Unfolding a Box

When a cereal box is empty, you can flatten it for recycling. Pull the glued flaps apart, open the box along its edges, and it lies flat on the table as one piece of card. The box was made this way, too: a factory printed and cut a flat shape, then folded it up and glued it. What does that flat shape look like?

A solid, opened out

In the Solid Shapes lesson we saw that a box is six faces joined at their edges. If you cut along just enough of the edges, the faces stay joined in one piece and open out flat.

The flat shape you get is called a net. Fold the net back up along the same edges and you have the solid again.

A cereal box being opened out: on one side the closed box, and on the other the flat piece of card it unfolds into, six rectangles joined at their edges, with the matching faces in the same colours

Every face of the solid is in its net, once. So a net of a cuboid has 6 rectangles, and a net of a square-based pyramid has 1 square and 4 triangles.

The nets of a cube

A cube has 6 square faces, so every net of a cube is 6 squares joined side to side. The best known looks like a cross: four squares in a row, with one more above and one below.

But that is not the only one. If you cut a cube open along different edges, you get different nets. Mathematicians have checked every way: a cube has exactly 11 different nets (turning a net round or flipping it over doesn't count as a new one).

The 11 different nets of a cube, each made of six squares joined side to side

Six squares that won't fold

Not every shape made of 6 squares is a net of a cube. Try folding six squares in one long row: the first four wrap round into a ring, and the last two land on top of squares that are already there. Two faces overlap, and two sides of the cube are left open.

A block of 2 rows of 3 squares doesn't work either. Look at a point inside it where four squares meet. At each corner of a cube only three faces meet, so four squares can't all fold round one corner: two of them would have to overlap.

Two shapes made of six squares that don't fold into a cube. A row of six squares, numbered 1 to 6: squares 1 to 4 fold into a ring, square 5 lands on square 1 and square 6 on square 2. A block of two rows of three, with a dot where four squares meet at one point

To check a net, imagine one square as the bottom and fold the others up round it, one at a time. If every square finds its own face, with none left over and none on top of another, it is a net of a cube.

Nets with curves

A cylinder has a net too. Cut off its two circle ends, then cut straight down its curved side and unroll it: the side becomes a rectangle. So the net of a cylinder is a rectangle and two circles. The long side of the rectangle is as long as the circumference of a circle end, since it wrapped once round it.

A tin can opened out into its net: a rectangle for the label, with a circle above it and a circle below it; the long side of the rectangle is the same length as the way round each circle

Try it at home

Open out an empty box. How many faces does its net have? Were there any extra flaps for the glue?
Draw the cross-shaped net of a cube on squared paper, 3 cm squares. Cut it out and fold it into a cube.
Peel the label off a tin can (ask first). What shape is it?

Answers: 1. 6 faces, usually with a few small flaps that are glued inside. 3. A rectangle.

Words to know

net - a flat shape that folds up into a solid shape

Big idea

A net is a solid opened out flat. Every face appears in it once. A cube has 11 different nets, but not every six squares will fold into a cube, and the net of a cylinder is a rectangle and two circles.