Face
Geometry & MeasurementA face is any flat surface of a 3-D solid; polyhedra are made entirely of polygonal faces.
Definition
A face is any flat side of a 3-D shape. A cube has $6$ faces, all squares. A triangular prism has $5$ faces: $2$ triangles and $3$ rectangles.
Example
Hold a die (cube): it has $6$ flat square faces. A cereal box (rectangular prism) also has $6$ faces, but they are rectangles of different sizes.
Key Insight
Count the faces to identify a solid. $4$ faces = tetrahedron (triangular pyramid). $5$ faces = square pyramid or triangular prism. $6$ faces = rectangular prism. $8$ faces = octahedron.
Definition
A face of a polyhedron is a polygonal region that forms part of its boundary. Polyhedra are classified by the number and shape of their faces. A face is bounded by edges and meets adjacent faces along edges. The five Platonic solids have faces that are all the same regular polygon.
Example
An octahedron has $8$ equilateral triangle faces. A dodecahedron has $12$ regular pentagon faces. By Euler's formula $F + V - E = 2$, a dodecahedron with $12$ faces and $20$ vertices must have $30$ edges: $12 + 20 - 30 = 2$.
Key Insight
Euler's formula $F + V - E = 2$ is a topological invariant of all convex polyhedra. It was one of the first results in topology, revealing that all convex solids share a hidden structural relationship regardless of their shape.
Definition
A face of a convex polytope $P$ in $\mathbb{R}^n$ is a subset of the form $P \cap \{x : c^Tx = \max_{y \in P} c^Ty\}$ for some nonzero vector $c$, i.e., the maximizers of a linear functional. Faces of a 3-D polyhedron are the 2-dimensional faces; edges are 1-dimensional faces; vertices are 0-dimensional faces. The face lattice of a polytope encodes all combinatorial structure.
Example
The face lattice of a cube has $1$ empty face, $8$ vertices, $12$ edges, $6$ faces, and $1$ full face (the cube itself), totaling $28$ elements. The dual polytope (octahedron) has the reversed face lattice: $1+6+12+8+1$.
Key Insight
The duality between the cube ($6$ faces, $8$ vertices) and octahedron ($8$ faces, $6$ vertices) is a general phenomenon: every convex polytope has a dual where faces and vertices are swapped. This duality is central to linear programming, where vertices of feasible polytopes correspond to basic feasible solutions.