In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object;^[1] a three-dimensional solid bounded exclusively by faces is a polyhedron.

In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).^[2]

In elementary geometry, a face is a polygon^{[note 1]} on the boundary of a polyhedron.^[2][3] Other names for a polygonal face include polyhedron side and Euclidean plane tile.

For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells.

More information Polyhedron, Star polyhedron ...

Regular examples by Schläfli symbol

Polyhedron	Star polyhedron	Euclidean tiling	Hyperbolic tiling	4-polytope
{4,3}	{5/2,5}	{4,4}	{4,5}	{4,3,3}
The cube has 3 square faces per vertex.	The small stellated dodecahedron has 5 pentagrammic faces per vertex.	The square tiling in the Euclidean plane has 4 square faces per vertex.	The order-5 square tiling has 5 square faces per vertex.	The tesseract has 3 square faces per edge.

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In higher-dimensional geometry, the faces of a polytope are features of all dimensions.^[2][4][5] A face of dimension k is called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. For any n-polytope (n-dimensional polytope), −1 ≤ k ≤ n.

For example, with this meaning, the faces of a cube comprise the cube itself (3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.

In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P.^[6] From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.^[4][5]

In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement for convexity is relaxed. Abstract theory still requires that the set of faces include the polytope itself and the empty set.

An n-dimensional simplex (line segment (n = 1), triangle (n = 2), tetrahedron (n = 3), etc.), defined by n + 1 vertices, has a face for each subset of the vertices, from the empty set up through the set of all vertices. In particular, there are 2^{n + 1} faces in total. The number of them that are k-faces, for k ∈ {−1, 0, ..., n}, is the binomial coefficient ${\displaystyle {\binom {n+1}{k+1}}}$.

There are specific names for k-faces depending on the value of k and, in some cases, how close k is to the dimensionality n of the polytope.

• Face lattice