Ta có :$\frac{{{V}_{D.AHK}}}{{{V}_{D.ABC}}}=\frac{SA}{SA}.\frac{SK}{SC}.\frac{DH}{DB}=\frac{1}{2}.\frac{DH.DB}{D{{B}^{2}}}=\frac{1}{2}.\frac{A{{D}^{2}}}{A{{D}^{2}}+A{{B}^{2}}}$

                     $=\frac{1}{2}.\frac{4{{a}^{2}}}{4{{a}^{2}}+3{{a}^{2}}}=\frac{2}{7}$

       ${{V}_{D.ABC}}=\frac{1}{3}DA.{{S}_{ABC}}=\frac{1}{3}2a.\frac{1}{2}2a.a\sqrt{3}=\frac{2{{a}^{3}}\sqrt{3}}{3}$

 Suy ra    ${{V}_{AHKD}}={{V}_{D.AHK}}=\frac{4{{a}^{3}}\sqrt{3}}{21}$.