geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a vertex is an

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

(shape) associated with a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

of an n-dimensional

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

. In two dimensions it refers to the angle formed by two intersecting lines, such as at a "corner" (

) of a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

. In higher dimensions there can be more than two lines ('' edges'') meeting at a vertex, making a description of the angle shape more complicated. In three dimensions and three-dimensional

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

, a vertex angle is a polyhedral angle or n-hedral angle. It is described by a sequence of ''n'' face angles, which are the angles formed by two edges of polyhedron meeting at the vertex, or by a sequence of ''n'' dihedral angles, which are the angles between two

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

sharing the vertex. The angle may be quantified using a single number by the interior

solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...

at the vertex (the ''

spherical excess Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...

''), which is related to the sum of the dihedral angles, or by the angular defect (or excess) of the vertex, which is related to the sum of the face angles, or other metrics such as the

polar sine In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin. Definition ''n'' vectors in ''n''-dimensional space Let v1, ..., v''n'' (''n'' ≥ 2) be non-zer ...

. The simplest type of polyhedral angle is a trihedral angle or trihedron (bounded by three planes), as found at the vertices of a Parallelepiped or

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

. For higher-dimensional

polytopes In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...

, a vertex angle can be quantified using a higher-dimensional solid angle, i.e. by the portion of the ''n''-sphere around the vertex that is interior to the polytope. Face and dihedral angles also generalize to higher dimensions. The term vertex angle is sometimes used synonymously with face angle, i.e. the angle between two edges meeting at a vertex. It may also refer to the (higher-dimensional) interior solid angle at a vertex.

Properties

A vertex angle in a

is often measured on the interior side of the vertex. For any simple ''n''-gon, the sum of the

interior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

s is π(''n'' − 2)

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...

s or 180(''n'' − 2) degrees. The face and dihedral angles of a polyhedral angle can be related to each other by interpreting the polyhedral angle as a ''spherical polygon'', whose side lengths are the face angles and whose vertex angles are the dihedral angles; the surface area of the polygon is the solid angle of the vertex (see

spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

, in particular the

spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "sph ...

). The higher dimensional analogue of a right angle is the vertex angle formed by mutually perpendicular edges, such as at the vertex of a hypercube. In three dimensions such an angle can be found in a

trirectangular tetrahedron In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the ''right angle'' of the trirectangular tetrahedron and the face opposite it is called the ''base''. Th ...

or a corner reflector.

References

Euclidean geometry Angle {{Elementary-geometry-stub

Properties

See also

References