Planar Partitions and Polygon Geometry

A simple polygon is a closed planar figure bounded by finitely many line segments -- its edges -- that meet only at their endpoints -- its vertices -- and do not cross.

Given a simple polygon with vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ listed in order around its boundary, its area is given by the Shoelace formula:

$$A = \dfrac{1}{2} \left| \sum\limits_{i=1}^{n} \bigl(x_i\, y_{i+1} - x_{i+1}\, y_i\bigr) \right|$$

where the indices are taken cyclically, so that $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.

For example, the triangle with vertices $(0,0)$, $(4,0)$, $(0,3)$ has area

$$\begin{align*} A &= \tfrac{1}{2}\bigl|(0 \cdot 0 - 4 \cdot 0) + (4 \cdot 3 - 0 \cdot 0) + (0 \cdot 0 - 0 \cdot 3)\bigr| \\[3pt] &= \tfrac{1}{2}\bigl|0 + 12 + 0\bigr| \\[3pt] &= 6. \end{align*}$$

In air-routing applications, sector boundaries are modeled as simple polygons, and the Shoelace formula is the workhorse for computing their areas.