Rectangle to Parallelogram

Date: 06/28/2002 at 06:52:07
From: Rose Wenzel
Subject: Parallelograms that are not rectangles

Hello.

I have a couple of questions.

First, when you press on the corners of a rectangle to make a
different parallelogram, what happens to the area and perimeter of
that figure?

Second, what happens when you have a family of parallelograms, with 
the first one being a rectangle and the others being "tilted or
extended to the right"?  The book says the area stays 20 square units,
but what about the perimeter?  

I am really confused with these issues.
Thank you.

Date: 06/28/2002 at 07:43:58
From: Doctor Ian
Subject: Re: Parallelograms that are not rectangles

Hi Rose,

In fact, a rectangle is a special case of a parallelogram, in 
which all the angles happen to be 90 degrees.  So you can use the 
same formula to compute the area of each, which is 

  area = base * height

As you 'squish' a rectangle into a parallelogram, the sides don't 
change length, so the perimeter (which is just the sum of the 
lengths of the sides) must stay the same; but the height 
decreases, so the area must decrease too. 

In fact, if you have a set of parallelograms with the same side 
lengths, the rectangle will be the one with the largest area; and 
the areas of the others can get as close to zero as you want, by 
making two of the angles very close to zero, and the other two 
angles very close to 180 degrees.  

To see why this works, take something like a cereal box and cut 
off the flaps at the ends.  Look through the box to see a 
rectangle.  Now start squishing the box as if you were going to 
throw it away, by making it as flat as possible.  The perimeter 
stays the same (since none of the sides are changing length), but 
the area smoothly decreases. 

I'm not entirely sure what your book is talking about - I suspect 
there are illustrations, which I would need to see - but it 
sounds to me as though there are two possibilities.  One is that 
the base and height stay the same, 

   +.....+          +.....+                +.....+
   .     .        .     .              .    .
   .     .      .     .            .     .
   +.....+    +.....+          +.....+

          These all have the same area,
          but different perimeters.

and the other is that the lengths of the sides stay the same, 

   +.....+           
   .     .          +.....+
   .     .        .     .              +     +
   .     .      .     .            .     .
   +.....+    +.....+          +.....+


          These all have the same perimeter,
          but different areas. 

The second case corresponds to what you should see in the cereal 
box experiment. 

From your initial statement, it sounds as though you may be 
confused about the basic ideas of perimeter and area.  If that's 
the case, you should probably take a look at 

  Area and Perimeter
  http://mathforum.org/library/drmath/view/57652.html 

Does this help? 

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