A net is what you get when you cut a solid along some of its edges and open it out flat with nothing overlapping. Every face is still there, still the same size, just lying down. Fold it back along the same lines and the solid returns.
Every cardboard box you have opened was made from a net. Flat card is cheap to print, cheap to stack and cheap to transport, and it only becomes a box at the last possible moment.
What makes a net valid
Not every arrangement of squares folds into a cube. A net has to satisfy three conditions.
| Condition | What it means |
|---|---|
| Right number of faces | A cube needs exactly six squares. Five leaves a hole, seven leaves a flap sticking out |
| All joined along edges | Every face must touch another along a full edge, not at a corner |
| Nothing overlaps when folded | Two faces must not end up in the same place |
Look for a strip of four squares in a straight line. Those four wrap round the sides like a belt. The remaining two are the lid and the base, and they must sit on opposite sides of that strip but not directly opposite each other.
Some arrangements look obviously wrong and fold perfectly; some look neat and fail. If you are not certain, do not argue about it. Copy it onto paper, cut it out and fold it. Ten seconds of folding settles what ten minutes of staring cannot.
How many nets does a cube have?
A cube can be opened out in exactly eleven different ways, counting two nets as the same if one is a rotation or mirror image of the other. That number cannot be guessed. It was found by cutting and folding systematically.
All eleven nets fold into the same cube, so mathematically they are equally good. In a factory they are not. A long thin net wastes card; a compact one fits more copies onto a sheet, and on a run of a million boxes that difference is real money.
Nets of the other solids
| Solid | Its net |
|---|---|
| Cuboid | Six rectangles, in matching pairs |
| Triangular prism | Three rectangles in a row, with a triangle above and below |
| Square pyramid | A square with a triangle on each of its four sides |
| Cylinder | A rectangle with a circle at each end |
| Cone | A circle for the base, with a sector of a larger circle wrapped round |
Take the label off a tin without tearing it and lay it flat. Its length is exactly the circumference of the circular end. To draw a correct net for a cylinder of radius 7 cm you need pi, and the rectangle comes out 44 cm long.
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Draw the net of a cuboid measuring 5 cm by 3 cm by 2 cm.
Draw the net of a cylinder with radius 7 cm and height 10 cm.
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A net is a solid cut along some edges and opened out flat with no overlaps.
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