Chapter 7: Identifying Solids
Copied from Year 7 SoW 2005
C u t t i n g C o r n e r s
A cube has
6 faces
I 2 edges and
8 vertices or corners
Edge
The vertices can be make a "truncated cube."
A truncated cube has more edges than the original chopped off to faces, vertices and cube.
1. W ork out how many faces (F), vertices (V) a solid has which is formed from a cube with chopped off. and edges (E) only one vertex
2. Suppose a cube has two of its vertices chopped off .
(a) What shape is each new face?
Is this always true? (b)
3. How many edges, faces and vertices does this new solid have?
4. Now copy and complete this table, by chopping off more of the original vertices of the -cube.
Number of vertices chopped from cube
0
Number of faces
F
6
Number of vertices
V
8
Number of edges
E
12
1
2
3
4
5. Find a relationship between V, F and E. Describe this relationship clearly in words or symbols.
(Make sure you look for a relationship that links all t h e t h r e e values together

6. You can chop off the vertices of an octahedron in a similar way. W hat shape is each new face?
7. Draw up a table similar to the one for the cube and complete it for the faces, edges and vertices of an octahedron.
8. Does the relationship yo u found for the cube solids work for the octahedron solids?
O P T I O N A L E X T E N S I O N
Attempt this section only i f you have completed and checked questions I to 8.
Tetrahedron
Square-based pyramid
9. Try other solids and check your relationship each time.

Name
Cuboid
Vertices removed
Faces
F
Vertices
V
Edges
E
Check.
Euler ?
0
1
2
6 8 12
Tetrahedron
Vertices removed
0
1
2
Faces
F
4
Vertices
V
4
Edges
E
Check...
Euler ?
6 
Square based pyramid
Vertices removed
Faces
F
Vertices
V
Edges
E
Check...
Euler ?
0
1
2
Octahedron
Vertices removed
5
Faces
F
5
Vertices
V
8
Edges
E

Check...
Euler ?
0
1
2
8 6 12 

Chapter 7: Identifying Solids
Aim:
Chapter 7 is the only opportunity in the 3 year ‘Key Maths’ scheme of work for students to explore polyhedra with a focus on volume an d surface area. This investigation can be used to familiarise students with polyhedra whilst discovering
Euler’s formula.
This is also a good investigation to use for differentiation as weaker students can focus on cutting the corners off a cube whilst others can extend this idea to other polyhedra.
Lesson Outline:
Students can use the worksheets copied above or the activity can be introduced more generally to students - encouraging them to become more responsible for setting out their investigation.
The Mathematics in progression:

Ensure students understand the key words: vertex/vertices, edge, face .

Most students will be able to spot that the columns F, V and E increase by one, two and three, respectively. More
F V E difficult is getting them to explain why this
0 6 8 12
1 7 10 15 occurs and in particular why the vertices increase by two and not three.
2 8 12 18
 Students should eventually look for Euler’s formula:
F+V-E=2
 The most able students should not only note Euler’s formula but may also try to explain why it works for convex polyhedra.
Further resources:
The website: mymaths.co.uk
has clear descriptions of faces, edges and vertices under ‘shape – 3D shapes’.
The website: http://mathworld.wolfram.com/Polyhedron.html
has clear java animations of many different polyhedra.
Real polyhedra are always going to be better than such virtual shapes and the mathematics department has plenty of hands-on resources for students to use.
A good website for trying to explain why Euler’s formula works is: http://www.mathsisfun.com/geometry/eulers-formula.html
It goes way beyond the understanding of many year 7 students but is still a great discussion.