Internal and external angles

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The corresponding internal (teal) and external (magenta) angles of a polygon are supplementary (sum to a half turn). The external angles of a non-self-intersecting closed polygon always sum to a full turn. Internal and external angles.png
The corresponding internal (teal) and external (magenta) angles of a polygon are supplementary (sum to a half turn). The external angles of a non-self-intersecting closed polygon always sum to a full turn.
Internal and external angles ExternalAngles.svg
Internal and external angles

In geometry, an angle of a polygon is formed by two adjacent sides. For a simple (non-self-intersecting) polygon, 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.

Contents

If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called 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 line extended from an adjacent side. [1] :pp. 261–264

Properties

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 directed angles. 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.

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References

  1. Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles , Prometheus Books, 2012.