过A作AE⊥A1B1交A1B1于E、作AF⊥A1D1交A1D1于F,
再作AO⊥平面A1B1C1D1交平面A1B1C1D1于O,连结EO、FO.
∵AA1＝AA1、∠AA1E＝AA1F＝60°、∠AEA1＝∠AFA1＝90°,∴AE＝AF.
∵O是A在平面A1B1C1D1上的射影,
∴EO、FO分别是AE、AF在平面A1B1C1D1上的射影,而AE＝AF,∴EO＝FO.
∵AE⊥A1E、AF⊥A1F,EO、FO分别是AE、AF在平面A1B1C1D1上的射影,
∴由三垂线定理的逆定理,有：EO⊥A1E、FO⊥A1F,又EO＝FO,
∴O在∠B1A1D1的平分线上.
∵∠AA1E＝60°、∠AEA1＝90°、AA1＝b,∴A1E＝b/2、AE＝√3b/2.
∵A1B1C1D1是正方形,A1O平分∠B1A1D1,∴∠OA1E＝45°.
∵∠OA1E＝45°、∠A1EO＝90°,∴EO＝A1E＝b/2.
∴AO＝√（AE^2－EO^2）＝√［（3/4）b^2－（1/4）b^2］＝（√2/2）b.
∵A1B1C1D1是边长为a为正方形,∴S(A1B1C1D1)＝a^2.
∴V(ABCD－A1B1C1D1)＝S(A1B1C1D1)×AO＝a^2×（√2/2）b＝（√2/2）a^2b.