Euler Formula For Polyhedra Calculator

Work out euler formula for polyhedra instantly with clear inputs, formula shown and shareable results.

About Euler Formula For Polyhedra Calculator

For every convex polyhedron Euler's formula gives V − E + F = 2. Other values of the characteristic indicate a surface with holes: a genus-g surface has χ = 2 − 2g, so a torus-like polyhedron gives χ = 0.

Formulas

Euler's polyhedron formula

V − E + F = 2

Genus

χ = 2 − 2g

Worked example

With the default values below, Euler Formula For Polyhedra Calculator returns Verdict of Valid convex polyhedron (χ = 2). Change any input above and every figure here is recalculated in your browser.

Inputs used

Vertices Vminimum 1: 8
Edges Eminimum 1: 12
Faces Fminimum 1: 6

Results

Verdict: Valid convex polyhedron (χ = 2)
Euler characteristic V − E + F: 2
Implied genus (number of holes): 0.0
Average sides per face 2E/F: 4.0000

Step by step

1Values used
2Euler's polyhedron formula
3Genus
4Verdict= Valid convex polyhedron (χ = 2)
5Euler characteristic V − E + F= 2
6Implied genus (number of holes)= 0.0
7Average sides per face 2E/F= 4.0000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Does a cube satisfy it?

Yes: 8 − 12 + 6 = 2.

Why 2E/F for average face sides?

Each edge is shared by exactly two faces, so summing sides over all faces double-counts every edge.