Take a cardboard cereal box, snip along some edges, and flatten it out. What you are left with is a net — the 3D shape opened up into one flat piece. Every cardboard box, drinks carton, and flat-pack shelf you have ever owned started life as a net: a flat outline that folds up into the solid.
A net is wonderfully useful because it shows all the faces at once, laid side by side where you can see and measure them. That makes it the perfect tool for working out a solid's surface area or for building a model from card.
A net is the flat 2D shape you get when you unfold a solid and lay all its faces out side by side, joined along their edges. Fold the net back up and you get the solid again. Each face of the solid appears exactly once in the net, so the number of pieces in the net equals the number of faces.
For example:
Step through the figure. First a cube flattens into its six squares; then a triangular prism unfolds into two triangles and three rectangles. Count the pieces — that is the number of faces.
A net folds up into a solid — here is the assembled cube that the flat net makes. Drag to rotate it and picture the six square faces of the net wrapping round into these six sides.
1) Which nets fold into a cube? Six squares can be arranged many ways, but not
all of them close up into a cube. A straight line of six squares (
2) Surface area of a cuboid from its net. A cuboid measures
Reading the areas straight off a net is exactly why nets make surface area easy — no hidden faces to forget.
3) Sketching a net. To draw the net of a triangular prism, start with a row of 3 rectangles (the long side faces) and attach a triangle to the top and bottom of the middle rectangle (the two ends) — 5 pieces in all. For a square-based pyramid, draw 1 square in the middle and fold a triangle up from each of its four edges — 1 square + 4 triangles = 5 pieces.
Two traps snare almost everyone with nets:
Nets are how the physical world gets packaged. Every drinks carton, pizza box and flat-pack wardrobe is designed as a flat net first — printed and cut from a single sheet, then folded up — because a flat sheet is cheap to print and stacks with almost no wasted material. Designers play a real game of geometry, tiling nets edge-to-edge so the cutting machine throws away as little card as possible.
The same "which flat shape folds into which solid?" question reaches into deep mathematics. Topology studies exactly how surfaces can be cut and folded; engineers use clever fold patterns (some borrowed from origami) to pack huge structures into tiny spaces — a satellite's solar panels launch folded flat and unfold like a net once in orbit. Even proteins, the tiny machines in your cells, work by folding a long flat chain into a precise 3D shape. Unfolding and refolding turns out to be one of nature's favourite tricks.