11.031. The center of a cube with edge length $a$ is connected to all its vertices. Determine the volume and surface area of each of the resulting pyramids.
## Solution.

Since the construction resulted in 6 identical pyramids (Fig. 11.26), each of them has a volume $V=\frac{a^{3}}{6}$ and a total surface area $S_{\text {full }}=a^{2}+4 \cdot \frac{1}{2} a \cdot O K$. But $O K=\sqrt{O P^{2}+P K^{2}}=\frac{a \sqrt{2}}{2}$, from which $S_{\text {full }}=a^{2}+a^{2} \sqrt{2}=a^{2}(1+\sqrt{2})$.

Answer: $\quad \frac{a^{3}}{6} ; a^{2}(\sqrt{2}+1)$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-669.jpg?height=402&width=386&top_left_y=1387&top_left_x=159)

Fig. 11.27