In their most general form, polygons are an ordered set of vertices, , , with edges joining consecutive vertices. Most frequently, one deals with simple polygons in which no two edges are allowed to intersect. (In fact, the adjective "simple" is almost always omitted, so the term "polygon" should be understood to mean "simple polygon" unless otherwise noted.)
A degenerate polygon is one in which some vertex lies on an edge joining two other vertices. This can happen in one of two ways: either the vertices and can be colinear or the vertices and can overlap (fail to be distinct). In either of these cases, our polygon of vertices will appear to have or fewer -- it will have "degenerated" from an -gon to an -gon. (In the case of triangles, this will result in either a line segment or a point.)
Angles in Polygons
In any simple convex polygon, the sum of the exterior angles is equal to .
The sum of interior angles can be given by the formula , where is the number of sides. Thus in regular polygons, any angle is .
|Number of Sides||Sum of Interior angles||Individual angle measure in regular polygon|
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