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Given *k*3 points in the plane, in a certain order,
we obtain a ** k-sided polygon** or

**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(

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.

*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.