What is the area of a polygon?

Finding the area of a polygon, when it is a rectagle or regular shape can be accomplished by analytic means, though sometimes with difficulty. Squares and hexagons are easy but heptagons require some trigonometry. When the shape is irregular, it is often necessary to decompose it into triangles and compute cross products to get areas.

The question we pose and then answer here is how to make the area measurement if we know only the coordinates of the vertices? Consider the example shown below.

There is a nifty formula, based on Greens theorem, or merely triangles and cross-products for determining the area. Form the table of the coordinate data as show below, with the first point repeated on the last row.

Now form the two products

The area of the polygon is given by

• What is remarkable is that this formula works for any polygon. The general formula for a polygon with vertices is given by
  
  

Let's check the formula for the rectangle shown

	The table is 1 1 3 1 3 2 1 2 1 1

We have that

which is of course the area.

Now let apply this formula to find the area of the five sided figure above. Follow the steps.

• Import the image.
  
  

• Select an origin (red cross) and a point (blue cross) on the -axis. Enter the scale. In this example, we've included a ruler to help out. Otherwise the answer will be accurate only up to a factor equal to the square of the scale ratio. Also, for increased accuracy, we put the origin and the -axis point on the ruler, as the formula is coordinate system independent. As the points selected are three units apart we enter that for the -axis value.
  
  

• Now select points - at each vertex and show points.
  
  

• Next select the points from the data box, copy and paste into the spreadsheet application. Of course, a calculator could be used here just as well. In a spreadsheet you will need to "parse" the data into two columns. (In Microsoft Excel this is accomplished by using the Data+Text_to_Columns keys.)
  
  

• Finally make the computation as given by the algorithm above. The area computed "seem" about correct. (Hint. Don't forget to repeat the first order pair in the last row.)