Euler's Formula for Polyhedron Calculator

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Euler's formula, V - E + F = 2, is one of the most celebrated results in topology and geometry, relating the number of vertices (V), edges (E), and faces (F) of any simply-connected (convex, hole-free) polyhedron. Named after Leonhard Euler, who described the relationship in the 18th century, it holds for every one of the Platonic solids (cube, tetrahedron, octahedron, dodecahedron, icosahedron) and every convex polyhedron in general, making it a fundamental check in solid geometry, graph theory, and computational geometry (mesh and 3D-model validation software often uses this exact formula to sanity-check that a model is a valid, hole-free solid).

How it works

Enter any two of vertices, edges, and faces and leave the third blank to solve for it — the calculator simply rearranges V - E + F = 2 depending on which variable is missing (for example, F = 2 - V + E). Alternatively, fill in all three known values to verify whether they actually satisfy the formula, which quickly confirms (or disproves) that a described shape could be a valid convex polyhedron.

Enter vertices (V) — leave blank to solve for it.
Enter edges (E) — leave blank to solve for it.
Enter faces (F) — leave blank to solve for it.
Click Calculate to see your results.

Examples

A cube

V = 8, E = 12, F = 6. Check: 8 - 12 + 6 = 2 ✓ — confirming a cube is a valid convex polyhedron.

A tetrahedron missing its face count

V = 4, E = 6, F = unknown. Rearranging: F = 2 - V + E = 2 - 4 + 6 = 4, matching a tetrahedron's actual 4 triangular faces.

A soccer ball (truncated icosahedron)

V = 60, E = 90, F = 32. Check: 60 - 90 + 32 = 2 ✓.

Common mistakes to avoid

Filling in all three values when trying to solve for one of them — leave the unknown one blank instead so the calculator knows which to solve for.
Applying the formula to a shape with holes (non-simply-connected) without adjusting for genus — the plain V-E+F=2 only holds for hole-free convex polyhedra.
Miscounting edges or vertices on a complex polyhedron, which throws off the whole check even though the formula itself is correct.

Frequently asked questions

It works for any convex (simply-connected, hole-free) polyhedron. Shapes with one or more holes through them, like a torus-shaped polyhedron (a "donut" made of flat faces), follow a different formula: V - E + F = 2 - 2g, where g is the number of holes (the genus).

It means the polyhedron's surface has no holes running through it and could, in principle, be continuously deformed into the shape of a sphere without tearing — a cube, pyramid, or any of the Platonic solids qualify, but a torus-shaped ring does not.

Yes — that's one of its most practical uses. If someone claims a polyhedron has, say, 6 vertices, 10 edges, and 5 faces, plugging into the formula (6-10+5=1≠2) immediately shows it can't be a valid convex polyhedron.

Leonhard Euler was an 18th-century Swiss mathematician who described this vertex-edge-face relationship, one of many foundational results across mathematics that bear his name.

Yes — the tetrahedron (V=4,E=6,F=4), cube (8,12,6), octahedron (6,12,8), dodecahedron (20,30,12), and icosahedron (12,30,20) all satisfy V-E+F=2, which is part of why there are exactly five of them.

Free Euler's Formula for Polyhedron Calculator