11. A sphere is tangent to all six edges of a regular tetrahedron. If the edge length of the tetrahedron is $a$, then the volume of the sphere is $\qquad$ .
The sphere is the inscribed sphere of the cube in which the regular tetrahedron is inscribed. Let its radius be $r$, then the edge length of the cube is $\frac{\sqrt{2}}{2} a$ $\Rightarrow r=\frac{\sqrt{2}}{4} a \Rightarrow V_{\text {sphere }}=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{\sqrt{2}}{4} a\right)^{3}=\frac{\sqrt{2} \pi a^{3}}{24}$.