Classifying three-dimensional shapes

Three-dimensional shapes - part 1

The first classification of three-dimensional shapes is to separate out those that have curved surfaces, such as a sphere (a perfectly round ball), a hemisphere (sphere cut in half), a cylinder (like a baked-bean tin) and a cone (see the nearest motorway). A shape that is made up entirely of flat surfaces (also called plane surfaces) is called a polyhedron (plural: polyhedral). How can you tell that a surface is a plane surface? Mathematically, the idea is that you can join up any two points on the surface by a straight line drawn on the surface. A spherical surface is not plane, for example, because two points can be joined only by drawing circular arcs. To describe a polyhedron we need to refer to the plane surfaces, which are called faces (not sides, incidentally), the lines where two faces meet, called edges, and the points where edges meet, called vertices (plural of vertex). The term ‘face’ should only be used for plane surfaces, like the faces of polyhedral. It is not appropriate, for example, to refer to a sphere ‘as a shape with one face’. A sphere has one continuous, smooth surface – but it is not ‘a face’.

Construction of some simple three-dimensional shapes from nets is an excellent practical activity for primary school pupils, drawing on a wide range of geometric concepts and practical skills.

As with polygons, the word regular is used to identify those polyhedral in which all the faces are the same shape, all the edges are the same length and all the angles between edges are equal. Whereas there are an infinite number of different kinds of regular polygons, there are, in fact, only five kinds of regular polyhedral. These are: (a) the regular tetrahedron (four faces, each of which is an equilateral triangle); (b) the regular hexahedron (usually called a cube; six faces, each of which is a square); (c) the regular octahedron (eight faces, each of which is an equilateral triangle); (d) the regular dodecahedron (twelve faces, each of which is a regular pentagon); and (e) the regular icosahedron (twenty faces, each of which is an equilateral triangle).