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When writing out dimensions in the form height x width x depth, what should you write for the units? For example, I can think of a few ways to approach this (assuming each dimension is measured in millimetres):

  • $1\times2\times3\,\mathsf{mm}$
  • $1\times2\times3\,\mathsf{mm}^3$
  • $1\,\mathsf{mm}\times 2\,\mathsf{mm}\times 3\,\mathsf{mm}$

Is one of these more generally accepted in engineering than others?

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    $\begingroup$ The title asks what is "correct", but the body of the question asks what is "generally accepted". Three of the four answers so far comment on what's "generally accepted", and they're right (IMO regrettably so, because the "generally accepted" form in the first bullet point is dimensionally incorrect.). If you want to know what's "correct", I suspect the place to look is the ISO 80000-3 standard, but I don't have a copy to hand to check. $\endgroup$ Oct 3 '20 at 15:39
  • $\begingroup$ There may be a good corollary question here - what if each dimension is measured in different units? If its metric then conversion is accurate and simple, but what if at least one was in another system? $\endgroup$
    – Criggie
    Oct 3 '20 at 23:52
  • $\begingroup$ @DanielHatton Good point about "correct" vs "generally accepted". I suppose if there was a correct way to do it, it would also be the generally accepted way. But if there was no defined standard (i.e. no correct way), we would be forced to use whatever is generally accepted. As such, I've changed my title to better reflect what I'm asking $\endgroup$
    – JolonB
    Oct 4 '20 at 2:43
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I 'll start with the one that you should DEFINITELY NOT use: $mm^3$ or $mm3$.

Probably the most widely used is the 1st. It's compact and economical and these are two of the most deciding factors in Engineering thought and practice.

To take the point one step further, if it's in an engineering drawing you don't even need to put units in. Unless, its explicitly stated - for mechanical engineering - in countries following the metric system, a dimension number always implies mm.

So that makes the third option, explicit but after a fashion "bloated" and redundant.

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In an earlier comment, I suggested that an answer to "what is the correct way..." (as opposed to "What is the generally accepted way...") might be found in the ISO 80000-3 standard. I had a look, and there's nothing relevant in ISO 80000-3. Nor can I find any other ISO standard that explicitly states a correct way of doing this. However, ISO, when they themselves need to specify the dimensions of an object, frequently use the format in OP's third bullet point, e.g. $1\,\mathsf{mm}\times 2\,\mathsf{mm}\times 3\,\mathsf{mm}$.

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    $\begingroup$ This is correct. ISO 80000-1 uses “80 mm × 25 mm” in one example but doesn’t explain this particular problem; however, it follows from the general rule that “quantities are multiplied and divided by one another according to the rules of algebra”. By way of comparison, the German standard DIN 1338 explicitly says that, for example, a paper size of “10,5 cm × 14,8 cm” is correct and “10,5 × 14,8 cm” is wrong. $\endgroup$
    – user28919
    Oct 4 '20 at 15:03
  • $\begingroup$ Thanks @FadedGiant. The information from DIN 1338 seems to be conclusive (albeit limited in geographical scope to Germany); although the example you found in ISO 80000-1 relates to the rather different situation where the intention is to multiply the numerical quantity values of the two dimensions to obtain the numerical quantity value of an area, rather than to specify the quantity values of the two dimensions individually. $\endgroup$ Oct 4 '20 at 15:27
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The first one is the only one that makes sense...

  1. Implies cubic units cubed

  2. Is needless

So obviously 1 is the clearest.

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    $\begingroup$ Nevertheless, obviously 1 is clearly wrong. And I would reject any report that uses such nonsense. $\endgroup$
    – user28919
    Oct 4 '20 at 15:05
  • $\begingroup$ But it actually is valid, because the other options imply intermediate results that are invalid. For example, if you take (3) and divide by 1mm, you get 6mm2, which has no physical meaning in three dimensions. $\endgroup$
    – Nrenene
    Oct 5 '20 at 4:24
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1x2x3 mm is usual. You might specify individual units if they used different multiples. For example, if you had a large sheet of thin material you could describe it as 1 m x 2 m x 3 mm. However in metric engineering drawings it is common to keep everything in mm and describe this as 1000 x 2000 x 3 (with a note in the corner of the drawing stating all dimensions in mm).

Your third format can be used for mixed units. Here in the UK I can buy wood sheet material in e.g. 8 ft x 4 ft x 18 mm sizes.

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  • $\begingroup$ Just a tip. Try to not use only numbers without unit (even if it's correct). If you write say 1x2x3 without units because the common sense in mechanical is 1x2x3 mm, if the reader is from civil or a layperson they can assume it is 1x2x3 cm. This happened with me years ago, after this I put unit without "mm" only in tech drawing, in lists or other medium I put 1x2x3 mm or 1mm x 2mm x 3mm. $\endgroup$
    – Leafk
    Oct 4 '20 at 17:29
  • $\begingroup$ I think its bad form to mix units. Especially if 2 of the units are the same, this may cause problems. @Leafk yeah to most people a mm is too small of a unit. $\endgroup$
    – joojaa
    Oct 5 '20 at 5:22
  • $\begingroup$ @joojar "I think it is bad form to mix units." So do I. However in the UK once you go outside engineering circles I'm afraid it's the reality in consumer circles, e.g. DIY supplies, and this is the standard way to do it. $\endgroup$
    – Graham Nye
    Oct 6 '20 at 11:11
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Cubic millimeters (mm3) would be used when describing volume of holding capacity.

In your situation your third option is correct, but use spaces: 1 mm x 2 mm x 3 mm, or 1 mm by 2 mm by 3 mm. Each number needs to have the unit follow it because 10 mm x 2 mm x 4 mm could also be written as 1 cm x 2 mm x 4 mm.

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There are two acceptable ways to let the the reader know exactly what is the units in an expression:

  1. Through problem statement in the beginning of the calculations. It can be simply stated "the units throughout is mm", or "All length in mm".

  2. Include the units in every step as in your example 3. Also, sometimes the units follows the answer only can be acceptable, as long as the reader knows the subject/objective oftype of the operation, and the operation involves one units only. For example 1 x 2 x 3 = 6 mm^3

From the engineering point of view, both your example 1 and 2 are unacceptable. They are considered sloppy, that prone to misinterpretation, and likely to cause unnecessary mistakes. When writing math expressions, clarity is the KEY requirement.

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