What this calculator does
Euler's formula for polyhedra works out finds the missing vertex, edge or face count of a convex polyhedron from V − E + F = 2. Enter your own figures above and the answer updates as you type: nothing is fixed in the code, so the result reflects exactly the numbers you supply.
The formula this calculator evaluates is printed under the tool and explained below, so you can check the working by hand or reuse it in a spreadsheet.
The formula
FormulaV − E + F = 2, for any convex polyhedron (V = vertices, E = edges, F = faces)
The inputs explained
| Field | What to enter |
|---|---|
| Solve for | Choose from Vertices (V), Edges (E), Faces (F). |
| Vertices (V) | A number. Starts at 8. |
| Edges (E) | A number. Starts at 12. |
| Faces (F) | A number. Starts at 6. |
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the answer changes with vertices (v)
Every other input is held at the calculator’s starting values while vertices (v) varies. Select any row to load that scenario into the calculator.
How the answer changes with vertices (v)
| Vertices (V) | Faces (F) | Check: V − E + F | Formula used |
|---|---|---|---|
| 4 | 10 | 2 (should equal 2 for a convex polyhedron) | V − E + F = 2 |
| 6 | 8 | 2 (should equal 2 for a convex polyhedron) | V − E + F = 2 |
| 8 | 6 | 2 (should equal 2 for a convex polyhedron) | V − E + F = 2 |
| 12 | 2 | 2 (should equal 2 for a convex polyhedron) | V − E + F = 2 |
| 16 | -2 | 2 (should equal 2 for a convex polyhedron) | V − E + F = 2 |
| 24 | -10 | 2 (should equal 2 for a convex polyhedron) | V − E + F = 2 |