240. Inscribed Spheres. If the faces of a certain hexahedron are equilateral triangles, congruent to the faces of a regular octahedron, then what is the ratio of the radii of the spheres inscribed in these polyhedra?
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240. Our hexahedron is composed of two regular tetrahedra of volume \( V_{t} \). Further, we note that \(\frac{V_{t}}{V_{0}} = \frac{1}{4}\) (see the solution to problem 113), where \( V_{0} \) is the volume of the octahedron. From this, denoting the volume of the hexahedron by \( V_{h} \), we get \(\frac{V_{h}}{V_{0}} = \frac{1}{2}\). Moreover, the surface area of the hexahedron is \(\frac{6}{8}\) of the surface area of the octahedron, so \(\frac{S_{h}}{S_{0}} = \frac{3}{4}\). Since \( V_{h} = \frac{r_{h} S_{h}}{3} \) and \( V_{0} = \frac{r_{0} S_{0}}{3} \), we find that \(\frac{r_{h}}{r_{0}} = \left(\frac{V_{h}}{V_{0}}\right) \left(\frac{S_{0}}{S_{h}}\right) = \frac{2}{3}\).

[X. Ives, A. M. M., 56, 693 (December 1949).]