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

**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:

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.

- is drawn in a counterclockwise direction.
- is drawn exactly once.
- The interior of is to the left of the direction of .

### Computing areas with Green’s Theorem

Now let’s do some examples.

and each draws the line as runs from to . Write:

For each of the integrands above, say , we will write and combine them into a single integral. Write with me

So the this is the area of the trapezoid.

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

and each draws the line as runs from to . Write:

For each of the integrands above, say , we will write and combine them into a single integral. Write with me

So the this is the area of the region.

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:

*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

The area of the polygon in question will be: The algorithm is called the “shoelace” algorithm because of the crisscrossing pattern you see above.

#### 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.

with . Moreover

this is just saying that the line integral along the perimeter of the polygon is the sum of the line integrals along the edges. Now write with me

So now we may write

Noting that the terms and will cancel with each other as we cycle through the sum, we find that

Since this value relies on being parameterized in a counterclockwise fashion, we take the absolute value to ensure a correct answer (just in case the young geometer accidentally parameterized in a clockwise fashion). Thus we have completed the explanation of the shoelace algorithm.

So the student concludes that the area is Is this correct?

For some interesting extra reading check out: