Euler's Polyhedron Formula Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
Euler's formula relates the vertices (V), edges (E), and faces (F) of any convex polyhedron: V − E + F = 2. For a cube (12 edges, 6 faces), this relation gives exactly 8 vertices — a result verifiable by directly counting a cube's corners.
Explanation
Discovered by Leonhard Euler in 1750, this formula reveals a remarkable topological property, valid for absolutely any convex polyhedron whatever its precise shape: the number of vertices minus the number of edges plus the number of faces is always exactly 2. This result is not specific to a particular figure — a cube, a pyramid, an icosahedron, or any other convex polyhedron, however complex, always satisfies this same relation, which makes it one of the first great results of topology, the branch of mathematics that studies the properties of shapes that remain invariant under continuous deformation. The formula recovers any of the three quantities (V, E, or F) from the other two: this calculator isolates the number of vertices, V=2+E−F, the most practical case when you know a solid's number of edges and faces but want to check or recover its number of vertices without counting them one by one. The five Platonic solids — the only possible regular convex polyhedra, known since Greek antiquity — perfectly illustrate this relation: the tetrahedron (4 vertices, 6 edges, 4 faces, each face a triangle whose area is calculated as for any triangle), the cube (8, 12, 6), the octahedron (6, 12, 8), the dodecahedron (20, 30, 12), and the icosahedron (12, 30, 20) each satisfy V−E+F=2 exactly, despite face counts ranging from 4 to 20. This relation even extends beyond classical geometry: it underlies important results in planar graph theory and computer graphics, wherever data structures representing meshed surfaces are handled.
Example: the cube
Inputs
Number of edges: 12. Number of faces: 6.
Calculation
V = 2 + 12 − 6 = 8.
Result
A cube has exactly 8 vertices — a result verifiable directly by counting its corners, confirming Euler's formula.
Frequently asked questions
Does this formula apply to any solid, even a non-convex one?
It applies to any polyhedron topologically equivalent to a sphere — which includes all convex polyhedra, but also some non-convex polyhedra as long as they have no "hole" passing through them (like a torus, the shape of a life ring). A polyhedron with one or more holes follows a generalized version of the formula, where the number 2 is replaced by a quantity related to the number of holes (the topological genus of the surface), a more advanced result beyond the scope of this calculator.
Why does the number 2 appear specifically in this formula?
This number 2 is what is called the Euler characteristic of the sphere, a topological invariant that depends only on the overall shape of the surface (here, a sphere), not on the details of its subdivision into vertices, edges, and faces — each face can be a polygon with a different number of sides without changing the final result. Any subdivision of a sphere into polygons, however different from another subdivision, will always give this same value of 2 — which is precisely what makes Euler's formula so universal and independent of the precise shape of the polyhedron studied.
Can this formula be used to check that a described polyhedron is geometrically possible?
It is a necessary but not sufficient check: if the numbers of vertices, edges, and faces claimed for a solid do not satisfy V−E+F=2, that solid certainly cannot exist as a regular convex polyhedron. But the converse is not guaranteed: satisfying this relation alone is not enough to prove that a corresponding polyhedron actually exists, other geometric constraints (such as the number of faces meeting at each vertex) also having to be met.