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:
“The burning of rebellious thoughts in the little breast, of internal hatred and opposition, could not long go on without slight whiffs of external smoke, such as mark the course of subterranean fire.”
—Harriet Beecher Stowe (1811–1896)
“Many are poets but without the name,
For what is poesy but to create
From overfeeling good or ill; and aim
At an external life beyond our fate,
And be the new Prometheus of new men.”
—George Gordon Noel Byron (1788–1824)
“It is a mistake, to think the same thing affects both sight and touch. If the same angle or square, which is the object of touch, be also the object of vision, what should hinder the blind man, at first sight, from knowing it?”
—George Berkeley (1685–1753)