4. A cube with edge $a$ is given. The faces of the cube serve as bases for six pyramids whose common vertex is the center of the cube. A sphere is inscribed in each of these pyramids. Find the volume of the body whose vertices are the centers of the spheres.
Solution. The body whose vertices are the centers of the spheres is composed of two congruent pyramids joined at their bases. The base of the pyramid is a square with side $2 R$ and height $\frac{a}{2}-R$, where $R$ is the radius of the spheres. Therefore, we have

$$
V=\frac{8 R^{2}}{3}\left(\frac{a}{2}-R\right)
$$

Thus, to find the volume of the body, we need to find the radius $R$ of the spheres.
![](https://cdn.mathpix.com/cropped/2024_06_05_969e4efbfe816ce80b6fg-4.jpg?height=274&width=864&top_left_y=350&top_left_x=440)

In the left drawing above, one of the six pyramids is represented, and in the right drawing, the section $A B C$ is shown, where $D$ is the center of the sphere. We have

$$
\begin{gathered}
P_{A C D}+P_{B C D}+P_{A B D}=P_{A B C}, \\
\frac{\frac{a \sqrt{2}}{2} R}{2}+\frac{\frac{a \sqrt{2}}{2} R}{2}+\frac{a R}{2}=\frac{a \cdot \frac{a}{2}}{2},
\end{gathered}
$$

from which we get $R=\frac{a}{2(\sqrt{2}+1)}$. Substituting into (1) we get $V=\frac{a^{3} \sqrt{2}}{3(\sqrt{2}+1)^{3}}$.

## Third Year