A polygon is bounded by edges; this square has 4 edges.

Every edge is shared by two faces in a polyhedron, like this cube.

Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract.

In
geometry, an **edge** is a particular type of
line segment joining two
vertices in a
polygon,
polyhedron, or higher-dimensional
polytope.^{
[1]} In a polygon, an edge is a line segment on the boundary,^{
[2]} and is often called a **polygon side**. In a polyhedron or more generally a polytope, an edge is a line segment where two
faces (or polyhedron sides) meet.^{
[3]} A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a
diagonal.

In
graph theory, an
edge is an abstract object connecting two
graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment.
However, any polyhedron can be represented by its
skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges.^{
[4]} Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by
Steinitz's theorem as being exactly the
3-vertex-connected
planar graphs.^{
[5]}

Any convex polyhedron's surface has Euler characteristic

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 edges is 2 less than the sum of the numbers of vertices and faces. For example, a
cube has 8 vertices and 6 faces, and hence 12 edges.

In a polygon, two edges meet at each
vertex; more generally, by
Balinski's theorem, at least *d* edges meet at every vertex of a *d*-dimensional convex polytope.^{
[6]}
Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge,^{
[7]} while in higher dimensional polytopes three or more two-dimensional faces meet at every edge.

In the theory of high-dimensional
convex polytopes, a *
facet* or *side* of a *d*-dimensional
polytope is one of its (*d* − 1)-dimensional features, a *
ridge* is a (*d* − 2)-dimensional feature, and a *
peak* is a (*d* − 3)-dimensional feature. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional
convex polyhedron are its ridges, and the edges of a
4-dimensional polytope are its peaks.^{
[8]}

**^**Ziegler, Günter M. (1995),*Lectures on Polytopes*, Graduate Texts in Mathematics, vol. 152, Springer, Definition 2.1, p. 51, ISBN 9780387943657.**^**Weisstein, Eric W. "Polygon Edge." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PolygonEdge.html**^**Weisstein, Eric W. "Polytope Edge." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PolytopeEdge.html**^**Senechal, Marjorie (2013),*Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination*, Springer, p. 81, ISBN 9780387927145.**^**Pisanski, Tomaž; Randić, Milan (2000), "Bridges between geometry and graph theory", in Gorini, Catherine A. (ed.),*Geometry at work*, MAA Notes, vol. 53, Washington, DC: Math. Assoc. America, pp. 174–194, MR 1782654. See in particular Theorem 3, p. 176.**^**Balinski, M. L. (1961), "On the graph structure of convex polyhedra in*n*-space",*Pacific Journal of Mathematics*,**11**(2): 431–434, doi: 10.2140/pjm.1961.11.431, MR 0126765.**^**Wenninger, Magnus J. (1974),*Polyhedron Models*, Cambridge University Press, p. 1, ISBN 9780521098595.**^**Seidel, Raimund (1986), "Constructing higher-dimensional convex hulls at logarithmic cost per face",*Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (STOC '86)*, pp. 404–413, doi: 10.1145/12130.12172, S2CID 8342016.