What is Length in a Rectangle?

Date: 05/31/99 at 17:22:02
From: Herb
Subject: What is length?

According to the Webster's Ninth Collegiate Dictionary, length means 
"the longer or longest dimension of an object."

So the length of a rectangle is the longest side whether it is 
vertical or horizontal. Correct?

Date: 06/01/99 at 13:05:35
From: Doctor Peterson
Subject: Re: What is length?

Hi, Herb.

Yes, I think you're right.

It happens that I just had a discussion of this issue with a teacher a 
few weeks ago, where the hard part to deal with was the word "width." 
It might be of interest to you. Here's what we said:

===================================================
Dear Dr. Math,

I am a sixth grade math teacher, and we are currently studying ratios.  
In a homework assignment two nights ago, the students were assigned 
problems concerning this concept, with three of the questions asking 
the students to write ratios of width to length on three different 
rectangles. Two of the rectangles had their short sides going right to 
left, with the long sides going up and down; the third rectangle 
showed the short sides going up and down and the long sides going 
right to left.  

The answers to the problems in the Teacher's Manual gave the answers 
as the width always being the short side, and the length always being 
the long side. However, this was not mentioned in the directions, so 
many of the students wrote the wrong answers; they were trying to stay 
consistent by using the idea that width is always "across," or left to 
right.

Our question is, if it's not clear in the instructions, how is one 
supposed to judge which is width and which is length?  I have asked 
all the math teachers in our building, and there just isn't a 
consensus. Some say that the short side is always the width, and the 
long side is always the length, regardless of "left to right" or "up 
and down."  Others say it's the other way around. Still others say it 
depends on the situation. 

Help! Is there a definitive answer to this? Is there a strict 
definition for width and length for two-dimensional objects? We 
anxiously await your answer.... :)

Sincerely,
Mrs. Fletcher's Sixth Graders  
===================================================

Isn't English wonderful! It makes math so much easier...

I looked up width in my dictionary, and it says "The measurement of 
the extent of something from side to side; the size of something in 
terms of its wideness." Length, on the other hand, is "(a) The 
measurement of the extent of something along its greatest dimension. 
(b) The measurement of the extent of something from back to front as 
distinguished from its width or height."

This gives me two ways to look at it. I can take "side to side" with 
reference to its position (from my perspective), and use definition 
(b) for length to say that length is whatever isn't width, so I have

    +---------------------------------+
    |                                 | l
    |                                 | e
    |                                 | n
    |                                 | g
    |                                 | t
    |                                 | h
    |                                 |
    +---------------------------------+
                  width

I don't like this, though; it does seem odd for the length to be both 
short and vertical! If I want "width" to mean across, I use "height," 
not length, for the other dimension.

Or I can use definition (a) of length to mean I have to look at it     
from its OWN perspective, the length being the long dimension (from 
its head to its tail, if it were an animal) and the width being "from 
side to side" across its own length:

    +---------------------------------+
    |                                 |
    |                                 | w
    |                                 | i
    |                                 | d
    |                                 | t
    |                                 | h
    |                                 |
    +---------------------------------+
                  length

If it's vertical, both approaches agree. If not, we have confusion.

The really bad case is in three dimensions. Does a rectangular prism 
have height, width, and length, or breadth, or depth, or - what do you 
call the dimension from front to back? We don't have any really 
good words for that.

Eric Weisstein's World of Mathematics provides definitions of length, 
depth, height, and width:

    http://mathworld.wolfram.com/Length.html
    http://mathworld.wolfram.com/Depth.html
    http://mathworld.wolfram.com/Height.html   
    http://mathworld.wolfram.com/Width.html   

    Length (Size)
    The longest dimension of a 3-D object.

    Depth (Size) 
    The depth of a box is the horizontal Distance from front to back
    (usually not necessarily defined to be smaller than the Width, the
    horizontal Distance from side to side).

    Height
    The vertical length of an object from top to bottom. 

    Width (Size)
    The width of a box is the horizontal distance from side to side 
    (usually defined to be greater than the Depth, the horizontal 
    distance from front to back).

This suggests a preference for absolute direction (from our 
perspective) rather than using the larger or smaller dimension to 
determine which is width, and a preference for height/width/depth for 
the three dimensions. Again, I don't know that this is a universally 
accepted definition, but I would tend to agree that width is 
horizontal, height is vertical, and length should not be used in 
combination with these; but when length and width are used together, 
it makes some sense to take length as the long dimension.

