Are Squares Rectangles? Are Rectangles Squares?

Date: 03/29/2004 at 17:16:48
From: Alisa
Subject: Can a rectangle be a square

Can a rectangle be a square?  I just was wondering because a square 
can be a rectangle, but can a rectangle be a square?

Date: 03/30/2004 at 09:01:48
From: Doctor Peterson
Subject: Re: Can a rectangle be a square

Hi, Alisa.

Actually, EVERY square is a rectangle, since the angles in a square 
are always right angles.  That's more than saying that a square CAN be 
a rectangle; it IS.

And since squares are rectangles, you know that some rectangles are 
squares--namely, the squares are!

These ideas confuse many people.  It may help to think about other 
categories, like dogs and animals.  Can a dog be an animal?  Of course 
--every dog is an animal!  Can an animal be a dog?  Yes, if you look 
at a dog, that is an animal that is a dog.  But not every animal is a
dog.  It's almost silly to ask and answer those questions.

It sounds less silly with squares and rectangles, but the idea is the 
same.  The trouble is that in everyday life, we usually use the word 
"rectangle" to refer to non-squares, because if the shape were a 
square, we would call it that.  So we teach children "this is a 
rectangle, that is a square--the rectangle's sides are different
lengths".  But when you grow up, it's important to think of the square 
as a SPECIAL rectangle, because it is all that a rectangle is, and 
more.

It's as if someone looked at your dog and said, "What a fine animal 
you've got!", and you answered, "That's not an animal--that's a 
pedigreed Labrador Retriever."  The specific type of animal matters to 
you, because it's special; but it really is still an animal too.  And 
a square is still a rectangle--but it's not JUST a rectangle, it's a 
(pedigreed) square.

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

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