Polygonarea

Polygonarea refers to the measure of the two-dimensional region enclosed by a polygon. For simple polygons (those that do not cross themselves), the area is well-defined and can be computed in several equivalent ways. For self-intersecting polygons, the area produced by certain methods may reflect overlaps and requires careful interpretation.

A common method is the shoelace formula. If a polygon has vertices (x0, y0), (x1, y1), …, (xn−1, yn−1) listed in order, the area is A = 1/2 |sum_{i=0}^{n−1} (xi yi+1 − xi+1 yi)|, with indices taken modulo n so that (xn, yn) equals (x0, y0). The result is the absolute value of a signed area, giving a nonnegative measure independent of the traversal direction.

Another approach is triangulation. Choose a fixed vertex v0 and form triangles (v0, vi, vi+1) for i = 1 to n−2. The total area is the sum of the areas of these triangles, where each triangle’s area can be computed from cross products: A_i = 1/2 |(vi − v0) × (vi+1 − v0)|.

Notes and variations: for polygons with holes, subtract the areas of the holes from the outer boundary area. For lattice polygons (vertices at lattice points), Pick’s theorem relates area to lattice points: A = I + B/2 − 1. In numerical work, ensure proper handling of floating-point errors and polygon orientation.

Applications of polygon area arise in geographic information systems, computer graphics, finite element analysis, and robotics, where knowing the size of planar regions is essential.