Green's Theorem and the Area of Polygons

I am an avid member of the Math.StackExchange community. We have recently reached a milestone, as our request to create a site blog has been approved by the Stack Exchange administration. I volunteered to write a post which I believe should be useful to competition programmers.

Using Green’s Theorem, this post derives a formula for the area of any simple polygon, dependent solely on the coordinates of the vertices. This is useful for some computational geometry problems in programming; for example, the formula can be used to compute the area of the convex hull of a set of points.