jmc

A full circle with radius 18 has circumference $2(\pi )(18)=36\pi$, so a 300-degree sector has arc length (shown in blue below) $$\frac{300^{\circ }}{360^{\circ }}\cdot 36\pi =30\pi .$$

When we fold the sector into a cone, the arc length of the sector becomes the circumference of the base of the cone, and the radius of the sector becomes the slant height of the cone.



Let the cone that is formed have height $h$ and radius $r$. Thus we have $$2\pi r=30\pi$$and $$r^2+h^2=18^2$$From the first equation we have $r=15$; from the second equation we have $h=\sqrt{18^2-15^2}=\sqrt{99}=3\sqrt{11}$.

Finally, the desired volume is $$\frac{1}{3}{r}^{2}h\pi =\frac{1}{3}(15^2)(3\sqrt{11})\pi =225\pi \sqrt{11}.$$So, dividing the volume by $\pi$ gives $\boxed{225\sqrt{11}}$.