3. Given a sphere that is tangent to all six edges of a regular tetrahedron with edge length $a$. Then the volume of this sphere is $\qquad$ .
3. $\frac{\sqrt{2}}{24} \pi a^{3}$.

Construct a cube with edge length $\frac{\sqrt{2}}{2} a$, and the radius of the sphere is $\frac{\sqrt{2}}{4} a$.