Finding Total Area of Several Rectangles

Date: 07/08/2005 at 21:59:22
From: Chris
Subject: Square footage

I am trying to calculate the square footage of all the lawn properties
I manage, and I am using general rectangles.  I multiplied the lengths
and widths of each rectangle individually on a spreadsheet and then 
summed all the square footages to get one number, but when I add up 
all the lengths and all the widths of the properties and multiply them 
together I get an entirely different number.  

Which number is correct?

75 X 100 = 7500
50 X 200 = 10000
90 X 175 = 15750
      total 33250

75+50+90 = 215
100+200+175 = 475
215*475 = 102125

Date: 07/08/2005 at 23:00:46
From: Doctor Peterson
Subject: Re: Square footage

Hi, Chris.

Adding the rectangles' areas is the correct way; adding their lengths
and widths gives the wrong number.

Let's take a simple example to see why.  Suppose you have two lawns,
one 4 by 6 and one 5 by 5.  I'll draw them this way:

  +---+---+---+---+---+---+
  |   |   |   |   |   |   |
  +---+---+---+---+---+---+
  |   |   |   |   |   |   |
  +---+---+---+---+---+---+ = 24
  |   |   |   |   |   |   |
  +---+---+---+---+---+---+
  |   |   |   |   |   |   |
  +---+---+---+---+---+---+---+---+---+---+---+
                          |   |   |   |   |   |
                          +---+---+---+---+---+
                          |   |   |   |   |   |
                          +---+---+---+---+---+
                          |   |   |   |   |   | = 25
                          +---+---+---+---+---+
                          |   |   |   |   |   |
                          +---+---+---+---+---+
                          |   |   |   |   |   |
                          +---+---+---+---+---+

The total area is 24 + 25 = 49.  But if you add the lengths and the
width, you get an 11 by 9 rectangle, whose area is 99:

  +---+---+---+---+---+---+-------------------+
  |   |   |   |   |   |   |                   |
  +---+---+---+---+---+---+                   |
  |   |   |   |   |   |   |                   |
  +---+---+---+---+---+---+                   |
  |   |   |   |   |   |   |                   |
  +---+---+---+---+---+---+                   |
  |   |   |   |   |   |   |                   |
  +---+---+---+---+---+---+---+---+---+---+---+
  |                       |   |   |   |   |   | = 99
  |                       +---+---+---+---+---+
  |                       |   |   |   |   |   |
  |                       +---+---+---+---+---+
  |                       |   |   |   |   |   |
  |                       +---+---+---+---+---+
  |                       |   |   |   |   |   |
  |                       +---+---+---+---+---+
  |                       |   |   |   |   |   |
  +-----------------------+---+---+---+---+---+

That's a lot of extra area you added on by adding the dimensions!  
With more than two rectangles, it gets worse.

The point of this experiment is, you can't add areas by adding their
individual dimensions.  You were right the first time.

If you have any further questions, feel free to write back.


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