What are Length and Width?

One of the recent discussions I showed last week dealt with the meaning of length, and I promised more about that. Here we will look at some older questions about the ambiguity of words  like length, width, depth, and height.

Is length the longest dimension?

Our first question is from 1999:

What is Length in a Rectangle?

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?

We’ll see that “length” is reasonably clear; it’s the other words used with it that cause trouble.

I answered by referring to a previous question that involved more subtlety than this one:

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:

Most of what follows will be that earlier discussion.

Length and width: colliding meanings

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.... :)

Mrs. Fletcher's Sixth Graders

Here is the book’s interpretation, where width is always the shorter side:

And here is what some students thought, where width is always horizontal:

I answered ironically:

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."

But what does “side to side” mean? That’s the question!

External perspective

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
    |                                 |

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.

… like this:

Sometimes formal definitions collide with our natural sense of meaning; the former is intended to describe the latter, because English starts with its speakers, not with lexicographers! And that can be very hard.

The key idea, I think, is that when we use “length” and “height” together, we imply that we are looking at it from our perspective, with fixed “up and down” vs. “left and right”. When we use “length” and “width” together, we probably mean something else:

Internal perspective

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
    |                                 |

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

That is, we would have no objection to this, either way:

    |                  |
    |                  |
    |                  | l
    |                  | e
    |                  | n
    |                  | g
    |                  | t
    |                  | h
    |                  |
    |                  |

There is no collision here.

Moving to three dimensions

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:

   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).

   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.

This is a little more subtle than the dictionaries.


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!

This last comment is important: Shapes don’t depend on orientation, something that was also mentioned here:

What is a Diamond?

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!

Definitions and context

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

It occurs to me now that fitting an object through a window depends on its width, in the sense of the narrower dimension, which is what limits it: If the narrowest dimension can’t fit, then nothing can. If only the length were too great, you could just turn it … which fits into the following.

I responded:

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.

This is where “word problems” get into trouble, using real-world situations and wording, and having to translate those into mathematical terms.

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.

I closed the 1999 answer:

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.

When it doesn’t matter

In 2002, we got a similar question that was added to this page:

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?

I replied, using a different dictionary:

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.

This is often the best answer: Don’t worry about it! (Of course, that wouldn’t work for the ratio problem.)

Height, length, width and depth

Here is another question, from 2003, about the 3-dimenstional form of the issue:

Length or Width?

Can you please explain the difference between height, length, width and depth?  For example, if you were measuring a door, what would you label each measurement?  Is it always the same for each thing?  For instance my brother says the longest measurement is length, but that is not the way my teacher explained it.  My parents say depth and my teacher says width.

Doctor Ian answered:

Hi Kelsey,

This might not be what you want to hear, but in the real world, those names are assigned somewhat arbitrarily.  

Usually 'height' means vertical distance... but if you turn something on its side, you might still use 'height' to refer to the dimension that _would_ be up in the normal orientation.  (For example, if your height is 5 1/2 feet, and you lie down, we would still say your height is 5 1/2 feet.)

If you're looking at something from the front, you'd probably label the dimensions this way:
              /               /|
             /               / +
            /               / / h
           /               / / t
          /               / / p
         +---------------+ / e
 height  |               |/ d

But what if it's a swimming pool?  Then I'd label it 

               /               /|
              /               / +
             /               / / h
            /               / / t
           /               / / g
          /               / / n
         +---------------+ / e
  depth  |               |/ l

because this use of 'depth' seems more natural.  (When we ask "How deep is your pool?", we mean in the vertical direction.)  Also, I would probably label it this way if it's just an open box, for much the same reason.  (If I'm going to stack things in the box, I'll stack them to some 'depth.')

This is a change of perspective in a different sense!

If I were looking at a door, use 'height' or 'length' to label the vertical dimension (usually, but not necessarily, the largest dimension--some garage or stable doors are wider than they are tall), 'width' to label the horizontal dimension, and 'depth' to label the dimension that I'm looking along (the one that goes from me through the door).

But as I said, these are arbitrary.  One question you might ask yourself is: What happens if you switch the names around?  The answer is: Nothing at all.  The object stays the same, and all the same formulas still work.  Since it makes no difference what you call them, there's no reason to get hung up about the names, is there?

Again, it doesn’t matter as far as the math is concerned (for finding volume, for example), only in terms of description.

Putting it in perspective

I also answered, almost simultaneously, referring to the discussion above:

Hi, Kelsey.

You can find a discussion (far too long!) of all these words here:

  What is Length in a Rectangle?

I would measure the height vertically and the width horizontally, in  the case of a door.  The thickness of the door might be called its depth, but "thickness" would be clearer in this case.

For a door, since the height is the longest dimension, I can't imagine calling that the width or depth!  Was that for something other than a door?

In general we use "length" for the longest dimension, but as in our example, sometimes it's not the relative length that matters, but the orientation (horizontal or vertical).

But this is really an English question, rather than math.  I would rather use a language that didn't force us to twist words around as we have to here; but other languages probably have different issues.

I’m guessing that Kelsey meant to say the thickness can be called depth or width.

And, yes, other languages definitely have different issues! But communication is tricky everywhere.

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