Take any convex polyhedron — count its vertices, its edges and its faces. Whatever shape you picked, the answer is always the same: V − E + F = 2.
Spin the solid
Octahedron
faces
8
8
edges
12
12
vertices
6
6
V − E + F = 6 − 12 + 8 = 2
drag to rotate
Spin the solid
Triangular prism
faces
5
5
edges
9
9
vertices
6
6
V − E + F = 6 − 9 + 5 = 2
drag to rotate
Euler's polyhedron formula
Check it on the Platonic solids
- Tetrahedron — V=4, E=6, F=4 → 4−6+4 = 2 ✓
- Cube — V=8, E=12, F=6 → 8−12+6 = 2 ✓
- Octahedron — V=6, E=12, F=8 → 6−12+8 = 2 ✓
- Dodecahedron — V=20, E=30, F=12 → 20−30+12 = 2 ✓
- Icosahedron — V=12, E=30, F=20 → 12−30+20 = 2 ✓
Euler's formula is true for any shape topologically equivalent to a sphere — even sliced or dented. For a torus (a donut), the answer is 0. The 2 is hiding the topology of the surface.