Areas of Figures Broken into Rectangles

Date: 10/17/2001 at 20:16:19
From: Rachel
Subject: Area of figures

On my sheet it tell me to "Calculate the area of each figure. First 
divide the figure into rectangles and squares."

I have divided them into the rectangles and squares, but I don't know 
how to find the area. Please help!

Thanks,
Rachel

Date: 10/18/2001 at 11:03:57
From: Doctor Ian
Subject: Re: Area of figures

Hi Rachel, 

Once you've divided something into rectangles, 

  +------+-------------+
  |  A   |             |
  |      |             |
  +------+     B       |
         |             |
         |             +-------------------+
         |             |      C            |
         +-------------+-------------------+

you can find the total area by adding up the areas of the individual 
rectangles.  

If you don't see why this is true, you might be a little confused 
about the meaning of 'area'. If that's the case, you might want to 
look at this answer from the Dr. Math archives: 

   Area and Perimeter
   http://mathforum.org/dr.math/problems/jessica.5.1.01.html   

Think of it this way. Suppose I need to paint the shape I've drawn 
above. To find out the total amount of paint I need, I would add up 
the amounts needed to paint the individual rectangles. Does that make 
sense? 

So, how do you find the area of a single rectangle? You find two 
adjacent sides (that is, two sides that touch at a corner) and 
multiply their lengths:

      3
  +--------+
  |        | 2        area = 3 * 2 = 6
  |        |
  +--------+

Note that a square is just a special kind of rectangle, in which each 
pair of adjacent sides has the same length. So the formula still 
works. 

You may be able to divide the same shape into rectangles in more than 
one way, for example:

  +--------------------+
  |                    |
  |                    |
  +------+             |
         |             |
         |             +-------------------+
         |                                 |
         +---------------------------------+


  +------+-------------+
  |  A   |             |
  |      |             |
  +------+     B       |
         |             |
         |             +-------------------+
         |             |      C            |
         +-------------+-------------------+

  +------+-------------+
  |         A          |
  |                    |
  +------+-------------+
         |             |
         |             +-------------------+
         |    B        |      C            |
         +-------------+-------------------+


  +--------------------+
  |           A        |
  |                    |
  +------+-------------+
         |    B        |
         +-------------+-------------------+
         |    C                            |
         +-------------+-------------------+

  +------------+-------+
  |  A         |  B    |
  |            |       |
  +------+-------------+
         |     C       |
         +---------+---+---------+---------+
         |     D   | E |  F      |   G     |
         +---------+---+---------+---------+

In each case, so long as the individual rectangles cover the whole 
shape, you'll get the same total area, no matter which rectangles you 
choose.  (Again, if you think of area in terms of paint, it should be 
clear why this has to be the case.)

Generally, you want to choose rectangles where you'll be told (or be 
able to figure out) the lengths. Often the difference between an easy 
solution and a hard solution is the way you decide to break a large 
shape into smaller shapes. 

It's really not that different from breaking up any large quantity 
into smaller quantities that are easy to find. For example, suppose 
you need to know the distance from Long Beach, CA to Chesterton, IN.  
You might not be able to find a table of distances that will give you 
that distance directly; but perhaps you can find three other tables 
that give you the distance from Long Beach to Los Angeles, the 
distance from Los Angeles to Chicago, and the distance from Chicago to 
Chesterton. Then you can add those three distances to get the total 
distance (assuming that you plan to drive through Los Angeles and 
Chicago on the way). 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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