Meters Squared vs. Square Meters

Date: 11/18/2001 at 20:31:58
From: Paul Thompson
Subject: Definition of meters squared

My wife is a teacher and we have had a disagreement over the 
definition of meters squared. She says that a rectangle 3 meters by 4 
meters equals 12 meters squared. I say it is 12 "square meters," and 
that 12 meters squared would be a square measuring 12 meters on each 
side (or a total of 144 square meters). Who is right? This is 
important because she doesn't want to incorrectly advise her students.

Date: 11/18/2001 at 22:47:26
From: Doctor Peterson
Subject: Re: Definition of meters squared

Hi, Paul.

It's written as 12 m^2 (that is, with an exponent of 2), but is better 
read as "12 square meters" than as "12 meters squared." 

The reason for this is as you say, that the latter sounds as if it 
means (12 m)^2, the square of 12 meters, rather than 12 (m^2), as it 
really is, 12 times one meter squared. 

The fact is, however, that you will find both forms used; I think it 
is generally agreed that "12 square meters" is better, in order to 
avoid that problem, but both are "correct." That is, the problem is 
not that "12 meters squared" MEANS the area of a 12-meter square, but 
that it can be taken either way, and is thus ambiguous. We simply want 
to avoid the ambiguity.

You can read about this here:

    Miles Squared vs. Square Miles
    http://mathforum.org/dr.math/problems/wlodarczak.7.09.99.html   

Looking for an official statement, however, I found this somewhat 
ambiguous rule:

    NIST Guide to SI Units
    9 Rules and Style Conventions for Spelling Unit Names
    http://physics.nist.gov/Pubs/SP811/sec09.html   

    9.6 Spelling unit names raised to powers
    When the names of units raised to powers are spelled out, 
    modifiers such as "squared" or "cubed" are used and are placed 
    after the unit name. 

    Example: meter per second squared (m/s^2) 

    The modifiers "square" or "cubic" may, however, be placed before
    the unit name in the case of area or volume. 

    Examples:
       square centimeter (cm^2)           cubic millimeter (mm^3)  
       ampere per square meter (A/m^2)    kilogram per cubic meter 
                                            (kg/m^3) 

All the examples except one use "square unit," and that example 
(a) uses seconds rather than meters, making the interpretation of 
"a one-second square" impossible, and (b) has it following other units 
rather than a number, so no confusion is possible ("12 newtons per 
meter squared" is likewise unambiguous). But the same page also says
 
    Note: These other spelling conventions are given for completeness;
    as indicated in Sec. 7.6, it is the position of this Guide that
    symbols for numbers and units should be used to express the values
    of quantities, not the spelled-out names of numbers and units.
    Reference [4] also requires that a symbol for a number be used
    whenever the value of a quantity is expressed in terms of a unit
    of measurement.

In other words, we are admonished not to _write out_ "12 meters 
squared," which might be ambiguous, but to use the symbols. When you 
are talking or reading aloud, you should be able to clarify the 
ambiguity in person. Nevertheless, I would strongly recommend reading 
the units in the unambiguous way.

I hope that answers the question!

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

Date: 04/18/2002 at 08:53:44
From: Kjerstin Epstein
Subject: Area

Why is area written two different ways, for instance square meters or 
m^2?
 

Date: 04/18/2002 at 09:35:08
From: Doctor Ian
Subject: Re: Area

Hi Kjerstin,

A 'square meter' is a particular unit of area, which is, as you might 
guess, equivalent to the area of a square that has a side length of 
1 meter. 

A 'meter squared' is the same thing, but is somewhat more general.  
Note that you can specify area in _any_ units you want, so long as you 
have a length multiplied by a length.  

Consider the following rectangle:

                3 feet
          +----------------+
  1 foot  |                |
          |                | 12 inches
          +----------------+
                1 yard

We can specify the area of this rectangle in any of the following 
ways:

       1 foot * 3 feet = 3 foot-feet 
                          (or foot^2, or square feet)

        1 foot * 1 yard = 1 yard-foot

     12 inches * 3 feet = 36 foot-inches

     12 inches * 1 yard = 12 yard-inches

  12 inches * 36 inches = 432 inch-inches 
                             (or inches^2, or square inches)

All of these 'units' are perfectly acceptable, although some of them 
are unusual. 

However, specialized units like this are sometimes selected in order 
to make quantities easy to visualize. For example, newspapers and 
magazines sell advertising space not in terms of 'square inches', but 
in terms of 'column-inches', which measure the area of a column - 
however wide that turns out to be for a given publication - one inch 
long.  

Similarly, for some applications involving large amounts of water, a 
measure of volume called an 'acre-foot' is used.  This is, as you 
might guess, the volume of something piled a foot deep over an acre of 
area. 

Sometimes, too, particular compound units are given special names 
because they are so often referred to, e.g., 'acre', 'gallon', 
'light-year', 'joule'.  I would say that units like 'square meter' and 
'cubic inch' fall into this category.  
  
On the whole, I find that meters^2 is easier to write (and m^2 is even 
easier), but square meters is easier to say, so that's how I use them.  

Does this help? 

- Doctor Ian, The Math Forum