# Other shapes

Solid shapes --- cube --- tetrahedron --- octahedron --- icosahedron --- dodecahedron --- other shapes --- Euler's formula --- glossary --- for teachers

### Platonic solids

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.

### Pyramid

The regular pyramid is a tetrahedron, which is made entirely of triangles, even its base. However, the pyramids in Egypt at Giza are square pyramids. Here is a net to make one for yourself. It is not a regular polyhedron, since it uses a square as well as triangles.

The volume of a pyramid is a third the area of the base times the height.

### Cuboid

The cuboid is similar to a cube, but is made of rectangles rather than squares. It is also known as a right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. (Sometimes the word 'cuboid' has a more general meaning.) There are 6 cuboid faces. Opposite sides are identical rectangles. A lot of food packages are cuboids. Here is a net to make a cuboid for yourself. It is not a regular polyhedron, since it uses rectangles which are not regular shapes. While all the angles of a rectangle are the same, it does not have all sides the same. Only opposite sides are.

If the sides of a cuboid are a, b and c, then the volume is abc.

### Cube Octahedron

A cube octahedron is an attractive shape with faces that are squares and triangles.
It has only 14 faces (6 squares and 8 triangles), so it is quite easy to make.
 Here is its net. Print it out, stick it on thin card, score along the lines and fold them, form the shape, then stick it together with small amounts of glue. For more details, see the notes for the net of a cube. A cube octahedron makes a good base for a star.

### Buckyball or Truncated Icosahedron

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.

 A buckyball has 32 faces, that is, 20 hexagons and 12 pentagons. This shape is called a buckyball after Richard Buckminster Fuller, who invented the geodesic dome. If you look at the football, you will see that it is not really a polyhedron with flat faces. It is made of leather which stretches slightly. So when it is stuffed or blown up, the centres of each face bulge out slightly. This makes a better sphere.

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.

### Star

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.

 Now make the points. You will need one for each face. Here are the nets for the cube octahedron, but you will need 6 of the four-sided points and 8 of the 3-sided points. These don't make a solid, and there are tabs round the hole at the bottom. If you want, you can make a taller point, which will make a more pointy star. Experiment for yourself! Remember that the base of the point must match the edge of the original shape, and the triangles must be isosceles (with two sides the same). Once you have made all the points, and allowed the glue to dry, carefully glue each one to each face of the original solid. Once finished and dry, you can paint it, or stick shiny foil on each point, or cover it with glitter.
 Any shape can be used as a base, but very simple shapes will not give a particularly convincing star, and complicated shapes will take a lot of work and gluing! Here is a very complicated star indeed, but I must admit that it was made from a kit from Tarquin.

### Sphere

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.