Find volume of the circumscribing tetrahedron Take four spheres each with a radius of 1.  Place them so they are mutually tangent.  Circumscribe a regular tetrahedron around the spheres.  Compute the volume of the tetrahedron.

The tetrahedron $T_c$ formed by the centers of the four spheres has sidelength $2$, height $h=2\sqrt{2\over3}$, and volume $V_c={2\sqrt{2}\over3}$. The faces of $T_c$ have  distance
$$d={h\over4}={1\over\sqrt{6}}$$
from the center $C$ of $T_c$. The faces of the circumscribing tetrahedron $T$ have the distance $d+1$ from $C$. It follows that the volume $V$ of $T$ is given by
$$V=\left({d+1\over d}\right)^3 V_c={38\over3}\sqrt{2}+12\sqrt{3}\ .$$
