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

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The special case of Green’s Theorem that can generate our formula for the area of a polygon has and . This yields

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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

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This second formula looks a little nicer to the eye, but I prefer the first formula for calculations by hand.

WoW! That is beautiful!