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 c ...

, an

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

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 toge ...

is formed by two sides of the polygon that share an endpoint. For a

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

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

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 ...

. If every internal angle of a simple polygon is less than π radians (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 polytope ...

. In contrast, an exterior angle (also called an or turning angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.Weisstein, Eric W. "Exterior Angle Bisector." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExteriorAngleBisector.htmlPosamentier, Alfred S., and Lehmann, Ingmar. ''

The Secrets of Triangles ''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, ...

'', Prometheus Books, 2012.

Properties

* The sum of the internal angle and the external angle on the same vertex is π radians (180°). * The sum of all the internal angles of a simple polygon is π(''n''−2) radians or 180(''n''–2) degrees, where ''n'' is the number of sides. The formula can be proved by using

mathematical induction Mathematical induction is a method for 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), ... all hold. Informal metaphors help ...

: 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 convex or non-convex polygon, if only one of the two external angles is assumed at each vertex, is 2π 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 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 are ...

and thus are equal.

Extension to crossed polygons

The interior angle concept can be extended in a consistent way to

crossed polygon A crossed polygon is a polygon in the plane with a turning number or density of zero, with the appearance of a figure 8, infinity symbol, or lemniscate curve. ''Crossed polygons'' are related to star polygons which have turning numbers greater ...

s 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, certain notable ones can arise through truncation operations ...

s by using the concept of

directed angles Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''Di ...

. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(''n''–2''k'')°, where ''n'' is the number of vertices, and the strictly positive integer ''k'' is the number of total (360°) revolutions one undergoes by walking around the

perimeter of the polygon A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pra ...

. In other words, the sum of all the exterior angles is 2π''k'' radians or 360''k'' 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 a ...

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. Polyg ...

s, ''k'' = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.

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