Area of a Rhombus and Area of a Square

Date: 11/16/2007 at 15:01:21
From: Linda
Subject: area of a rhombus vs. area of a square

If a rhombus is a figure with four equal sides, isn't that the same 
as a square "pushed over"?  Why do we have to use base times 
perpendicular height to find the area, rather than side times side 
as in a square? 

If I calculate the area of a 4cm square, it is 4x4.  If I pushed this 
square over a bit, wouldn't I then have a rhombus?  And then the area 
would have to remain 4x4.  But I know the area formula for a rhombus 
says otherwise.  I get that there is a difference between the slant 
height and the perpendicular height, but wouldn't the two areas still 
have to be the same?  I'm sure there's a simple explanation, but I 
can't find it.  Thanks!

Date: 11/16/2007 at 15:33:15
From: Doctor Peterson
Subject: Re: area of a rhombus vs. area of a square

Hi, Linda.

As you push the square over, you are keeping the (now slanted) side 
the same length, so it is a rhombus; the height is reduced, right?

Now push it farther and farther until the top reaches the ground-- 
it's now a flat line rather than a rhombus.  Its area is now zero, 
right?  Can you see that as you kept pushing, the area must have been 
getting smaller all the time, and not staying the same?

Here's another way to see it.  This is a rhombus:

       +----------+
      /          /
     /          /
    /          /
   /          /
  +----------+

We can cut off a right triangle on the left and paste it onto the 
right to make a rectangle:

       +----------+        +----------+
      /:         /:        |          |
     / :        / :   -->  |          |
    /  :       /  :        |          |
   /   :      /   :        |          |
  +----+-----+....+        +----------+

But the height of this rectangle is less than the height of the 
original square, so its area is less. In fact, the area of the rhombus 
(or any parallelogram) is the base times the height, not the base 
times the other side.

Ultimately, your error was in assuming that the area is determined 
only by the sides, and can't vary when you change the angles between 
them.

Does that help?

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

Date: 11/20/2007 at 13:22:44
From: Linda
Subject: Thank you (area of a rhombus vs area of a square)

Thanks so much!  I get it now!