3. A sphere is tangent to the six edges of a regular tetrahedron. If the edge length of the tetrahedron is \(a\), then the volume of the sphere is \(\qquad\)
(2000 National High School League Question)
3. $\frac{\sqrt{2}}{24} \pi a^{3}$. Let the center of the sphere inscribed in the regular tetrahedron $ABCD$ be $O$, the radius of the sphere be $r$, the volume be $V$, the center of face $BCD$ be $O_{1}$, and the midpoint of edge $BC$ be $E$. Then $AO_{1}=\frac{\sqrt{6}}{3} a$, $OB=\frac{\sqrt{6}}{4} a$, $r=\frac{\sqrt{2}}{4} a$, hence $V=\frac{\sqrt{2}}{24} \pi a^{3}$.