A polyhedron is a three-dimensional solid made from flat polygon faces that meet along straight edges. Cubes, pyramids, prisms, and many crystals are examples of polyhedra. Counting faces, edges, and vertices helps reveal the structure hidden inside these shapes.
Euler's formula, F - E + V = 2, is one of the most elegant links between geometry and topology.
Key Facts
- Euler's formula for any convex polyhedron is F - E + V = 2.
- F means the number of faces, E means the number of edges, and V means the number of vertices.
- For a cube, F = 6, E = 12, V = 8, so F - E + V = 6 - 12 + 8 = 2.
- For a triangular pyramid, F = 4, E = 6, V = 4, so F - E + V = 4 - 6 + 4 = 2.
- An edge is counted once even though it belongs to two faces.
- Euler's formula applies to convex polyhedra and many sphere-like polyhedra, but not to shapes with holes.
Vocabulary
- Polyhedron
- A solid three-dimensional shape made of flat polygon faces, straight edges, and vertices.
- Face
- A flat polygon region on the surface of a polyhedron.
- Edge
- A straight line segment where two faces of a polyhedron meet.
- Vertex
- A corner point where edges of a polyhedron meet.
- Convex polyhedron
- A polyhedron with no dents, where any line segment connecting two points inside the solid stays inside the solid.
Common Mistakes to Avoid
- Counting each shared edge twice is wrong because one edge belongs to two faces but is still only one edge of the solid.
- Confusing faces with surfaces is wrong because only flat polygon regions count as faces in a polyhedron.
- Using Euler's formula on a doughnut-shaped solid is wrong because F - E + V = 2 does not apply to polyhedra with holes.
- Forgetting hidden edges or vertices in a 3D drawing is wrong because the full solid must be counted, not just the visible front side.
Practice Questions
- 1 A cube has 6 faces and 8 vertices. Use Euler's formula to find the number of edges.
- 2 A convex polyhedron has 9 faces and 14 vertices. How many edges does it have?
- 3 A student counts a rectangular prism and gets F = 6, E = 10, and V = 8. Explain how Euler's formula shows that the count must be incorrect, and identify what was likely missed.