P5 Maths — Nets of Solids

1. What a net is

A net is what a solid looks like when it is opened out flat. Fold every flap back up and the net becomes the solid again — no flaps left over, and no two flaps landing on the same face.

So a net of a cube has six squares, one for each face, and they must be arranged so that they fold up without overlapping. A net of a square pyramid has one square and four triangles.

2. The eleven nets of a cube

There are exactly eleven, counted up to turning and flipping the paper. They fall into four families, and sorting them that way is the fastest way to remember them.

1-4-1 — four squares in a straight line, one square on one side and one on the other (6 of the eleven)

1-3-2 — three squares in a line, one square on one side and two on the other (3 of the eleven)

2-2-2 — three pairs, stepped like a staircase (1 of the eleven)

3-3 — two rows of three, overlapping in one column (1 of the eleven)

How to use this: if a six-square shape is not one of these eleven, it is not a net of a cube. Most exam questions are the reverse of this — you are shown a shape and must decide — so the four families are the answer key.

3. Five rules that settle any cube-net question

Rule 1 — Count the squares.

A cube has 6 faces, so the shape must have exactly six equal squares. Five squares, or six squares plus an extra one, is not a cube net.

Rule 2 — Look for a 2 × 2 block.

If any four squares form a square block, the shape is never a net of a cube — those four would fold onto only three faces. This one trap answers many exam questions on its own.

Rule 3 — No five squares in a straight line.

At most four squares can lie in one straight line. (Fold a strip of five and the first and fifth squares cover each other.)

Rule 4 — In a straight line, the squares two apart become opposite faces.

Number a strip of four 1, 2, 3, 4: faces 1 and 3 end up opposite each other, and 2 and 4 end up opposite each other. The two flaps sticking off the strip also end up opposite each other. If two squares would be forced onto the same face, it is not a net.

Rule 5 — Fold it in your head, one square at a time.

Roll the cube one square at a time, naming the face each square becomes. If you reach six different faces and none repeats, it is a net.

4. Worked examples

The cross — a net

Four squares in a straight line, with one square above the second and one below it. Number the strip 1, 2, 3, 4: squares 1 and 3 become opposite faces, and 2 and 4 become opposite faces. The square above and the square below fold onto the last two faces. Six different faces, nothing repeated, so this is a net of a cube. Verdict: YES

Five in a row — not a net

The first square and the fifth square both become the top face, so only five faces are covered. Rule 3: never more than four squares in a straight line. Verdict: NO

A 2 × 2 block — not a net

Four of the squares form a 2 × 2 block. Folding them gives only four different faces instead of six, so two faces of the cube are left empty. Rule 2: a 2 × 2 block is always wrong. Verdict: NO

Both extra squares on the same side — not a net

The strip of four is fine, but the two squares above it fold onto the same face, so one face is covered twice and another is left empty. This is the trap that looks almost right. Verdict: NO