Euler’s polyhedron formula holds for every convex polyhedron: a cube (8 − 12 + 6), a tetrahedron (4 − 6 + 4), a soccer ball (60 − 90 + 32). It was one of the first results of topology.
How to use the Euler’s Formula for Polyhedra Calculator
- Find
- Faces F
- Vertices V
- Edges E
Results update as you type, and everything is calculated in your browser.
How it works
Choose which quantity is missing and enter the other two.
Inputs explained
| Field | What to enter |
|---|---|
| Find | Choose one: Faces F, Vertices V, Edges E |
| Faces F | A number |
| Vertices V | A number |
| Edges E | A number |
Example values
How the main result changes when Faces F changes and everything else stays at the example values above:
| Faces F | Missing value |
|---|---|
| 3 | 9 |
| 4.5 | 11 |
| 6 | 12 |
| 7.5 | 14 |
| 9 | 15 |
| 12 | 18 |
Formula
V − E + F = 2
Worked example
Inputs — Find: Edges E; Faces F: 6; Vertices V: 8.
Results — Missing value = 12.
Key terms
- Euler characteristic
- V − E + F.
Frequently asked questions
Does Euler's formula hold for all polyhedra?
For convex ones (and any polyhedron topologically like a sphere).
What about a torus-shaped polyhedron?
V − E + F = 0.
How many edges does a dodecahedron have?
30 (V = 20, F = 12).
What is the Euler characteristic?
χ = V − E + F, a topological invariant.