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| Types of angles |
|---|
| 2D angles |
| Spherical |
| 2D angle pairs |
Adjacent |
| 3D angles |
| Solid |
Ingeometry, anangle of apolygon is formed by two adjacentsides. For asimple polygon (non-self-intersecting), regardless of whether it isconvex or non-convex, this angle is called aninternal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle pervertex.
If every internal angle of a simple polygon is less than astraight angle (πradians or 180°), then the polygon is calledconvex.
In contrast, anexternal angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and aline extended from an adjacent side.[1]: pp. 261–264
The interior angle concept can be extended in a consistent way tocrossed polygons such asstar polygons by using the concept ofdirected angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by180(n − 2k)°, wheren is the number of vertices, and the strictly positive integerk is the number of total (360°) revolutions one undergoes by walking around theperimeter of the polygon. In other words, the sum of all the exterior angles is2πk radians or360k degrees. Example: for ordinaryconvex polygons andconcave polygons,k = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.
Consider apolyhedron that istopologically equivalent to asphere, such as anyconvex polyhedron. Any vertex of the polyhedron will have severalfacets that meet at that vertex. Each of these facets will have an interior angle at that vertex and the sum of the interior angles at a vertex can be said to be the interior angle associated with that vertex of the polyhedron. The value of2π radians (or 360 degrees) minus that interior angle can be said to be the exterior angle associated with that vertex, also known by other names such asangular defect. The sum of these exterior angles across all vertices of the polyhedron will necessarily be4π radians (or 720 degrees), and the sum of the interior angles will necessarily be2π(n − 2) radians (or360(n − 2) degrees) wheren is the number of vertices. A proof of this can be obtained by using the formulas for the sum of interior angles of each facet together with the fact that theEuler characteristic of a sphere is 2.
For example, arectangular solid will have three rectangular facets meeting at any vertex, with each of these facets having a 90° internal angle at that vertex, so each vertex of the rectangular solid is associated with an interior angle of3 × 90° = 270° and an exterior angle of360° − 270° = 90°. The sum of these exterior angles over all eight vertices is8 × 90° = 720°. The sum of these interior angles over all eight vertices is8 × 270° = 2160°.