
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 $\vec {F}:\R ^2\to \R ^2$, if $\curl \vec {F} = 1$, then the left-hand side of the conclusion of Green’s Theorem gives the area of the region $R$:

So now the question becomes, which vector fields have $\curl \vec {F} = 1$? Here are three basic candidates:

• $\vec {F}(x,y) = \vector {0,x}$
• $\vec {F}(x,y) = \vector {-y,0}$
• $\vec {F}(x,y) = \vector {-y/2,x/2}$
The key idea that connects the three vector fields above is:
Their curl is $1$. 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 $1$, 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 $(x,y)$-coordinates:

Here we see a polygon with $n$ 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 $\answer {47}$ 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?

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