Height and Length…Width and Length…Height and Width?

I’ve actually had this conversation with a few previous colleagues. It arose due to the wording in some questions in a maths text book and some of the terminology I heard kids using when talking about measuring area/perimeter of 2D shapes (predominantly quadrilaterals.

The main issue is what do you call each side? From a very cursory glance on the internet I have found most people use length and then either width or height. I will try to articulate why I hate using the word length in this context. But before I do that I will explain the words I would use with regards to measuring in various dimensions.

1D- length e.g. a piece of string, a line

2D- height and width (being vertical and horizontal respectively with regards to your orientation with the paper) e.g. most 2D shapes (some of which have multiple widths e.g. a trapezius)

3D- height, width and depth (height and width being used to measure the face of the object and depth being the extra measurement, front to back if you will) e.g. cubes and that jazz

4D- fuck knows…

1000px-dimension_levels
x,y,z being width, height and depth respectively

Now, as I’ve pointed out, I would measure anything that is 1 dimensional as a length, and as height, width and depth on their own are 1 dimensional, they are all lengths. They need one another to create the extra dimensions. Whether you measure left to right, back to front, top to bottom, any one of these measurements on their own is one dimensional and it is the relationship between these measurements that creates further dimensions.

Summary: height, width and depth are all lengths so referring to the side of an object as ‘length’ is ambiguous and could be used for any thing.

My main reason for wanting some sort of concrete definition is that it makes it simpler with regards to textbook questions and things like this. This primary concern arises mainly due to the fact that I’m a teacher and unfortunately doesn’t apply universally which I will explain later on.

I decided to do a little browsing and see what other people think. Generally people seem to use either height or width in conjunction with length, in their eyes, length meaning the longest side of a 2D shape. Some other people, like myself, prefer to use height and width but it seems to be a minority (based on a very little research). I don’t really like this method as it means you could use height and width interchangeably but they have completely different meanings!

Now the fact that there isn’t a general consensus really annoys me as maths, predominantly anyway, is based on a set of rules that we created. Why isn’t there a rule for this? The problem arises due to the use of language. Height has a very real meaning in a physical world, but on a piece of paper it requires extra information to determine what it is. For example you could label from left to right as height of an object if right and left were both labelled ground and sky. Maths happens to be very reliant on words despite the fact that it in itself is a language.

Within maths, I would argue that depth means front to back, the z axis, but in the real world it could also mean the depth of a pool or ocean, despite the fact that that is a vertical measurement and so could be measured as height. But referring to a pool as being 2 meters tall (or -2 meters tall) just doesn’t seem right. This leads to my next point which is the problem of theory vs practical. In theory, having width, height and depth represented in the x,y and z is perfectly fine as these dimensions are all represented, for the most part, the same way in 2 dimensions. If you take this out of the theoretical and in to the practical it becomes more confusing as your orientation with regards to the object will ultimate change what is depth or width.

Height, I’m sure we can all agree, is up and down in relation to the ground. With that out of the way we’re left with depth and width, now depending on what side of the object I am facing I could refer to the sides as both width and depth if I was so inclined. Some examples I would argue have a pre-defined depth or width, for example a wardrobe. A wardrobe has a very clear front and therefore you can use the, front to back, method for defining depth, but what if the object doesn’t have a clear front or back? Without a set way to look at the object you cannot say which side is the width and which is the depth.

To conclude I would argue that, at least in the theoretical field, we could just pick definitions with regards to the x,y and z axis. If only to make it easier for students to follow. Following this if you were studying a field that applies these areas of maths then maybe further explanation with regards to orientation to the object, or even new terminology might arise. I don’t know, that’s up to you.

Alongside this, which I’m sure won’t happen, I guess it’s really up to teachers to make sure that questions are worded with such clarity that mistakes cannot be made due to ambiguity (which is explained well here).

I certainly don’t have an answer, just my own solution to the problem, but obviously I’m hardly a mathematical scholar and have only the authority I give myself on the matter. Do you have any ideas? How do you refer to the different sides of an object?

Here are some pictures to show how confusing it could be given the variety of combinations.

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