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Base or Width?

Date: 06/11/2002 at 10:12:47
From: Alexis Harrison
Subject: Measuring

What is the difference between length*width and base*height? 
They look the same to me, except the figure being measured is rotated. 

Why do you use two different formulas to measure a parallelogram and
square? 

Thanks, I'm really confused.


Date: 06/11/2002 at 12:38:10
From: Doctor Peterson
Subject: Re: Measuring

Hi, Alexis.

These are largely just different terms for the same things, but named 
differently to remind you of some important details.

For a rectangle, we usually refer to the length and width; properly, 
these are the longer side and the shorter side, respectively, but 
that really makes no difference. If you set the rectangle 
horizontally, there is no reason not to call them the "width" and 
"height":

    +---------+
    |         |
    |         |h
    |         |
    +---------+
         w

The area is the product of these.

For a parallelogram, the important thing to remember is that the 
"height" is no longer a side of the shape, but the "base" is:

        +---------+ ---
       /         /   |
      /         /    h
     /         /     |
    +---------+ -------
         b

The height is the distance between the top and bottom, measured 
perpendicular to them. The base is the length of the bottom. If we 
had said "width", you might think it meant the greatest side-to-side 
distance, which is not the base:

    |<-----w?---->|
    |             |
    |   +---------+ ---
    |  /         /   |
    | /         /    h
     /         /     |
    +---------+ -------
         b

But for a rectangle (which, after all, is a kind of parallelogram!), 
there is no confusion over the meaning of "width", so you can say 
that the area is the base times the height or the width times the 
height.

That's why we use the words we do. Does that help? If you think about 
it, you will realize that you can learn ONLY the formula for the 
parallelogram, and you can find the area of rectangles and squares as 
well -- three for the price of one. Realizing that is a big help!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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