Slant height of cone $=$ radius of circle from which sector is cut

                             $l=10cm$

Area length of ${144}^0$ sector $=$ $2\pi \times radius\times \frac{{144}^0}{{360}^0}$

                                              $=$ $2\pi \times 10\times \frac{144}{360}=8\pi$

Circumference of the base of cone $=$ arc length $=$ $8\pi$ cm

$\therefore$ $2\pi r=8\pi \Rightarrow r=4cm$

$h=\sqrt{l^2-r^2}=\sqrt{{10}^2-4^2}=\sqrt{100-16}=\sqrt{84}=2\sqrt{21}$

$\therefore$ Volume of cone $=$ $\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \frac{22}{7}\times 4\times 4\times 2\sqrt{21}$

                                $=$ $\frac{704\sqrt{21}}{21}$