### 11.203 Through every three vertices of a cube, located at the ends

 of each triplet of edges converging at one vertex, a plane is drawn. Find the volume of the body bounded by these planes, if the edge of the cube is $a$.
## Solution.

A

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-261.jpg?height=359&width=378&top_left_y=1134&top_left_x=187)
We connect the center of each face of the given cube with the centers of all adjacent faces. The resulting body $K N L Q M P=T$ is the desired one. To verify this, it is sufficient to consider any face $K L Q$: it lies in the plane $\triangle A B C$, whose vertices are the vertices of a triplet of edges of the given cube converging at one vertex $D$.
$M N L Q$ is a square with side $\frac{a}{\sqrt{2}}\left(Q L=\frac{1}{2} A B\right)$, and the height $h$ dropped from $K$ to $M N L Q$ is $\frac{a}{2} \Rightarrow$

$$
\Rightarrow V_{T}=2 V_{K M N L Q}=2 \cdot \frac{1}{3} h \cdot S_{M N L Q}=\frac{a^{3}}{6}
$$

Answer: $\frac{a^{3}}{6}$.