Square Centimeters and Centimeters Squared

Date: 07/09/2004 at 18:30:23
From: Alana
Subject: Square centimeter abbreviation: 1 cm^2 vs. 1^s cm

If it's pronounced "1 square centimeter" why is it abbreviated 1 cm^2?
My Dad said that 1 centimeter squared is much bigger than 1 square
centimeter.  I don't get it.

What's most confusing about it is that when it is not abbreviated it 
is square centimeter, but when it is abbreviated it is centimeter 
squared.  Why don't they just say 1^s cm?  The ^s would stand for 
square and it would be much easier to understand.

Date: 07/09/2004 at 23:00:36
From: Doctor Peterson
Subject: Re: Square centimeter abbreviation: 1 cm^2 vs. 1^s cm

Hi, Alana.

Actually, a "centimeter squared" is the _definition_ of a "square 
centimeter" (though I think I know why your father said what he did, 
and I'll mention that later).  Let me go through the ideas involved 
slowly.

The name "square centimeter" came before the notation, which is why 
we read it differently than it looks in symbols; but there's also 
good reason not to read it as "centimeters squared", though there's 
good reason for the symbolic form we use.

An area is measured as if we were cutting out lots of little paper 
squares one centimeter on a side, and seeing how many it takes to 
cover a region.  We call each of those a "square centimeter" because 
it is a square corresponding to the linear unit called a centimeter.

Eventually someone (I don't know when or where) realized that if you 
treat units as if they were variable names (in algebra), multiplying 
and dividing them along with the numbers, you could easily see the 
units used in the result of a calculation.  For example, if you 
multiply a time in hours by a speed in kilometers per hour, you get 
the distance in kilometers:

                                km
  4 hr * 5 km/hr = 4 * 5 * hr * -- = 20 km
                                hr

since the "hr" in the numerator and denominator "cancel".

When you do that with an area, say of a 4 by 5 centimeter rectangle, 
you get this:

  4 cm * 5 cm = 4 * 5 * cm * cm = 20 cm^2

while we know that the area is 20 square centimeters.  That is, the 
"square centimeter" comes out as "centimeters squared".  What does 
that mean?  It means that a square centimeter can be thought of as the 
product of a centimeter times a centimeter; which of course it is, 
since you find the area of a one-centimeter square by squaring the 
length of its side, which is 1 cm.  So the unit we've long called a 
square centimeter can be thought of as the square of a centimeter, and 
written as cm^2.

Now, why not switch over and just read it as "centimeters squared"?  
This is where your father's comment comes in.  Suppose I called the 
area of that rectangle above "20 centimeters squared".  We've just 
observed that one centimeter squared is the area of a one-centimeter 
square.  So it sounds like 20 centimeters squared should be the area 
of a 20-centimeter square.  But that's a lot larger; it's actually 
400 cm^2, since you have to multiply 20 by 20 to find the area.  So 
if we read the area that way, we would have to keep in mind that we 
mean

  20 (centimeter squared)s

rather than

  (20 centimeters) squared.

That just doesn't make good English. So we keep the old wording

  20 square centimeters

which can only be interpreted one way.

Here are some pages you and your father will find interesting:

  Defining a Square Centimeter
    http://mathforum.org/library/drmath/view/60392.html 

  Meters Squared or Square Meters?
    http://mathforum.org/library/drmath/sets/select/dm_m_squared.html 

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

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