Rhombus and Square Comparison

Date: 01/14/2004 at 22:19:25
From: Meredith 
Subject: Are All Rhombi Squares

Last time my math class got together, a new question arose.  If a
rhombus can have obtuse and acute angles, how could a square possibly
turn into one?

One boy thought a square can be a rhombus and a rhombus can be a
square.  Others only agreed to the first (like me), and still more 
agreed neither was possible.  So it's kind of undecided.  Do you have
an answer?

Date: 01/15/2004 at 09:39:25
From: Doctor Peterson
Subject: Re: Are All Rhombi Squares

Hi, Meredith.

This is a matter of definition.  I'll give you an example of a similar 
kind of definition (simplified):

  a mammal is a four-legged animal with hair and milk

  a dog is a mammal that barks

So every dog is a mammal, but not every mammal is a dog--some are
cats, cows, or whatever.  That doesn't mean that a dog is also a cow! 
Rather, dogs and cows are two different kinds of mammals.

Similarly,

  a rhombus is a quadrilateral whose four sides are all congruent

  a square is a quadrilateral whose four sides are all congruent
    and whose angles are all right angles

In other words,

  a square is a rhombus that is also a rectangle

These are both rhombuses:

  +--------+
  |        |        +--------+
  |        |       /        /
  |        |      /        /
  |        |     /        /
  +--------+    +--------+

But only this is a square:

  +--------+
  |        |
  |        |
  |        |
  |        |
  +--------+

So every square is a rhombus, because it's a quadrilateral with four
congruent (equal) sides.  But there are rhombuses that are not square,
because their angles are not right angles.

The fact that there are rhombuses without right angles doesn't mean
that a square has to somehow "turn into" a shape without right angles,
any more than a dog has to "turn into" a cow in order to be a mammal.
It's just that the definition of a rhombus doesn't mention its angles
at all!  The angles matter in asking whether it is a square, but for
the rhombus only the sides matter.

Does that help?

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