令B1D1的中点为E.利用赋值法,令AA1＝1.
∵ABCD－A1B1C1D1是正方体,
∴AA1＝A1B1＝A1D1＝1、∠AA1B1＝∠AA1D1＝∠B1A1D1＝90°,
∴AB1＝AD1＝B1D1＝√2.
∵∠B1A1D1＝90°、E∈B1D1且B1E＝D1E,∴A1E＝B1D1/2＝√2/2.
∵A1B1＝A1D1、E∈B1D1且B1E＝D1E,∴A1E⊥B1D1.
∵AB1＝AD1、E∈B1D1且B1E＝D1E,∴AE⊥B1D1.
由A1E⊥B1D1、AE⊥B1D1,得：∠AEA1＝二面角A－B1D1－A1的平面角.
∵ABCD－A1B1C1D1是正方体,∴AA1⊥平面A1B1D1,∴AA1⊥A1E,
∴tan∠AEA1＝AA1/A1E＝1/（√2/2）＝√2.
∴∠AEA1＝arctan√2.
∴二面角A－B1D1－A1的大小为arctan√2.