How this calculator works
For any convex polyhedron, the number of faces F, edges E and vertices V are tied together by Euler's formula: F - E + V = 2. A cube has 6 faces, 12 edges and 8 vertices, and 6 - 12 + 8 = 2. A tetrahedron has 4 of each face and vertex and 6 edges, which also balances. Because the relation is linear, supplying two counts fixes the third, so the calculator can solve for whichever one is missing.
Formula
F - E + V = 2; F = 2 + E - V; V = 2 + E - F; E = F + V - 2.
Worked example
A cube with 6 faces and 8 vertices must have 8 + 6 - 2 = 12 edges. Checking a tetrahedron with 4 faces, 6 edges and 4 vertices gives 4 - 6 + 4 = 2, so the counts are consistent. A set such as 5 faces, 8 edges and 4 vertices gives 1, which no convex polyhedron can have.
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Frequently asked questions
Does Euler's formula work for every solid?
It works for convex polyhedra and for any polyhedron that can be deformed to a sphere without tearing. Solids with a hole, such as a doughnut-shaped torus, give a different characteristic, usually zero.
Why must faces and vertices be at least four?
Three faces cannot enclose a solid volume, and the smallest convex polyhedron, the tetrahedron, already has four faces and four vertices. A computed count below four means the combination cannot exist.
What does the Euler characteristic mean?
It is the number V - E + F. For a convex polyhedron it is exactly 2. Other values reveal a surface with a different topology, such as a hole or a non-closed surface.