Example 3 A sphere is tangent to all six edges of a regular tetrahedron. If the edge length of the regular tetrahedron is $a$, find the volume of this sphere.
Solve As shown in Figure 16-4, let the center of the sphere be $O$, the radius be $r$, and the volume be $V$. The center of $\triangle B C D$ is $O_{1}$, and the midpoint of edge $B C$ is $E$.
Then $A O_{1}=\sqrt{a^{2}-O_{1} B^{2}}=\sqrt{a^{2}-\frac{1}{3} a^{2}}=\frac{\sqrt{6}}{3} a$
From $O B^{2}=O_{1} O^{2}+O_{1} B^{2}=\left(O_{1} A-(O B)^{2}+O_{1} B^{2}\right.$
we get $\quad \frac{2}{3} a^{2}-\frac{2 \sqrt{6}}{3} a \cdot O B+\frac{1}{3} a^{2}=0$
Thus $\quad O B=\frac{3}{2 \sqrt{6}} a=\frac{\sqrt{6}}{4} a$
Therefore $\quad r=O E=\sqrt{O B^{2}-B E^{2}}=\sqrt{\frac{3}{8} a^{2}-\frac{1}{4} a^{2}}=\frac{1}{2 \sqrt{2}} a$
So $V=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi \cdot \frac{1}{16 \sqrt{2}} a^{3}=\frac{\sqrt{2}}{24} \pi a^{3}$