2.32. The centers of the faces of a regular tetrahedron serve as the vertices of a new tetrahedron. Find the ratio of their surface areas and the ratio of their volumes.
2.32. Let the volumes and surface areas of the given and new tetrahedra be denoted by \( V_{1}, S_{1} \) and \( V_{2}, S_{2} \) respectively. Let \( AB = a \) (Fig. 2.27); then \( DK = \frac{a}{2} \). Since \( \triangle DSK \sim \triangle MSN \), we have \( DK : MN = SD : SM = \frac{3}{2} \), from which \( MN = \frac{a}{3} \). Finally, we obtain

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-472.jpg?height=517&width=504&top_left_y=107&top_left_x=302)

Fig. 2.27

$$
V_{1} : V_{2} = a^{3} : \left( \frac{a}{3} \right)^{3} = 27 : 1, \quad S_{1} : S_{2} = a^{2} : \left( \frac{a}{3} \right)^{2} = 9 : 1
$$

Answer: \( 9: 1 ; 27: 1 \).