In higher-dimensional geometry the faces of a polytope are features of all dimensions. 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 include the empty set, its vertices (0-faces), edges (1-faces) and squares (2-faces), and the cube itself (3-face).

All of the following are the faces of a 4-dimensional polytope:

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". From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.

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.

పీక్ (peak) లేదా ("n"-3)-ముఖం
"k"-ముఖం
ఫాసెట్ (Facet) లేదా ("n"-1)-ముఖం
బహుభుజి ముఖం