Polygonal face
In elementary geometry, a face is a polygon on the boundary of a polyhedron.^{[2]}^{[3]} Other names for a polygonal face include side of a polyhedron, and tile of a Euclidean plane tessellation.
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.
Some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets (flat polygons formed by coplanar vertices which do not lie in the same face of the polyhedron).
Number of polygonal faces of a polyhedron
Any convex polyhedron's surface has Euler characteristic
- $V-E+F=2,$
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.