A Proof of Simple Polygonal Area via Green’s Theorem

24 Aug

Enjoyed Bryce Summers’ math proof via Green’s Theorem. His WordPress site has a number of interesting math, computer science and algorithm related blogs.



In this blog post, I will prove that a very elegant theorem for the area of a simple polygon based on Green’s theorem is true. Please note that Green’s theorem is a standard theorem from multi variable calculus. This theorem is very useful for computing the area of polygons in computer programming applications and in physics simulations.

Problem Statement

Let a fixed, yet arbitrary simple polygon P be defined by a list of points L in the euclidean plane.

Let $latex L_{k}$ denote the kth point in L and let $latex L_{n + 1} = L_{1}$

Let Area(l) be the area of the polygon represented by the given list of points.

For any point p $latex in$ L let the Cartesian x and y coordinates of p be denoted by p.x and p.y respectively.

Let Area(a, b) $latex = frac{1}{2}(a.x + b.x)(a.y – b.y)$, where a and b are…

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