In geometry, an interior angle (or internal angle) is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave of the polygon, this angle will be an angle on the 'inner side' of the polygon. A polygon has exactly one internal angle per vertex.
If every internal angle of a simple, closed polygon is less than 180°, the polygon is called convex.
In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple, closed polygon and a line extended from an adjacent side.
The sum of the internal angle and the external angle on the same vertex is 180°.
The sum of all the internal angles of a simple, closed polygon is 180(n-2)° where n is the number of sides. The formula can by proved using mathematical induction and starting with a triangle for which the angle sum is 180°, and then adding a vertex and two sides, etc.
The sum of the external angles of any simple closed (convex or concave) polygon is 360°.
The concept of 'interior angle' 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 k = 0, 1, 2, 3 ... represents the number of total revolutions of 360° one undergoes walking around the perimeter of the polygon. In other words, 360k° represents the sum of all the exterior angles. For example, for ordinary convex and concave polygons k = 1, since the exterior angle sum = 360° and one undergoes only one full revolution walking around the perimeter.
Famous quotes containing the words internal, external and/or angle:
“What makes some internal feature of a thing a representation could only its role in regulating the behavior of an intentional system.”
—Daniel Clement Dennett (b. 1942)
“Given that external reality is a fiction, the writer’s role is almost superfluous. He does not need to invent the fiction because it is already there.”
—J.G. (James Graham)
“I fly in dreams, I know it is my privilege, I do not recall a single situation in dreams when I was unable to fly. To execute every sort of curve and angle with a light impulse, a flying mathematics—that is so distinct a happiness that it has permanently suffused my basic sense of happiness.”
—Friedrich Nietzsche (1844–1900)