Area of an Irregular Polygon

Date: 03/29/2001 at 11:37:19
From: Erin Cooper
Subject: Irregular polygons

Can you tell me the formula for finding the area of irregular 
polygons?

Date: 03/29/2001 at 12:23:44
From: Doctor Peterson
Subject: Re: Irregular polygons

Hi, Erin.

It depends very much on what you know about the polygon, and what 
kinds of math you know. There is a nice formula if you know the 
coordinates of the vertices, which you can find here by clicking on 
Two Dimensions: Polygons:

  http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html   

Another nice formula works if all the vertices are at "lattice points" 
(integer coordinates) and you can count the number of lattice points 
in the polygon; search our archives at  for "lattice points" to find 
Pick's rule.

  http://mathforum.org/mathgrepform.html   

Yet another set of formulas would be used by a surveyor, someone who 
measures the lengths of the sides and the angles between them and uses 
trigonometry; these can be found here by selecting Relations in 
Oblique Triangles:

  http://mathforum.org/dr.math/faq/formulas/faq.trig.html   

At your level, I would expect that things are not quite so irregular 
that you need these sorts of formulas. Perhaps your polygons have all 
right angles, and you can break them down into right triangles and 
rectangles. If you need help with some specific problems, write back 
and show me a sample problem and how far you can get in solving it.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/29/2001 at 20:05:37
From: David Cooper
Subject: Re: Irregular polygons

Thank you for helping with my math problem. Here is a problem that we 
were given at school. 

            28.5                   13.5
     +----------------+         +-------+
     |                |         |13.8   |
     |                +---------+       |
     |12.0                13.0          |26.5
     |                                  |
     |                                  |
     +----------------------------------+
                      55.0

Thank you,
Erin

Date: 03/29/2001 at 22:29:15
From: Doctor Peterson
Subject: Re: Irregular polygons

Hi, Erin.

I'm not sure how to interpret the 12.0, since it looks as if the two 
top edges are level. I'll assume the picture should be more like this:

                                   13.5
                                +-------+
                                |       |
            28.5                |       |
     +----------------+         |13.8   |
     |                |         |       |26.5
     |                +---------+       |
     |12.0                13.0          |
     |                                  |
     |                                  |
     +----------------------------------+
                      55.0

You can generally approach this sort of problem either by cutting the 
shape into smaller pieces and adding their areas, or by starting with 
a larger shape and subtracting pieces from it. Here's one way to do it 
by addition:

                                   13.5
                                +-------+
                                |       |
            28.5                |       |
     +----------------+         |13.8   |
     |                |         |       |26.5
     |                +---------+       |
     |12.0            |   13.0  |       |
     |                |         |       |
     |                |         |       |
     +----------------+---------+-------+
                      55.0

You just have to figure out the height of that middle section, which 
you can do by subtracting 13.8 from 26.5. (Look at the right-hand 
rectangle to see why.)

Here's another way, using subtraction:

                                   13.5
     +--------------------------+-------+
     |                          |       |
     |      28.5                |       |
     +----------------+---------+13.8   |
     |                |         |       |26.5
     |                +---------+       |
     |12.0                13.0          |
     |                                  |
     |                                  |
     +----------------------------------+
                      55.0

First find the area of the whole rectangle, then subtract from that 
the area of the small rectangles. You can find the width of the top 
part by adding 28.5 and 13.0, and its height by subtracting 12.0 from 
26.5. The height of the smallest piece is a little tricky, but once 
you have the height of the top piece, you can subtract.

The hard part of these problems is to figure out which sides to add or 
subtract to get another. If you don't see it quickly, just try filling 
in all the sides that aren't already labeled, starting with the sides 
of rectangles opposite labeled sides. That's the key: opposite sides 
of a rectangle have the same length. Usually once you've labeled 
everything you can do immediately, you will quickly see how to get the 
ones you need. For example, here's the second method with everything 
labeled based on opposite sides:

                          55.0
         +--------------------------+-------+
         |?                        ?|  13.5 |
         |      28.5         13.0   |       |
         +----------------+---------+13.8   |
     26.5|                |        ?|       |26.5
         |                +---------+       |
         |12.0                13.0          |
         |                                  |
         |                                  |
         +----------------------------------+
                          55.0

I hope this helps. If you need more help, write again and let me know 
where you're stuck.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/