Associated Topics || Dr. Math Home || Search Dr. Math

### 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!
```
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search