3. The volume of the circumscribed sphere of a regular tetrahedron with edge length $a$ is $V$, then $V=$.
3. Let the four vertices of a regular tetrahedron be $A, B, C, D$. The center is $O$, and extend $AO$ to intersect $\triangle BCD$ at $H$.
Then $AO=R, OH=\frac{1}{3} R$.
$$
\begin{array}{l}
\Rightarrow BH=\sqrt{BO^{2}-OH^{2}}=\frac{2 \sqrt{2}}{3} R \\
\Rightarrow BC=\sqrt{3} \cdot BH=\frac{2 \sqrt{6}}{3} R=a,
\end{array}
$$

So $R=\frac{3}{2 \sqrt{6}} a=\frac{\sqrt{6}}{4} a$,
So $V=\frac{4}{3} \pi R^{3}=\frac{\sqrt{6}}{8} \pi a^{3}$.