How to teach networks and Euler's formula
Year 10 (ages 15 to 16)
A network (or graph) models connections between things using vertices (points) and edges (connecting lines). Euler's formula, V - E + F = 2, relates the number of vertices, edges and faces of any connected planar network.
How to teach it
- Introduce networks with a real example (a map of train stations, a social network of friendships) before any formula.
- Count vertices, edges and faces together on a simple drawn network, including the outer face, before stating Euler's formula.
- Verify the formula on several small examples (a triangle, a square, a network with one internal division) to build confidence it always holds.
- Practise rearranging the formula to find whichever one value (V, E or F) is missing, given the other two.
- Connect networks to real situations: interpreting what a network diagram represents, not just calculating with it.
Worked example
A connected planar network has 6 vertices and 9 edges. Find the number of faces F = 2 - V + E = 2 - 6 + 9 = 5
Common mistakes
- Forgetting to count the outer (unbounded) region as one of the faces.
- Rearranging Euler's formula incorrectly, e.g. adding instead of subtracting a term.
- Miscounting vertices or edges on a busy diagram, especially where edges cross without an actual vertex.
- Applying Euler's formula to a network that is not connected or not planar, where it does not directly hold.
Frequently asked questions
What is Euler's formula for networks?
For a connected planar network (a graph drawn without edges crossing), V - E + F = 2, where V is the number of vertices, E the number of edges, and F the number of faces (including the outer, unbounded region).
What counts as a 'face' in Euler's formula?
Every region enclosed by edges, PLUS the single outer region surrounding the whole network. A simple triangle has 2 faces: the inside and the outside.
How do you use Euler's formula to find a missing value?
Rearrange the formula for whichever value is missing: F = 2 - V + E, E = V + F - 2, or V = E - F + 2, then substitute the two known values.
What year is interpreting networks taught?
In the Australian Curriculum this is a Year 10 skill (AC9M10SP02): interpreting networks and network diagrams that model relationships in real situations.
Practise with free worksheets
Printable worksheets with answer keys that are never wrong.