Internal and external angles
Types of angles |
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2D angles |
Exterior |
2D angle pairs |
Adjacent |
3D angles |
Dihedral |
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 internal 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 π radians (180°), then the polygon is called convex.
In contrast, an exterior angle (also called an external angle or turning angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.[1][2]: pp. 261-264
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: 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 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 polygons by using the concept of . In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, 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. In other words, the sum of all the exterior angles is 2πk radians or 360k degrees. Example: for ordinary convex polygons and concave polygons, k = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.
References[]
- ^ Weisstein, Eric W. "Exterior Angle Bisector." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExteriorAngleBisector.html
- ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
External links[]
- Internal angles of a triangle
- 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.
- Angle
- Euclidean plane geometry
- Elementary geometry
- Polygons