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:
“I maintain that I have been a Negro three times—a Negro baby, a Negro girl and a Negro woman. Still, if you have received no clear cut impression of what the Negro in America is like, then you are in the same place with me. There is no The Negro here. Our lives are so diversified, internal attitudes so varied, appearances and capabilities so different, that there is no possible classification so catholic that it will cover us all, except My people! My people!”
—Zora Neale Hurston (1891–1960)
“It can be fairly argued that the highest priority for mankind is to save itself from extinction. However, it can also be argued that a society that neglects its children and robs them of their human potential can extinguish itself without an external enemy.”
—Selma Fraiberg (20th century)
“So much symmetry!
Like the pale angle of time
And eternity.
The great shape labored and fell.”
—N. Scott Momaday (b. 1934)