According to cube properties, AA 1⊥ bottom→ aa1⊥ b1d1.

∫b 1d 1⊥a 1c 1，a 1c 1∑AC→AC⊥b 1d 1。

∴D 1B 1⊥ airplane A 1C 1CA.

Take the midpoint O 1D 1.

A, o, c, A 1, O 1, C 1 are all on the ACC 1A 1 plane.

Take the H point on AA 1 as OH⊥AO 1.

Three lines can be used to prove that OH is vertical AB 1D 1.

Then, the height and bottom area of the triangular pyramid O-AB 1D 1 can be easily obtained, and V= (bottom area * height) /3.

Method 2: Overall redundancy

The volume of triangular pyramid O-AB 1D 1 =V cube -V triangular prism BCD-b1c1d1-v triangular pyramid b1-aob-VD1-.

The bottoms and heights of the next three triangular pyramids are equal.

So v = a * a * a * (1/2-1/4) = (a 3)/4.