Example 2. For a regular quadrilateral frustum $\mathrm{ABCD}$, the side length of the lower square base is 2, and all other edges are 1. $\mathrm{M}$ is the intersection point of the diagonals of the side face $\mathrm{BB}^{\prime} \mathrm{C}^{\prime} \mathrm{D}$ (Figure 6). Find the lengths of $\mathrm{A}^{\prime} \mathrm{M}$ and $\mathrm{DM}$.
Solve: Unfold the sides $\mathrm{AA}^{\prime} \mathrm{B}^{\prime} \mathrm{B}$, $\mathrm{BB}^{\{\prime} \mathrm{C}^{\prime} \mathrm{C}$, and $\mathrm{CC}^{\prime} \mathrm{D}^{\prime} \mathrm{D}$ onto a plane (Figure 7). $\mathrm{O}$ is the intersection point of the lateral edges.

According to the plane geometry knowledge of the unfolded surface, $\mathrm{A}^{\prime} \mathrm{B}$ and $\mathrm{MB}$ are perpendicular to $\mathrm{BH}^{\prime}$, $\mathrm{MC} \perp \mathrm{CC}^{\prime}$, and $\mathrm{A}^{\prime} \mathrm{B} = \sqrt{3}$, $\mathrm{MB} = \mathrm{MC} = \frac{\sqrt{3}}{3}$. Let $\mathrm{DQ}$ be the distance from $\mathrm{D}$ to $\mathrm{C}^{\prime} \mathrm{C}$. Thus, $\mathrm{DQ} = \frac{\sqrt{3}}{2}$, $\mathrm{CQ} = \frac{1}{2}$.

The dihedral angles between the two sides of the regular frustum are all equal, denoted as $\theta$, and it is known that $\cos \theta = -\frac{1}{3}$. Therefore,
$$
\begin{array}{l}
=2 \text {, } \\
\end{array}
$$
$$
\begin{aligned}
\mathrm{DM} & =\sqrt{\mathrm{CQ}^{2} + \mathrm{DQ}^{2} + \mathrm{MC}^{2} - 2 \mathrm{DQ} \cdot \mathrm{MC} \cdot \cos \theta} \\
& =\frac{\sqrt{15}}{3} .
\end{aligned}
$$