The signed area of a polygon with vertices numbered through can be calculated exactly by the formula , where and . If the polygon is simple (non-intersecting sides), with the vertices numbered in a counterclockwise direction, the signed area is the area. This formula is surprisingly useful in surveying, architecture, and many other applications.
The formula for the area of a simple polygon can be elegantly derived using Green's theorem and extended to moments of the region. S. F. Bockman, "Generalizing the Formula for Areas of Polygons to Moments," Amer. Math. Monthly, 96(2), 1989 pp. 131-132.
For more information, see: S. Wagon, Mathematica in Action, 2nd ed., New York: Springer, 1999.