Let be polygon with its ordered vertices (the vertices you take in order as you travel around it) be .
Also say they are taken in a counterclockwise fashion.
Let be the line segment from to for . But for define .
Define by .
Let denote the interior of the polygon, then by Green's theorem:
Thus, the area is equal to:
For define for . Notice that parametrizes .
Now, , so , while .
So if you add these up including you get:
Work on simplying that expression now.