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I have in mind that volume is the amount of room or space a 3-d object takes up - its "outsideness" and that capacity with the amount of room or space a 3-d object can hold. Then I start thinking about that capacity piece and think terms like "volume by weight" and start getting a bit boggled. THEN I start thinking of liquids and then I get more befuddled. Just looking for a good, quick, succinct way to explain this to my 5th graders. Perhaps the concepts ARE indeed situational specific? Perhaps I'm making too much of this?

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    $\begingroup$ "volume by weight"? I think that would confuse nearly anyone. $\endgroup$ – Jessica B Feb 17 '15 at 7:48
  • $\begingroup$ Does this BBC Skillswise page help? $\endgroup$ – Benjamin Dickman Feb 17 '15 at 9:00
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    $\begingroup$ Can you clarify what you mean by the terms "capacity" and "volume"? If you mean the same thing that @BenjaminDickman's page is referring to, then "capacity" simply means "the maximum volume that a container can hold". In other words, "capacity" is just the name of a specific volume--the volume that completely fills a container. $\endgroup$ – msouth Feb 17 '15 at 20:03
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    $\begingroup$ While I think that one can make the (arbitrary) distinction between capacity as the possible volume a container can hold (and perhaps using $l$ vs. $m^3$ to differentiate), I'd avoid all of this and say volume = capacity. $\endgroup$ – Jasper Feb 17 '15 at 21:16
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    $\begingroup$ I'm surprised no one has mentioned this yet, but in mathematical discussions one needs to be careful about using the word "capacity" because it has long been used in a technical sense in potential theory (an abstract off-shoot of things relating to Fourier series and potential as used in physics). That said, probably no one here was confused as to whether you meant capacity notions in potential theory! $\endgroup$ – Dave L Renfro Feb 17 '15 at 22:30
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The situation does indeed change your interpretation of the terminology.

Volume is a general concept of the amount of 3D space something takes up.

For a solid object like a brick, or for a liquid there is no ambiguity and you can simply do a calculation (like $V = \pi r^2 h$ for a solid cylinder), or possibly compare its mass to the mass of a known volume (eg $1$ cm$^3$ weighs $2$ g, so $50$ g must be $25$ cm$^3$).

However, for an object with a hollow or hole in it, there is some ambiguity as to which part of the object you are referring to when you say "the volume".

For example, with a coffee cup, you could interpret its volume as the amount of space it takes up including the space inside it, or as the amount of space just the porcelain takes up. In that case you should always specify which you mean (for example, "the volume of porcelain", or "the volume of the cup including the space inside it"). Interestingly, you couldn't find the volume of the cup including the space inside it simply by weighing it, because the space wouldn't contribute anything to the weight. You'd either have to use a formula, or find a way to find the volume of porcelain and space separately and combine them.

If what you want is "the volume of the space inside the cup", then there is a special word for this and it is "capacity". This word allows us to not have a whole sentence to specify the part we want the volume of, but it is still a volume.

So in short, volume is a measure of 3D space, but if an object isn't the same throughout you have to specify which bits you want; and capacity is a kind of volume, but it refers specifically to the space inside something.

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Volume counts the amount of cubes in a shape and can be measured in $cm^3$

Capacity measures the amount of liquid filling the cubes in units of $ml$ for example.

The connection is that $1cm^3=1ml$

If you had a $cm^3$ cup it could be refilled about $330$ times from a can of coke.

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    $\begingroup$ I've never encountered this idea of volume=cm^3, capacity=ml, and I'm skeptical that it is widely recognized. $\endgroup$ – Ben Crowell Feb 17 '15 at 20:12
  • $\begingroup$ Volume counts cubes and can be measured in $cm^3$ capacity can be seen as the amount of liquid the filling cubes. Capacity can be measured in ml. I'm not sure what is wrong with my answer. $\endgroup$ – Karl Feb 17 '15 at 20:41
  • $\begingroup$ Edited answer to include cubes and the liquid filling them. $\endgroup$ – Karl Feb 17 '15 at 20:58
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    $\begingroup$ i agree that I've understood capacity to indicate the liquid that can be held in a given cavity. $\endgroup$ – JoeTaxpayer Feb 18 '15 at 3:15
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    $\begingroup$ @BenCrowell I have encountered this idea before in a maths textbook. Whether it's widely recognized or not I don't know. This article <mathforum.org/library/drmath/view/60595.html> also acknowledges Karl's view before advocating a view which is kind of similar to DavidButlerUofA's. $\endgroup$ – David Ebert Feb 18 '15 at 8:16

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