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.