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### 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

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/
```
Associated Topics:
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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