Cube of a Binomial — (a + b)³

Just like (a + b)² can be seen as a square split into four area pieces, (a + b)³ can be seen as a cube split into eight volume pieces. This page lets you explore that cube directly: rotate it, then move each piece across one at a time to watch the formula build itself up.

Splitting the Cube

Take a cube whose every edge is (a + b) long. Split each of its three edges — length, width and height — into an a-part and a b-part. That divides the whole cube into a 2×2×2 grid of 8 smaller boxes:

• 1 box of size a×a×a — volume a³
• 3 boxes of size a×a×b (one for each pair of edges) — volume a²b each
• 3 boxes of size a×b×b — volume ab² each
• 1 box of size b×b×b — volume b³

Add up all 8 volumes and nothing has been added or removed from the original cube — only cut apart — so a³ + 3a²b + 3ab² + b³ must equal the same volume as the whole cube, (a + b)³.

Try It Yourself

Drag either cube with your mouse to rotate it and look at it from any angle. Then click a piece in the left cube to send it into its matching empty slot on the right — the dashed outline always shows you exactly which piece is missing from the left cube and which one has arrived on the right.

Why This Proves the Identity

Nothing about cutting a solid into pieces and moving them around changes their total volume. The 8 boxes above are exactly the same material as the original (a + b)³ cube, just relabelled by which edge-part (a or b) each one occupies along each direction. So:

Practice: Cube of a Binomial