geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

of a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

is formed by two adjacent sides. For a

simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...

(non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than a straight angle (

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. It is defined such that one radian is the angle subtended at ...

s or 180°), then the polygon is called

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.Posamentier, Alfred S., and Lehmann, Ingmar. '' The Secrets of Triangles'', Prometheus Books, 2012.

Properties

* The sum of the internal angle and the external angle on the same vertex is radians (180°). * The sum of all the internal angles of a simple polygon is radians or degrees, where is the number of sides. The formula can be proved by using

mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...

: starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on. * The sum of the external angles of any simple polygon, if only one of the two external angles is assumed at each vertex, is radians (360°). * The measure of the exterior angle at a vertex is unaffected by which side is extended: the two exterior angles that can be formed at a vertex by extending alternately one side or the other are vertical angles and thus are equal.

Extension to crossed polygons

The interior angle concept can be extended in a consistent way to crossed polygons such as

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

s by using the concept of directed angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by , where is the number of vertices, and the strictly positive integer is the number of total (360°) revolutions one undergoes by walking around the

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

of the polygon. In other words, the sum of all the exterior angles is radians or degrees. Example: for ordinary

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

s and

concave polygon A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180° degrees and 360° degrees exclusive. ...

s, , since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.

Extension to polyhedra

Consider a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

that is topologically equivalent to a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, such as any

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

. Any vertex of the polyhedron will have several facets that meet at that vertex. Each of these facets will have an interior angle at that vertex and the sum of the interior angles at a vertex can be said to be the interior angle associated with that vertex of the polyhedron. The value of radians (or 360 degrees) minus that interior angle can be said to be the exterior angle associated with that vertex, also known by other names such as

angular defect In geometry, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the ''exces ...

. The sum of these exterior angles across all vertices of the polyhedron will necessarily be radians (or 720 degrees), and the sum of the interior angles will necessarily be radians (or degrees) where is the number of vertices. A proof of this can be obtained by using the formulas for the sum of interior angles of each facet together with the fact that the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

of a sphere is 2. For example, a rectangular solid will have three rectangular facets meeting at any vertex, with each of these facets having a 90° internal angle at that vertex, so each vertex of the rectangular solid is associated with an interior angle of and an exterior angle of . The sum of these exterior angles over all eight vertices is . The sum of these interior angles over all eight vertices is .

References

External links

Internal angles of a triangle
- Provides an interactive Java activity that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons. {{DEFAULTSORT:Internal And External Angle Angle Euclidean plane geometry Elementary geometry Polygons