Question 1. $O$ is the centroid of the equilateral $\triangle ABC$, and $D$, $E$, and $F$ are the midpoints of $OA$, $OB$, and $OC$, respectively. The regular triangular pyramid $\mathrm{V}-\mathrm{APC}$ and the regular triangular prism $\mathrm{DEF}-\mathrm{D}^{\prime} \mathrm{E}^{\prime} \mathrm{F}^{\prime}$, both with height $\mathrm{h}$ (where $\mathrm{V}$ is inside the plane $\mathrm{D}^{\prime} \mathrm{E}^{\prime} \mathrm{F}^{\prime}$), have $\triangle \mathrm{ABC}$ and $\triangle \mathrm{DEF}$ as their bases, respectively. Let the side length of $\triangle \mathrm{ABC}$ be $\mathrm{a}$. Draw the 3D diagram and find:
1) The volume of the part of the triangular prism $\mathrm{DEF}-\mathrm{D}^{\prime} \mathrm{E}^{\prime} \mathrm{F}^{\prime}$ that is outside the pyramid $\mathrm{V}-\mathrm{ABC}$.
2) If $\mathrm{M}$ is the midpoint of $\mathrm{AB}$, find the length of the segment of $\mathrm{F}^{\prime} \mathrm{M}$ that lies inside the triangular prism $\mathrm{DEF}-\mathrm{D}^{\prime} \mathrm{E}^{\prime} \mathrm{F}^{\prime}$.
To draw a three-dimensional figure, one must first study the intersection lines of the prism and pyramid. According to the problem, it is easy to know that the lateral edges of the prism $\mathrm{DD}^{\prime}, \mathrm{EF}, \mathrm{FF}^{\prime}$ and the corresponding lateral edges of the pyramid $\mathrm{AV}, B V, C V$ are respectively in the planes VOA, VOB, VOC. Thus, the corresponding lateral edges intersect, with the intersection points being P, Q, R (as shown in Figure 1). Therefore, the intersection lines of the corresponding lateral faces of the prism and pyramid are $P Q, Q R, R P$. According to the problem, it is easy to know that $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ are the midpoints of the lateral edges of the prism and pyramid, and thus the section $P Q R$ is parallel to the top and bottom bases of the prism, and $\triangle \mathrm{PQR}$ is an equilateral triangle.
$$
=\frac{\mathrm{h}}{2} \mathrm{~S}_{\triangle \mathrm{DEF}}=\frac{\sqrt{3}}{32} \mathrm{a}^{2} \mathrm{~h}.
$$
$\therefore$ The volume of the part of the triangular prism outside the section $\mathrm{V}-\mathrm{ABC}$ is $\frac{2}{3} V_{P Q R}-D^{\prime}{ }_{E}{ }^{\prime} \mathrm{P}^{\prime}=\frac{V^{\prime}}{4.8} \mathrm{a}^{2} h_{2}$.
2) According to the problem, we need to first find the two intersection points of $\mathrm{F}^{\prime} \mathrm{M}$ with the prism $\mathrm{DEF}$ - $\mathrm{D}^{\prime} \mathrm{E}^{\prime} \mathrm{F}^{\prime}$. Clearly, $\mathrm{F}^{\prime}$ is one of the intersection points. As shown in Figure 1, the other intersection point is on the intersection line $\mathrm{NN}^{\prime}$ of the planes $M C F F^{\prime} N^{\prime}$ and $\mathrm{DD}^{\prime} \mathrm{E}{ }^{\prime} \mathrm{E}$, which is the intersection point $\mathrm{T}$ of $\mathrm{F}^{\prime} \mathrm{M}$ and $N^{\prime}$. To facilitate the calculation of $\left|\mathrm{F}^{\prime} \mathrm{T}\right|$, the front view of the section $\mathrm{MCF}^{\prime} \mathrm{N}^{\prime}$ is drawn (as shown in Figure 2). $\because \mathrm{V}, \mathrm{O}$ are the centroids of $\triangle D^{\prime} F^{\prime} F^{\prime}$ and $\triangle A B C$, and $\mathrm{N}, \mathrm{F}$ are the midpoints of $\mathrm{MO}$ and $\mathrm{O}$ respectively,
$$
\begin{array}{l}
\therefore\left|F^{\prime} T\right|:\left|F^{\prime} M\right|=|F N|:|F M| \\
=3: 4, \\
\begin{array}{l}
=F^{\prime} M \mid=\sqrt{|F M|^{2}+\left|F F^{\prime}\right|^{2}} \\
\quad=\frac{1}{3} \sqrt{3 \mathrm{a}^{2}+9 \mathrm{~h}^{2}},
\end{array} \\
\begin{array}{l} 
\therefore\left|\mathrm{F}^{\prime} T\right|=\frac{3}{4}\left|\mathrm{~F}^{\prime} \mathrm{M}\right| \\
\quad=\frac{1}{4} \sqrt{3 \mathrm{a}^{2}+9 \mathrm{~h}^{2}} .
\end{array}
\end{array}
$$