Ta có: V_ACMNPQ = V_EAMNC – V_EACPQ

\[{V_{EACPQ}} = \frac{1}{3}d(E,(ACPQ)).{S_{ACPQ}} = \frac{1}{3}d(E,\,(ACD)).\left( {{S_{ACD}} - {S_{DPQ}}} \right)\]

\[{V_{EACPQ}} = \frac{1}{3}d(B,(ACD))\,.\,\left( {{S_{ACD}} - \frac{1}{9}{S_{ACD}}} \right) = \frac{8}{9}{V_{ABCD}}\]

(Vì P, Q là trọng tâm của ΔBCE và ΔABE)

\[{V_{ACMNPQ}} = \frac{V_{ABCD}} = \frac \cdot \frac{{{a^3}\sqrt 2 }} = \frac{{11\sqrt 2 {a^3}}}\]

Vậy \[{V_{ACMNPQ}} = \frac{{11\sqrt 2 {a^3}}}\].