11.183. A regular triangular prism is described around a sphere, and a sphere is described around it. Find the ratio of the surfaces of these spheres.
Solution.

Let $r$ and $R$ be the radii of the inscribed and circumscribed spheres (Fig. 11.73); then $B D=3 r, A D^{2}+B D^{2}=A B^{2}=4 A D^{2}$, $B D^{2}=3 A D^{2}, A D^{2}=3 r^{2}$.

From $\triangle A K D$ we find that

$$
K A^{2}=K D^{2}+A D^{2}=r^{2}+3 r^{2}=4 r^{2}, \mathrm{a}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0882.jpg?height=456&width=459&top_left_y=382&top_left_x=804)

From $\triangle O K A$ we find that $O A^{2}=O K^{2}+K A^{2}=r^{2}+4 r^{2}=5 r^{2}=R^{2}$. Denoting the surfaces of the inscribed and circumscribed spheres by $s$ and $S$, we have $s=4 \pi r^{2}$, $S=4 \pi R^{2}$, then $S: s=R^{2}: r^{2}=5: 1$.

Answer: $5: 1$.