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

- Scope of site
- Making polyhedral models
- Other activities
- Mathematical language
- Useful links
- Book suggestions

This website is about simple solid or three dimensional shapes, otherwise known as polyhedra. It concentrates on the Platonic polyhedra (tetrahedron, cube, octahedron, dodecahedron and icosahedron), which are the only regular convex (sticking-out) polyhedra, although it mentions a few more.

Some of the concepts will be within the grasp of Junior school children (age 7-11), although the site as a whole would be rather overwhelming for them. Still Primary school teachers could use the page, either to get ideas for a lesson in solid shapes, or to improve their own knowledge of the subject! It will also be useful for Primary school children of above-average ability. It can be hard to find work for these children while the rest of the class is happily looking at the net of a cube, and this gives some more complicated shapes, and extra concepts to keep them quiet.

For Secondary school, this will be more in the area of recreational mathematics, as a fun activity, or for a maths club. Below, there are links to websites and books which are more challenging if you want to explore further. The subject of polyhedra fascinates mathematicians.

Making up solid shapes from a flat piece of paper intrigues children, and it can become a craft activity if you colour the faces in various ways. The website gives nets for several simple polyhedra. It also describes making a cube or a tetrahedron from a cylinder, which is often used in packaging. It shows how you can make a star by sticking pyramids on the faces of an ordinary polyhedron.

I would suggest that any teacher thinking of making models tries them out first, to see where the potential problems are. It is very important to make sure that the net is accurate if you're drawing your own. The cutting out should also be accurate, which means reasonably good quality scissors which cut cleanly (not always a feature of school scissors!) You should also 'score' along any folds, by drawing along them with a pen or point of some sort quite heavily. Then crease the paper before starting to fold up the figure. When gluing, the usual mistake is to use too much rather than too little. It can help to wait before the glue is 'tacky' before pressing down on the fold. Glue one seam at a time, and make sure that it is holding before going onto the next (easier if you haven't used too much glue!) The last seams are the worst, since you can no longer get a finger inside to apply pressure on both sides of the seam, so you can only push (gently) on the whole polyhedron. You will find it a lot easier to colour the shapes before gluing them, or even cutting them out, although this requires a good ability to visualise the 3-D shape from the 2-D net. Still, that can be an exercise in itself! The designs could emphasise the shape of the polyhedron, or make other designs (like dice dots for a cube), or just be imaginative.

Making shapes from nets means glue, and that means mess, so a teacher may prefer a more cerebral activity. There are three interactive activities which choose the correct nets for cubes, tetrahedra and octahedra. There is also a discussion of Euler's formula. The webpage gives the 'answer', but teachers could read this, then set their own activities, such as getting children to draw up a table of edges, vertices and faces for various polyhedra, and then trying to figure out the relationship between them. I would suggest that you have some solids for them to look at to do this task. You could even make it a graph activity (plot Vertices + Faces against Edges), or an IT activity, as it involves a table, which could be a spread-sheet.

There are also snippets of information about the different polyhedrons, which might be used in projects.

Through this webpage, I have tried to use the correct mathematical language, such as vertex (vertices) for the corner of a polyhedron. This is partly to distinguish between the corner of a polygon (flat shape) and the corner of a polyhedron (solid shape), otherwise one talks of three corners meeting at a corner, which is confusing, to say the least! It is also a good idea to expose children to the correct terms, since they will have to use them later. However, Primary school teachers may prefer to use simpler term, in which case they will have to 'translate' this page. There is a glossary to help teachers to do this. It is important, whichever terms you use, not to get the 2-D and 3-D terms muddled up, so if you don't use the precise terms, make sure that you use different simple words, perhaps angle (2-D) and corner (3-D), or qualify which is which, as in flat shape and solid shape.

Other websites:

- Mathworld on polyhedra - this is technical - below are relevant sections:
- Wikipedia
- Rob's Polyhedron Models - photos of lots of interesting polyhedra

My own websites:

## Useful books |

This is a story imagining what it would be like to live in two dimensions, with visits to one and zero dimensions:

These books give nets for more complicated polyhedra (including some very complicated ones!):

- Mathematical Models by H. Martyn Cundy, A.P. Rollett - buy UK or USA
- Polyhedron Models by Magnus J. Wenninger - buy UK or USA

I can also recommend Tarquin publications.

© Jo Edkins 2007 - Return to Solids index