# 5 Polygons

Given k 3 points in the plane, in a certain order, we obtain a k-sided polygon or k-gon by connecting each point to the next, and the last to the first, with a line segment. The points are the vertices and the segments are the sides or edges of the polygon. When k=3 we have a triangle, when k=4 we have a quadrangle or quadrilateral, and so on (see table of regular polygons ). Here we will assume that all polygons are simple: this means that no consecutive edges are on the same line, and no two edges intersect (except that consecutive edges intersect at the common vertex). See Figure 1. Figure 1: Two simple quadrilaterals (left and middle) and one that is not simple (right). We will not consider non-simple polygons.

When we refer to the angle at a vertex we have in mind the interior angle (as marked in the leftmost polygon in Figure 1). We denote this angle by the same symbol as the vertex. The complement of is the exterior angle at that vertex; geometrically, it is the angle between one side and the extension of the adjacent side. In any k-gon, the sum of the angles equals 2(k-2) right angles, or 2(k-2)×90°: for example, the sum of the angles of a triangle is 180°.

The area of a polygon whose vertices have coordinates , for , is the absolute value of where in the sumation we take and . In particular, for a triangle we have In oblique coordinates with angle between the axes, the area is as given above, times sin .

If the vertices have polar coordinates , for , the area is the absolute value of where we take and .

Formulas for specific polygons in terms of side lengths, angles, etc. are given in the next sections.

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Silvio Levy
Wed Oct 4 16:41:25 PDT 1995

This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.