Internal and external angles

"Interior angle" redirects here. For interior angles on the same side of the transversal, see Transversal line.

In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (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.

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

If every internal angle of a simple polygon is less than a straight angle (πぱい 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.

Properties

• The sum of the internal angle and the external angle on the same vertex is πぱい radians (180°).
• The sum of all the internal angles of a simple polygon is πぱい(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 polygon, if only one of the two external angles is assumed at each vertex, is 2πぱい 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 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 2πぱいk 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.

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.