9. Suppose the slant height of a cone is a constant $a$, then the maximum value of its volume is

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9. $\frac{2}{27} \sqrt{3} \pi a^{3}$ Let the base radius be $r$, then the height is $\sqrt{a^{2}-r^{2}}$,
$V=\frac{1}{3} \pi r^{2} \sqrt{a^{2}-r^{2}}=\frac{1}{3 \sqrt{2}} \pi \sqrt{r^{2} \cdot r^{2} \cdot\left(2 a^{2}-2 r^{2}\right)} \leqslant \frac{1}{3 \sqrt{2}} \pi \sqrt{\left(\frac{2 a^{2}}{3}\right)^{3}}=\frac{2 \sqrt{3}}{27} \pi a^{3}$, when $r=\sqrt{\frac{2}{3}} a$, the equality holds.