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Polyhedron Vertices, Edges & Faces Relation

Solve the Euler V − E + F = 2 relation for a missing count in an applicable closed polyhedron.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For a cube, 8 vertices and 6 faces give 12 edges because 8 + 6 − 12 = 2.

V − E + F = 2; E = V + F − 2

Worked example

Enter these known values and leave the other values blank.

Vertices (V)
8
Faces (F)
6
Edges (E)
12

Assumptions and limitations

  • The relation with right-hand side 2 applies to convex polyhedra and more generally suitable closed polyhedral surfaces topologically equivalent to a sphere.
  • Handles, holes, open boundaries or disconnected surfaces require a different topological treatment. Do not apply this formula indiscriminately to them.
  • Counts must be positive whole numbers. Satisfying the equation alone does not prove that a proposed set of counts can form a polyhedron.
  • This calculator checks the numerical relation; it does not construct or inspect a 3D shape.

Common questions

Does this test whether a polyhedron exists?

No. It solves one necessary counting relation for the stated class of surfaces. A list of counts can satisfy the arithmetic without describing a realizable shape.

What counts does a tetrahedron have?

It has 4 vertices, 6 edges and 4 faces. The equation gives 4 − 6 + 4 = 2.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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