Suppose that the vertices of our polygon--when traversed counter-clockwise around the polygon--have coordinates (x

1, y

1), (x

2, y

2), ...., (x

n, y

n) for some positive integer n which is at least 3 (so if n=3 you have a triangle, if n=4 a quadrilateral, etc).

As a beautiful consequence of

Green's Theorem, areas that are enclosed by a curve (on a plane) can be expressed in terms of the

line integral around the curve. So, it turns out that the area of our polygon above can be written as:

(1/2)*[(x

2y

1-x

1y

2)+(x

3y

2-x

2y

3)+(x

4y

3-x

3y

4)+...+(x

1y

n-x

ny

1)].

(Sorry for the crappy subscripts.)

What's also kinda neat is that, as a corollary, the quantity above is never negative (it's an area!), which isn't immediately obvious, since it contains a lot of sums and differences.