In other words, when length is used, it should be the long dimension, 
and position is irrelevant; when length is not mentioned, the 
dimensions are all relative to our perspective, and relative sizes are 
irrelevant. I think the statement of the problem was poorly worded, 
and should have been clarified, but their interpretation of it makes 
the best sense. In particular, in talking about the shape of a 
rectangle, it's right to ignore position and ask for the ratio of its 
long to its short dimension, since that helps us recognize similar 
rectangles. This would be even clearer if one of the three rectangles 
had been tilted 45 degrees!

As to how you're supposed to judge things like this when they don't 
tell you - to each his own. One of the things math teaches us is the 
importance of clarity in language, and the need to add extra words or 
special definitions to clarify what English leaves obscure. I wouldn't 
count wrong those who took width as horizontal (I probably would have 
been one of them, before thinking this out), but I think this should 
be a memorable lesson for all of you. I love seeing this kind of 
argument, because you can't really lose!

==================================

Mrs. Fletcher replied:

Thanks so much for all that information!

Actually, I told the students (apparently incorrectly), that with the 
absence of clarity, I personally would have assumed that width is 
horizontal, and that length is... well... the other one.... :)  I used 
the example of the doorway into the classroom, which has a window over 
its top. The door is taller than it is "wide" (there we go again), and 
it occurred to me that if someone tried to bring something into the 
room that wouldn't fit horizontally, we would describe that object as 
being too "wide." With that in mind, the window at the top of the door 
is horizontally longer than the vertical sides, but since the door 
beneath it is the same size horizontally, then it follows that the 
window is "wider than it is long." Again, we're assuming that the word 
"height" isn't part of the vocabulary at the moment. Did I confuse you 
with all that?

It seems to me that when the only two words we have available to make 
our point are "width" and "length," we have to come up with one 
definition that works for all cases. I just naturally used the one 
where something doesn't fit through the door because it's too "wide." 
With that in mind, then the "width" of the window at the top is wider 
than it is long. Or something like that. 

Anyway, thanks for the help... we'll write again when we have another 
one for you... kids can come up with 'em, can't they? Have a good 
one!

Mrs. Fletcher
===================================================

You nicely illustrates the importance of context in language. As I 
said, before I thought it through carefully I would have joined you in 
assuming that width means horizontal; and certainly in the context of 
an object fitting through a door, or of describing a window, that 
would be exactly right. In these contexts the position of the 
rectangle is fixed, and it seems that the positional definition of 
width takes precedence in that case. Width as narrow dimension is 
applicable only in cases where the object itself is the frame of 
reference, where it is thought of as movable and is the focus of our 
attention. On a page of abstract rectangles, there are no cues to tell 
us which way to take it, so I think both are valid.

In math, we try to avoid letting words depend on context, so in more 
advanced fields we define special terms very carefully. In elementary 
math, we don't have the freedom to choose our own terms, especially 
when we deal with real-world applications, so we have to be all the 
more careful.

It's interesting that even kids naturally try to attain consistency. 
Sometimes the only way to do that is to make an arbitrary convention; 
but to make it a convention we have to share it with others. Your 
class has become a model of the mathematical community, looking for 
ways to define terms in order to communicate clearly. I love it!

Thanks for an interesting math/language issue -- I enjoy this kind of 
topic.

===========================================

That's probably much more than you wanted, Herb, since your question 
was about length rather than width, but it does suggest some of the 
complexity of language.

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

Date: 03/13/2002 at 14:49:14
From: Mr. LeRossignol
Subject: Naming rectangle dimensions

Is the longer side of a rectangle always considered the length, so 
that by default the shorter side is always considered the width? Or is 
it arbitrary, meaning the length can be either the short or the long 
side?

Date: 03/13/2002 at 15:15:10
From: Doctor Peterson
Subject: Re: Naming rectangle dimensions

Hi, Mr. LeRossignol.

Properly speaking, in English "length" means either the longest 
dimension, or the primary dimension in some other sense. Here is part 
of the definition from my American Heritage dictionary:

    2.a. The measurement of the extent of something along its
    greatest dimension. b. The measurement of the extent of
    something from front to back as distinguished from its width
    or height.

But since English lacks a general word without reference to relative 
size or orientation, in math we often use "length and width" without 
any distinction. For instance, in the formula for the area of a 
rectangle, it makes no difference which is bigger, so "l" and "w" in 
my mind are just arbitrary labels for the two dimensions.

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