Recognition of Solids
119. The Magic of Polyhedra · V + F – E = 2 always holds
V + F – E always equals 2 for any polyhedron.
Euler's formula for any convex polyhedron: V + F − E = 2, where V is vertices, F is faces, E is edges. Cube: 8 + 6 − 12 = 2 ✓. Tetrahedron: 4 + 4 − 6 = 2 ✓. A beautiful invariant of 3D solids.
What this lesson covers
Try to break it
Drag the yellow Perspective handle to skew the cube's view. No matter how you tilt or rotate it, V = 8, E = 12, F = 6. Compute V − E + F — it always equals 2. Euler's formula refuses to break.
How you build it
Construct a triangular prism and check V + F − E = 2.
- Place point A for the first corner of the base triangle.
- Place point B for the second corner of the base triangle.
- Place point C for the third corner of the base triangle.
- Draw segment AB of the base triangle.
- Draw segment BC of the base triangle.
- Draw segment CA to close the base triangle.
- Place point D directly above A for the top triangle.
- Place point E directly above B for the top triangle.
- Place point F directly above C for the top triangle.
- Draw the vertical edge AD.
- Draw the vertical edge BE.
- Draw the vertical edge CF.
- Draw segment DE of the top triangle.
- Draw segment EF of the top triangle.
- Draw segment FD to complete the top triangle and the prism.
The proof, step by step
Prove that V + F − E = 2 for the polyhedron.
- For a cube, count the vertices: V = 8.
- Count the faces: F = 6.
- Count the edges: E = 12.
- Compute V + F - E = 8 + 6 - 12 = 2. The formula holds for this cube.
Worked example
A polyhedron has 12 vertices and 20 faces. Find the number of edges using Euler's formula.
Using V + F – E = 2, we get 12 + 20 – E = 2. So, 32 – E = 2, which gives E = 30.
- 28
- 30 — correct
- 32
- 24