To make a net of a cube, first look at one, such as a dice. How many faces does it have? Six, so make sure that your net has six squares. Now you must work out a way to arrange six squares so they will fold up into a cube. There are eleven different ways to do this, apart from rotations (turn it round) and reflections (turn it over). See if you can find them all below. The correct nets will change colour as you click on them. Click on New go for another go.

The easiest way to find a net is to think of a cube as four sides, a top and a bottom. Arrange four squares in a line. These are the sides. Now put the top square on one side of this line, and the bottom on the other. It doesn't really matter where on each side, they all work. There are 6 possible arrangements (apart from rotations and reflections). There are other layouts which work, but you need to think about them. Arrange three squares around a point. These will form a vertex (or corner). Arrange another three similarly. These will form the opposite vertex. Now lay one group alongside the other. There are 3 of these arrangements. The last 2 arrangements are harder to see. One is two lines of three, staggered. Both rows forms U-shapes, which fit into each other. The last arrangement is a T-shape, which forms an empty box, and one more square to make the top.

If the sides of a cube are length a, then the volume is a³ or a times a times a.

In fact, most food packages are cuboids rather than cubes.

There are three different solids that you can make with triangles. The first is the triangular pyramid, made of four triangles. Usually pyramids have three triangles and a square, such as the Great Pyramid in Giza, Egypt. This is an interesting shape, but it isn't a regular shape, since it uses a square as well as triangles. To stop confusion between the two sorts of pyramids, mathematicians use the word tetrahedron to describe a triangular pyramid. 'Tetra' means four, and the tetrahedron has four sides. If you were going to be very pedantic, you could describe a cube as a hexahedron, but people tend not to! The mathematical word for a solid shape like these is a polyhedron (poly means many). By the way, the plural is polyhedra. A regular tetrahedron has its faces as equilateral triangles, where all sides are equal, and the angle between them is 60°

See if you can work out which nets will make a tetrahedron. There are two correct nets, and they will change colour as you click on them. Click on New go for another go.

The volume of a pyramid is a third the area of the base times the height. This is true of a tetrahedron as well as a Egyptian pyramid.

A tetrahedron has three triangles at each vertex (corner). Unlike a cube, it is possible to have more than three triangles at the vertex of a regular solid. These make an octahedron or anicosahedron.

A tetrahedron has three faces meeting at each point, similar to a cube. We saw that you cannot make a polyhedron with four squares meeting at each point, but you can do it with four triangles. This makes a shape called an octahedron. You can tell from its name that it has eight faces (similar to an octagon, which is a flat shape with eight angles).

See if you can find all the nets for an octahedron below. There are eleven correct nets, and they will change colour as you click on them. Click on New go for another go.

Of course, for an octahedron, you must have eight triangles in the net. There are also four triangles round a point. These are hard to work out round the edge, but if five or six triangles are together, you know that something is wrong.

Octahedra are not usually used as packaging! But in fact an octahedron makes an attractive box. You usually think of an octahedron on its point, as this makes its shape obvious. However, if you lay it flat on one face, then it has not only a flat bottom but a flat top as well. However, the sides are not vertical. You need to think how to make it into a box with a lid. Perhaps you could make two octahedra, each with one side removed and one bigger than the other. Then the bigger one could fit over the smaller one. It must only be bigger by a very small amount, or it will rattle.

An octahedron has four triangles at each vertex (corner). It is possible to have a different number of triangles at the vertex of a regular solid. These make a tetrahedron or an icosahedron.

A tetrahedron is made of triangles with three triangles at each point. An octahedron has four triangles at each point. Can we fit five triangles at each point? Yes, we can, and it's called anicosahedron. It has 20 faces.

There are 43380 distinct nets for the icosahedron, so I don't expect you to find them all! An icosahedron can be thought of as ten triangles going round the 'equator', with five at the 'north pole' and five at the 'south pole'. Thinking of this helps us to work out a net.

You might like to think of a colour scheme for your finished shape. It's a lot easier if you colour it in before you stick it together, or even before you cut it out. Try to imagine what the finished shape will look like when colouring it in. You could try to draw lines that run over edges. That's easy if the faces are together in the net, less easy if there are gaps! Do you want straight lines or curvy ones? Can you draw a line which will end up going right round the shape? How about colouring all the bits near a point in the same colour? You could paint the 'equator' one colour, and the two 'poles' another. There are diamonds on the finished surface - try finding them and painting them. An icosahedron is fairly close to a sphere, so if you are really ambitious, you could try drawing the world on it.

Then when you stick it together, you can see if your shape's design looks anything like you imagined it would!

So far, we have found a tetrahedron with three triangles at each vertex (corner), an octahedron with four triangles and an icosahedron with five triangles. If you try to fit six triangles round a point, it becomes flat, so there are no more.

A dodecahedron's faces are pentagons (5 sides). There are 12 faces, and 3 faces meeting at each vertex (corner).

A dodecahedron is the only regular polygon which uses pentagons, as it is impossible to fit more than 3 pentagons round a vertex. There are no polyhedrons which use only hexagons, as three hexagons at a vertex would make a flat surface. However, a buckyball uses both hexagons and pentagons.

There are five Platonic solids: cube, tetrahedron, octahedron, icosahedron and dodecahedron. These are convex regular polyhedra. Convex means that the vertices (corners) stick out rather than in. A regular polyhedron has all its faces and angles between them the same.

There are other solids which are not so regular which are well-known.

There is a story that a scientist discovered what the molecule of a new form of carbon looked like. He found that it was an interesting shape, a bit like a ball, but made of hexagons and pentagons arranged in a regular pattern. He was very excited and rang up a friend who was a mathematician to boast of this new shape that he'd found. The mathematician told him to look at a soccer ball! Even footballers can't get away from mathematics.

Here is a net of a buckyball. See the notes for the net of a cube to see how to print this net and make your own buckyball. I'm afraid that I've left the tabs out of this one. Add them on every other side of the edges of the net. I suggest that you do NOT start on this net first! Try a simpler one to get used to the idea.

It is easy to make an attractive star. Start with a shape such as an octahedron or a cube octahedron. Make this shape up (the nets are provided on this site) and wait for it to dry.

It is impossible to make a perfect sphere (ball or globe) from a flat sheet of paper. Paper can curve in one direction, but cannot curve in two directions at the same time. So all spheres made from paper or card will be approximations. Probably the best way to make a sphere is to make a polyhedron with a large number of sides. A football is a buckyball, for example, and you can make a ball from a dodecahedron or an icosahedron. In these cases, the material of the surface stretches a little to make a better sphere, since the faces are not flat but bulge out in the centre.

Another way to make a sphere is with pointed ellipses. Globes can be made this way, since the edges of the net run along longitudes. This would be easier if you were sticking the map of the globe onto an existing ball, but I think it would be tricky to make a sphere like this with just this net. All those points meeting at the 'poles' would be very difficult to stick together. It would be a good idea to have a small disc of paper to stick over each pole to hold them together. I've left out the tabs as well, as I'm not sure where they would go.

If you want to make a globe, here are some websites to help you.

Polyhedron	Faces	Vertices (corners)	Vertices + Faces	Edges
Tetrahedron	4	4	8	6
Cube	6	8	14	12
Octahedron	8	6	14	12
Icosahedron	20	12	32	30
Dodecahedron	12	20	32	30

Some of these websites are designed for children, parents and teachers; others are for general use.

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