Nets of polyhedra — Mathematics, 11–13
Unfold a solid flat, fold it back, and see that one solid can have several nets. Mathematics, 11–13 years.
A solid opened flat
A net is what you get when you cut the surface of a polyhedron along some edges and lay it flat in one piece. Every face appears once. When you fold it back, sides that meet become one edge. A cube's net has 6 squares. The same polyhedron can have different nets: the cube has 11 (nets that are only turned or flipped count as the same).
The whole surface at a glance
A net shows every face of the solid at the same time, which is how a box is cut out of cardboard before it is folded. It also lets you connect the flat drawing with the solid. Sides that will meet become the same edge, and you can mark them with the same colour. Parallel and perpendicular faces can be found on the net, and checked by folding it.
Counting edges on a cube net
Net: a row of 4 squares (A, B, C, D), one square above B and one below B. There are 6 × 4 = 24 sides. The 5 folds between squares use 5 × 2 = 10 sides, so 14 sides are left on the border. They are glued in pairs: 14 ÷ 2 = 7 edges. Total: 5 + 7 = 12 edges, as in the cube. Parallel faces: A and C, B and D, and the top and bottom squares.
Six joined squares are not always a cube
It is tempting to think that any 6 squares joined by their sides fold into a cube. Try a straight row of 6. Folding, squares 1 to 4 go round the cube like a belt, but square 5 lands on top of square 1 and square 6 on top of square 2. Two faces are covered twice and two are left empty. Only 11 arrangements of 6 squares work. When in doubt, fold it, or use coloured sides.
Boxes that arrive flat
Many cardboard boxes arrive flat and you fold them: the flat piece is a net. To make a paper pyramid with a square base you draw 1 square and 4 triangles, one on each side of the square. That net has 5 faces, and folding it gives 8 edges. Before you cut, you can plan the whole model on paper.
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