Mathematics Year 7: Faces, edges, vertices and Euler's formula
Mathematics, Year 7 (Portugal): Count many polyhedra, find what never changes, and use it to check your own counts.
A number that never changes
Count the vertices (V), edges (E) and faces (F) of several polyhedra and write them in a table. Cube: 8, 12, 6. Tetrahedron: 4, 6, 4. Octahedron: 6, 12, 8. Pentagonal prism: 10, 15, 7. A pattern appears: V − E + F is always 2. This is Euler's formula, written V − E + F = 2, or V + F = E + 2. You found it the way rules are found: by looking at many cases.
More relations hide in the table
The formula links the three numbers: know two and you find the third. There is more. Each edge lies between two faces, so adding up the sides of all the faces counts every edge twice: sum of sides = 2 × E. If all faces are triangles, 3 × F = 2 × E, so F is even and E is a multiple of 3. In a prism E = 3 × n is a multiple of 3, and in a pyramid E = 2 × n is even.
A wider table, and a football
(V, E, F): tetrahedron 4, 6, 4; cube 8, 12, 6; octahedron 6, 12, 8; dodecahedron 20, 30, 12; icosahedron 12, 30, 20. Always V − E + F = 2, e.g. 20 − 30 + 12 = 2. Now a test on the truncated icosahedron (12 pentagons, 20 hexagons). F = 12 + 20 = 32. Sides: 12 × 5 + 20 × 6 = 180, so E = 180 ÷ 2 = 90. Every vertex has 3 faces, so V = 180 ÷ 3 = 60. And 60 − 90 + 32 = 2.
When the sum is not 2
You count a solid and get V = 8, E = 13 and F = 6: 8 − 13 + 6 = 1. “Euler's formula failed!” It did not: it holds for every polyhedron you study, so the mistake is in the count. With 8 vertices and 6 faces the formula asks for E = 12. Use it to check your counts. But a 2 does not prove the counts right, because two mistakes can cancel each other.
Checking 3D models
Anyone who builds 3D models on a computer works with lists of vertices, edges and faces. Counting the three and testing V − E + F = 2 is a quick check that a piece has not gone missing. You can also plan a model: a polyhedron with 12 vertices and 30 edges must have F = 2 + 30 − 12 = 20 faces, as the icosahedron does. One formula saves you from counting everything one by one.
Test what you learned
A cube has 8 vertices, 12 edges and 6 faces. Which relation is true for the cube, and for all the polyhedra you studied?
- a) V + E = F + 2
- b) E + F = V + 2
- c) V + E + F = 2
- d) V + F = E + 2
Correct answer: d) V + F = E + 2 — For the cube 8 + 6 = 14 and 12 + 2 = 14. It also works for the tetrahedron: 4 + 4 = 6 + 2. This is Euler's formula.
A polyhedron has 8 faces, all triangles, and Euler's formula holds for it. How many vertices does it have?
- a) 6
- b) 12
- c) 18
- d) 24
Correct answer: a) 6 — Yes: 8 × 3 = 24 sides, so 12 edges, and V = 2 + 12 − 8 = 6. This is the case of the octahedron.
Léa counts a solid: V = 8, E = 13, F = 6, so V − E + F = 1. What should she do?
- a) Conclude that Euler's formula does not work for this solid.
- b) Accept the counts: 1 is close enough to 2.
- c) Decide the formula only works for regular polyhedra, so this solid must be irregular.
- d) Count again: one count is wrong. If V and F are right, E must be 12.
Correct answer: d) Count again: one count is wrong. If V and F are right, E must be 12. — Right: 8 − E + 6 = 2 gives E = 12, so she probably counted one edge too many. Euler's formula works as a check.
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