Example $\mathbf{5}$ Three $12 \times 12$ squares are each divided into two pieces, $A$ and $B$, by a line connecting the midpoints of two adjacent sides, as shown in Figure 5. These six pieces are then attached to the outside of a regular hexagon and folded into a polyhedron. Find the volume of this polyhedron.
【Analysis】It's not easy to draw an intuitive diagram directly, but the square piece of paper gives some inspiration. We might as well draw a cube first, and then try to mark some midpoints.

Solution As shown in Figure 6, it is easy to see that the figure formed by connecting the six midpoints is a regular hexagon. This regular hexagon section precisely divides the cube into two desired polyhedra. Therefore, the volume of the required polyhedron is half of the cube's volume, i.e.,
$$
\frac{1}{2} \times 12^{3}=864 \text {. }
$$