Task B-3.3. (20 points) From a cube with an edge length of $3 \mathrm{~cm}$, all eight vertices have been cut off so that the new body has all edges of equal length. What is the volume of the newly formed body?

![](https://cdn.mathpix.com/cropped/2024_05_30_9c56249806d755c5106ag-11.jpg?height=508&width=539&top_left_y=223&top_left_x=770)
Solution. From the equations $b+2c=a$ and $b=c\sqrt{2}$, we get $c=\frac{3(2-\sqrt{2})}{2} \text{ cm}$.

The volume of each of the eight tetrahedra is

$$
V_{T}=\frac{c^{3}}{6}=\frac{9(10-7 \sqrt{2})}{8} \text{ cm}^{3}
$$

The volume of the cube is $V_{K}=a^{3}=27 \text{ cm}^{3}$.

Finally, the volume of the body is

$$
V=V_{K}-8 \cdot V_{T}=63(\sqrt{2}-1) \text{ cm}^{3}
$$