Euler's Formula for Polyhedra Calculator — V − E + F = 2
Enter any two of a polyhedron's vertices (V), edges (E) and faces (F) to compute the missing value using Euler's formula V − E + F = 2.
Euler characteristic = 2 — valid convex polyhedron
V−E+F = 2
- 1
Compute E = V + F − 2
8 + 6 − 2 = 12 - 2
Euler characteristic V − E + F
8 − 12 + 6 = 2χ = 2 confirms a valid convex (genus-0) polyhedron.
How does this calculator work?
Euler's polyhedron formula V − E + F = 2 holds for any convex (genus-0) polyhedron. Given any two of vertices, edges and faces, the third follows: E = V + F − 2, V = 2 + E − F, or F = 2 + E − V. The quantity χ = V − E + F is the Euler characteristic; for a torus χ = 0 and for a sphere χ = 2.
Formula
How this is calculated
Euler's polyhedron formula, published by Leonhard Euler in 1758, states that for any convex polyhedron (and more generally any simply-connected polyhedron homeomorphic to a sphere), the number of vertices V, edges E and faces F satisfy V − E + F = 2. The quantity V − E + F is called the Euler characteristic χ, and χ = 2 for any surface topologically equivalent to a sphere.
The formula is a topological invariant: it holds regardless of the shape, size or exact geometry of the polyhedron. Classic examples — cube (8 − 12 + 6 = 2), tetrahedron (4 − 6 + 4 = 2), octahedron (6 − 12 + 8 = 2), dodecahedron (20 − 30 + 12 = 2) — all satisfy it.
The formula does not hold for non-convex polyhedra with holes (genus ≥ 1). For a polyhedron with g holes (a 'torus-like' surface), χ = V − E + F = 2 − 2g, so a torus gives χ = 0. This calculator checks whether the entered values satisfy χ = 2, signalling a valid convex (genus-0) polyhedron.
Frequently asked questions
For any convex polyhedron, the number of vertices V, edges E and faces F satisfies V − E + F = 2. This topological identity was proved by Euler in 1758 and holds regardless of the shape or size of the polyhedron.
It holds for convex polyhedra and any polyhedra homeomorphic to a sphere (genus 0). For polyhedra with holes — like a torus — the formula generalises to V − E + F = 2 − 2g, where g is the number of holes (genus).
The simplest polyhedron is the tetrahedron: 4 vertices, 6 edges, 4 faces. Any valid convex polyhedron requires V ≥ 4, E ≥ 6, F ≥ 4 — entering smaller values violates geometric constraints regardless of whether they satisfy the formula arithmetically.
TG we-Calculate Editorial Team. (2026). Euler's Formula for Polyhedra Calculator — V − E + F = 2 [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/eulers-formula-for-polyhedron-calculator
TG we-Calculate Editorial Team. "Euler's Formula for Polyhedra Calculator — V − E + F = 2." TG we-Calculate. 2026. https://we-calculate.com/calculator/eulers-formula-for-polyhedron-calculator.
TG we-Calculate Editorial Team, "Euler's Formula for Polyhedra Calculator — V − E + F = 2," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/eulers-formula-for-polyhedron-calculator
@misc{wecalculate_eulers_formula_for_polyhedron_calculator, title = {Euler's Formula for Polyhedra Calculator — V − E + F = 2}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/eulers-formula-for-polyhedron-calculator}}, year = {2026}, note = {TG we-Calculate} }
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