Ta có $\widehat{\left( \left( C'AI \right),\left( ABC \right) \right)}=\widehat{CIC}={{60}^{o}}$

Mặt khác  $\tan \widehat{CIC'}=\frac{CC'}{CI}\Rightarrow CC'=CI.\tan \widehat{CIC'}=\frac{a\sqrt{3}}{2}$

Ta có ${{S}_{ANI}}=\frac{1}{4}{{S}_{ABC}}=\frac{1}{4}.\frac{{{a}^{2}}\sqrt{3}}{4}=\frac{{{a}^{2}}\sqrt{3}}{16}$

$\Rightarrow {{V}_{C'.NAI}}=\frac{1}{3}CC'.{{S}_{NAI}}=\frac{1}{3}.\frac{a\sqrt{3}}{2}.\frac{{{a}^{3}}\sqrt{3}}{2}=\frac{{{a}^{3}}}{32}$