For any solid with flat faces and straight edges (a polyhedron), the faces (F), vertices (V) and edges (E) are linked: F + V − E = 2. Check it on a cube: 6 + 8 − 12 = 2. So if you know two of the counts, you can find the third. To find edges, add the faces and vertices, then take 2. To find faces, take the vertices from the edges, then add 2. To find vertices, take the faces from the edges, then add 2. Always check by putting all three numbers back into the formula: the answer must be exactly 2.
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Words to know
- polyhedron
- a solid whose faces are all flat polygons
- Euler's formula
- F + V − E = 2, true for every polyhedron without holes
Worked examples
A cube has 6 faces and 8 vertices. Use Euler's formula F + V − E = 2 to find how many edges it has.
- 1.E = F + V − 2.
- 2.E = 6 + 8 − 2 = 12.
= 12
A polyhedron has 12 edges and 6 vertices. How many faces does it have?
- 1.F = E − V + 2.
- 2.F = 12 − 6 + 2 = 8.
= 8
A polyhedron has 12 vertices. It has 18 more edges than vertices. How many faces does it have?
- 1.Edges: 12 + 18 = 30.
- 2.F = 30 − 12 + 2 = 20.
= 20
For parents
Method: Rearrange F + V − E = 2 and check the answer by substituting it back.
What to say: Which count is missing? After you find it, do F + V − E. Did you get 2?
Common mistakes:
- Leaves out the 2 in F + V − E = 2, as if faces and vertices added up to the edges exactly.
- Moves the 2 across the equals sign without changing its operation (adds 2 where it should subtract, or the reverse).
- Uses the 'add two counts and take 2' pattern for faces or vertices too, instead of rearranging the formula.
- Finds the count described in words (the first step) and gives it as the answer.