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According to Wikipedia, the definition of Volume is given as "Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains."

How then, is it possible for us to find the volume of an n dimensional object, such as an n-cube (as given here) or for an n dimensional sphere (as given in this Wikipedia article)?

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    $\begingroup$ Wikipedia is defining the common, ordinary notion of volume. One can also generalize this notion to higher dimensions. $\endgroup$ – David G. Stork Sep 14 '18 at 1:06
  • $\begingroup$ @DavidG.Stork how does one think about volume in multiple dimensions? For instance, say that features used for classification are used to form a multi-dimensional matrix. I understand that we cannot visualize the matrix, but accept the fact that such a structure exists mathematically. However, what does its volume represent? $\endgroup$ – rahs Sep 14 '18 at 1:22
  • $\begingroup$ Start here: en.wikipedia.org/wiki/Hypercube $\endgroup$ – David G. Stork Sep 14 '18 at 1:23
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The Wikipedia definition of volume is based on the ambient space being ordinary $3-$space. When we move to higher dimensions we may talk of $n-$volume to emphasize the fact that we are talking about that. A four dimensional brick has a $4-$volume that is computed by multiplying the four dimensions. Other four dimensional shapes can have their $4-$volumes computed by integration, just like more complicated shapes in $3$ dimensions.

It is quite possible to speak of the area of a cube-it is the surface area. A $4-$dimensional hypercube has eight cubical faces, you can speak of the total $3-$volume of those faces. You can also speak of the total area of the square faces of the eight cubes and so on. We don't have everyday words for the content of things in higher dimensions because most people don't worry about them.

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