Finding the area of an irregular polygon via geometry can be a bit of a chore, as the process depends heavily on the shape of the polygon. It turns out, however, that there’s a formula that can give you the area of any polygon as long as you know the polygon’s vertices. That formula is based on Green’s Theorem.
Green’s Theorem is a powerful tool in vector analysis that shows how to convert from a line integral around a closed curve to a double integral over the region enclosed by the curve and vice versa. More specifically, the circulation-curl form of Green’s Theorem states that, if D is a closed region with boundary curve C and C is oriented counterclockwise, then
The special case of Green’s Theorem that can generate our formula for the area of a polygon has and . This yields
Since the double integral here gives the area of region D, this equation says that we can find the area of D by evaluating the line integral of counterclockwise around the curve C. The boundary curve for a polygon consists of a finite set of line segments, though, and so to obtain our formula we just need to find out what is for a line segment L that runs from a generic point to another generic point . Let’s do that now.
We need a parameterization of L. That requires a point on the line segment L and a vector in the direction of L. The starting point and the vector from the starting point to the ending point work well, giving us the parameterization
With and , we have . This gives us the integral
In other words, the line integral of from to is the difference in the y coordinates times the average of the x coordinates.
Let’s put all this together to get our formula. Suppose we have a polygon D with n vertices. Start with any vertex. Label it . Then move counterclockwise around the polygon D, labeling successive vertices , and so forth, until you label the last vertex . Finally, give the starting vertex a second label of . Then the area of D is given by
In words, this means that to find the area of polygon D you can just take the difference of the y coordinates times the average of the x coordinates of the endpoints of each line segment making up the polygon and then add up.
This formula isn’t the only such formula for finding the area of a polygon. In fact, a more common formula that can also be proved with Green’s Theorem is
This second formula looks a little nicer to the eye, but I prefer the first formula for calculations by hand.