the area of the quadrilateral whose vertices are the constructed points.
6. The area of the inner quadrilateral can be obtained by subtracting the areas of four triangles from the area of the larger square. Each pair of such triangles forms a square with a side length of $4 \mathrm{~cm}$, and the area of this square is $16 \mathrm{~cm}^{2}$. Thus, the area of the desired quadrilateral is $32 \mathrm{~cm}^{2}(8 \times 8-16 \times 2=32)$.

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-168.jpg?height=522&width=520&top_left_y=710&top_left_x=768)

Answer: $32 \mathrm{~cm}^{2}$.