Editing Internal and external angles

{{short description|Supplementary pair of angles at each vertex of a polygon}}
{{Redirect|Interior angle|interior angles on the same side of the transversal|Transversal line}}
{{More citations needed|date=November 2023}}
[[File:Internal and external angles.png|thumb|upright=1.25|The corresponding internal (teal) and external (magenta) angles of a polygon are supplementary (sum to a half [[Turn (angle)|turn]]). The external angles of a non-self-intersecting closed polygon always sum to a full turn.]]
{{Angles}}
[[Image:ExternalAngles.svg|thumb|upright=1.25|right|Internal and external angles]]

In [[geometry]], an [[angle]] of a [[polygon]] is formed by two adjacent [[edge (geometry)|sides]].  For a [[simple polygon]] (non-self-intersecting), regardless of whether it is [[Polygon#Convexity and non-convexity|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 (geometry)|vertex]].

If every internal angle of a simple polygon is less than a [[straight angle]] ([[pi|{{mvar|π}}]] [[radian]]s or 180°), then the polygon is called [[convex polygon|convex]].

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 [[Extended side|line extended from an adjacent side]].<ref name=PL>Posamentier, Alfred S., and Lehmann, Ingmar. ''[[The Secrets of Triangles]]'', Prometheus Books, 2012.</ref>{{rp|pp. 261–264}}

==Properties==
* The sum of the internal angle and the external angle on the same vertex is {{mvar|π}} radians (180°).
* The sum of all the internal angles of a simple polygon is {{math|''π''(''n'' − 2)}} radians or {{math|180(''n'' − 2)}} degrees, where {{mvar|n}} is the number of sides. The formula can be proved by using [[mathematical induction]]: 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 {{math|2''π''}} 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 polygon]]s such as [[star polygon]]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 {{math|180(''n'' − 2''k'')°}}, where {{mvar|n}} is the number of vertices, and the strictly positive integer {{mvar|k}} is the number of total (360°) revolutions one undergoes by walking around the [[perimeter of the polygon|perimeter]] of the polygon. In other words, the sum of all the exterior angles is {{math|2''πk''}} radians or {{math|360''k''}} degrees. Example: for ordinary [[convex polygon]]s and [[concave polygon]]s, {{math|1=''k'' = 1}}, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.

==Extension to polyhedra==
{{further|Descartes' theorem on total angular defect}}

Consider a [[polyhedron]] that is [[Homeomorphism|topologically equivalent]] to a [[sphere]], such as any [[convex polytope|convex polyhedron]].  Any vertex of the polyhedron will have several [[facet (geometry)|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 {{math|2''π''}} 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]].  The sum of these exterior angles across all vertices of the polyhedron will necessarily be {{math|4''π''}} radians (or 720 degrees), and the sum of the interior angles will necessarily be {{math|2''π''(''n'' − 2)}} radians (or {{math|360(''n'' − 2)}} degrees) where {{mvar|n}} 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]] 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 {{math|1=3 × 90° = 270°}} and an exterior angle of {{math|1=360° − 270° = 90°}}.  The sum of these exterior angles over all eight vertices is {{math|1=8 × 90° = 720°}}.  The sum of these interior angles over all eight vertices is {{math|1=8 × 270° = 2160°}}.

==References==
{{Reflist}}

==External links==
*[http://www.mathopenref.com/triangleinternalangles.html Internal angles of a triangle]
*[http://dynamicmathematicslearning.com/star_pentagon.html Interior angle sum of polygons: a general formula] - Provides an interactive Java activity that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons.

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[[Category:Angle]]
[[Category:Euclidean plane geometry]]
[[Category:Elementary geometry]]
[[Category:Polygons]]