Finding the Area of a Polygon with the Shoestring Algorithm
For any closed polygon with vertices (x0, y0), (x1, y1), (x2, y2), (x3, y3) through (xn, yn), the area
of the figure is equal to
Of course, this is a mouthful to say and write, so let’s take a look at this method in practice to see
just how easy it is. Consider the triangle with vertices (1,1), (1,5) and (3,3). Obviously, it would
be easy to verify that the area of this triangle is 4 square units. But the shoestring algorithm
would work as follows to give the result:
1. Write the ordered pairs on top of one another to form two columns, one with the x-values
and one with the y-values. At the end, repeat the first ordered pair. This has been done
below; note that the first ordered pair (1,1) was repeated at the bottom.
2. Multiply diagonally down and to the right. Note below that this gave the three products 5,
3 and 3. Repeat this process multiplying diagonally down and to the left.
3. Add the products in each column. Below, this gave the sums 19 and 11, shown in bold.
4. Finally, take half the positive difference between the two sums. This gives the number of
square units in the area of the polygon.
1
1
1 ← 1
5 → 5
15 ← 3
3 → 3
3 ← 1
1 → 3
19
11
The name for this algorithm is likely obvious; it comes from the form of the “laces” that connect
the numbers to be multiplied.
A more complex example will help to exemplify the rule further. Consider the quadrilateral with
points (1,2), (5,1), (6,3) and (3,5). The area of this quadrilateral would be found as follows:
1
2
10 ← 5
1 → 1
6 ← 6
3 → 15
9 ← 3
5 → 30
5 ← 1
2 → 6
30
52
The area is ½ (52 – 30) = 11 square units.
You can check it for yourself using grid paper. Of course, it’ll take a whole lot longer than this
method took. :-)
The only thing to note is that the polygon must be closed and it cannot intersect itself. The two
polygons shown below would not work using this method:
The following Web site provides a semi-proof of the technique about halfway down the page:
http://astronomy.swin.edu.au/pbourke/geometry/polyarea/