11.074. The height of the pyramid is 8 m. A plane parallel to the base is drawn at a distance of 3 m from the vertex. The area of the resulting section is $4 \mathrm{~m}^{2}$. Find the volume of the pyramid.
Solution.

The volume of the pyramid $V=\frac{1}{3} S H$ (Fig. 11.67). Pyramid $S A B C D E F$ is similar to pyramid $S A_{1} B_{1} C_{1} D_{1} E_{1} F_{1}$ with a similarity coefficient $k=\frac{S O}{S O_{1}}=\frac{8}{3}$.

Then the areas of their bases are in the ratio $\frac{S_{A B C D E F}}{S_{A_{1} B_{1} C_{1} D_{1} E_{1} F_{1}}}=k^{2}$. Hence, $S_{A B C D E F}=\frac{64}{9} \cdot 4=\frac{256}{9}$ ( $\left.\mathrm{cm}^{2}\right)$. Therefore, $V=\frac{1}{3} \cdot \frac{256}{9} \cdot 8 \approx 75.85 \mathrm{~cm}^{3}$.

Answer: $\approx 75.85 \mathrm{~cm}^{3}$.