A planimeter computes the area of a region by tracing the boundary.

Green’s Theorem may seem rather abstract, but as we will see, it is a fantastic tool for computing the areas of arbitrary bounded regions. In particular, Green’s Theorem is a theoretical planimeter. A planimeter is a “device” used for measuring the area of a region. Ideally, one would “trace” the border of a region, and the planimeter would tell you the area of the region.

How is Green’s Theorem a planimeter?

Recall Green’s Theorem:

Given a vector field , if , then the left-hand side of the conclusion of Green’s Theorem gives the area of the region :

So now the question becomes, which vector fields have ? Here are three basic candidates:

The key idea that connects the three vector fields above is:
Their curl is . They are conservative fields. They are gradient fields. When used in combination with Green’s Theorem, they help compute area.

Once we have a vector field whose curl is , we may then apply Green’s Theorem to use a line integral to compute the area.

Computing areas with Green’s Theorem

Now let’s do some examples.

Finally, what do you do if you have a very strangely shaped curve? You approximate it with a polygonal curve. Check out next example.

Green’s Theorem gives a fairly easy method for computing any the area of any polygonal region. Any region with a “smooth” border can be approximated by a polygonal region. The upshot? Green’s Theorem is a powerful tool for computing area.

The shoelace algorithm

Green’s Theorem can also be used to derive a simple (yet powerful!) algorithm (often called the “shoelace” algorithm) for computing areas. Here’s the idea: Suppose you have a two-dimensional polygon, where the vertices are identified by their -coordinates:

Here we see a polygon with vertices, and the “dashed-line” means that there could be more to this polygon than “meets the eye.” So to compute the area, here is a neat trick. Write all of the coordinates in a column, writing the starting coordinate twice, both at the beginning and at the end:

Now multiply entries of the columns diagonally down from the left to the right and add them together

to obtain: Now, multiply entries of the columns diagonally down from the left to the right and subtract them from our previous sum to obtain: The area of the polygon in question will be: The algorithm is called the “shoelace” algorithm because of the crisscrossing pattern you see above.
Compute the area of the following polygon: The area is square units.

Why does the shoelace algorithm work?

Now we are going explain why the shoelace algorithm works via Green’s Theorem. The restrained young mathematician may protest that we are using a “crane to crush a fly,” but whatever. We like Green’s Theorem.

A student is working with a pentagon: and using the shoelace algorithm: computes the area of the pentagon as: So the student concludes that the area is Is this correct?
Yes No
What is the correct answer?

For some interesting extra reading check out: