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Confused about the language to use with 3D shapes?

Thursday, 28 April 2022
A question that I am regularly asked concerns the correct language to use when teaching the properties of 3D shapes. This is a typical question I had from one school:
Spheres are often referred to having one face, cones two faces and cylinders three faces. Mathematically a face on a 3D shape is part of a plane and therefore is flat. Does this means that much of the language that has been used is inaccurate?

They went on to say that their staff had discussed their method for explaining that a sphere has a curved surface; a cone has 1 face and 1 curved surface etc. However, the definition of an edge is that it is where 2 faces meet. They were unsure how this could be explained and asked if I had any suggestions.

3D shapes

What a great question! I'm asked quite a lot about this and I remember when writing a maths dictionary some years ago that it was a point of contention between maths academics at the time. Some of the issues are because we want it defined for primary school use to make it accessible to our children.

It is best to consider this by sorting shapes by their properties. Start with any 3D or solid shape and they can be classified as polyhedra and non-polyhedra.

A polyhedron is any solid shape which has faces that are polygons, joined at their edges to make a solid. As a polygon is a closed shape made with straight sides, the edges of a polyhedron are therefore straight, with plane faces, not curved. You could then further classify and sort the shapes in this set - cubes, cuboids, pyramids, prisms, platonic solids etc. The relationship between the vertices, edges and faces (Euler's Formula) is only valid for different polyhedra.

Now we can look at the other set of non-polyhedra, and these will include spheres, cones, cylinders and hemispheres. As these are not made with faces of polyhedra, then they include curved surfaces, faces and curved edges. This is where definitions get a little loose at primary level, but if you are consistent and make a clear distinction between surfaces and faces as well as edges and curved edges then this should be fine. 


So, going back to the question from the school, they were just about spot on with their thoughts on this in their discussion: 

  • a sphere has a single curved surface and no edges or vertices.
  • a cone has one curved surface, a flat face with a curved edge (usually, but not always, a circle) and an apex. To make it clearer, the face can be called a base and the point is the apex, not a vertex.
  • a cylinder has one curved surface, two curved edges (always circles) and no vertices.
  • a hemisphere has one curved surface, a flat face and a curved edge (always a circle).

What is fantastic is that schools are having these types of discussions in staff meetings, in an attempt to get accuracy and consistency in the use of mathematical language and the related teaching methods. This is sure to have a real impact on the quality of teaching and learning in the classroom.
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