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:
“If the Revolution has the right to destroy bridges and art monuments whenever necessary, it will stop still less from laying its hand on any tendency in art which, no matter how great its achievement in form, threatens to disintegrate the revolutionary environment or to arouse the internal forces of the Revolution, that is, the proletariat, the peasantry and the intelligentsia, to a hostile opposition to one another. Our standard is, clearly, political, imperative and intolerant.”
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“Modesty is the only sure bait when you angle for praise.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)