Nets

To make a solid from flat material, you need a flat shape that folds up into it. That shape is called a net. Seen the other way, a net is what you get when you unfold a solid onto a flat surface.

Two practical points. The material must be strong enough to stand up. And the faces need something to hold them together: tape for cardboard, or small extra flaps for glue on lighter paper. You have seen those flaps on packaging boxes.

But the net itself is only the unfolded shape. Flaps help with making the box; they are not part of the net.

A cube can be unfolded in more than one way. A cube has exactly 11 different nets (counting two as the same if one is just a turn or a flip of the other). Try to find all 11 with paper squares.

A regular tetrahedron (a triangular pyramid whose faces are all equilateral triangles) has just 2 nets.

Curved solids unfold too. Take off a cylinder's two circles and cut the curved part straight down its height. You get a rectangle with a circle at each end. The rectangle's width is the circumference of the circle, and its height is the cylinder's height.

Cut a cone along a slant line and unroll it, and you get a sector of a circle (a slice of a round shape), plus the base circle. Why a sector? Every point on the base circle is the same distance from the tip O. After unrolling they are still all that distance from O, and points all the same distance from one point lie on a circle's arc.

Two more solids. An octahedron is two square pyramids joined at their square bases; like the cube, it has 11 nets. A dodecahedron has faces that are all pentagons. Mathematicians have counted its nets exactly: 43,380.