Classifying Quadrilaterals

Date: 01/25/2006 at 09:11:07
From: adrienne
Subject: Why is a square a rectangle

My teacher says that a square is also a rectangle.  I don't understand
how that can be since the sides are the same length.

Date: 01/25/2006 at 10:12:45
From: Doctor Peterson
Subject: Re: Why is a square a rectangle

Hi, Adrienne.

In everyday usage, we would point to a square and say "that's a 
square, not a rectangle."  That's because we generally name everything 
as specifically as possible.  "That's not a lady, that's my wife!"

In math and science, we have a slightly different perspective on 
words: we want each term to apply to anything for which it makes 
sense, and we want each definition to be as straightforward as 
possible.  So, for example, we define a rectangle simply as "a figure 
with four straight sides, all of whose angles are right angles".  If 
you compare a square with this definition, you see that it fits: it 
does have four sides, and it does have four right angles.  So it is a 
rectangle.  It's MORE than a rectangle, of course; in addition to the 
requirements of a rectangle, it also has four EQUAL sides.  But that 
doesn't make it NOT a rectangle, only a more SPECIAL rectangle.

We make definitions like this (called "inclusive definitions") for a 
reason.  When we state a theorem about rectangles, we want to be able 
to apply it in every case where it is true.  Since any fact about a 
shape that depends only on the fact that it has right angles will 
apply not only to "mere rectangles", but also to squares (special 
rectangles), it makes sense to use one word to cover them all, rather 
than having one word for "non-square rectangles", and another for 
squares.  If nothing else, this makes it a lot easier to state 
theorems.

The same is true in other fields of science.  We use inclusive 
definitions in naming animals, for example: a terrier is a special 
kind of dog, and a dog is a special kind of mammal, and a mammal is a 
special kind of vertebrate.  You wouldn't say "that's a terrier, not a 
mammal" just because "terrier" is a more specific term than "mammal"; 
we need to have a word that covers all mammals, so that we can talk 
about facts that are true of all of them.  In the same way, a square 
is just a "species" of rectangle.

Just as we can make a whole classification tree for animals, we can 
classify shapes using these inclusive names.  Here is a classification 
of the main types of quadrilaterals (four-sided figures):

                 quadrilateral
                   /       \
                  /         \
                 /           \
               kite        trapezoid
                |           /     \
                |          /       \
                |         /         \
                |  parallelogram   isosceles
                |     /   \        trapezoid
                |    /     \         /
                |   /       \       /
              rhombus       rectangle
                   \          /
                    \        /
                     \      /
                      square

Each figure is a special case of the figure(s) above it.  Without 
inclusive naming, we would have to write theorems like "if you 
connect the midpoints of successive sides of a quadrilateral or 
kite or trapezoid or parallelogram or rhombus or rectangle or 
square, then the resulting figure will be a parallelogram or rhombus 
or rectangle or square."  Using inclusive definitions, the theorem is 
just "if you connect the midpoints of successive sides of a 
quadrilateral, then the resulting figure will be a parallelogram." 
That saves a lot of trees!

The following page talks about both the general issue of inclusive 
definitions, and the specific issue of naming quadrilaterals, 
including squares and rectangles:

  Quadrilateral Classification  
http://mathforum.org/library/drmath/sets/select/dm_quad_classify.html 

If you have any further questions, feel free to write back.


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