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If you have a one dimensional object, it would just have length. If you had a two dimensional object, it would have area, and it's edges would have length. If you had a three dimensional object, it would have volume, it's faces would have area, and it's edges would have length. If I had a four dimensional object, it would have what, and it's what would have volume, and it's faces... etc.

Additionally, in English, are there prefixes and roots for going even further for these two terms. I googled this, and I wasn't able to hit the right terms to lead me to the awnser.

Grant Davis
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  • I'm pretty sure we still just call it volume or area, with the understanding (from context) that we're working in n dimensions: "n-dimensional volume". As an aside, the "dimensional analysis" tag refers to something else, I would just leave it with the terminology tag alone. – Keshav Mar 19 '18 at 03:07

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Mathematicians are apparently not as clever as you think. Rather than make up new words, they instead use "volume" to mean all of these things. Length is $1$-dimensional volume, area is two dimensional, etc. There may be a special word for four dimensional volume somewhere, but it's not in common use.

To generalize what you mean when you say "faces," you could call it the boundary.

Matt Samuel
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The notion of volume generalises to $n$-dimensions: $$V_{2k}(R)=\frac {\pi^k}{k!} R^{2k}$$ is the volume of the $2k$-ball of radius $R$. In odd dimensions it's $$V_{2k+1}(R)=\frac {2^{k+1}\cdot \pi^k}{(2k+1)!!}R^{2k+1}$$

*Note that the expressions go to zero as $n$ goes to $\infty $.

For volumes of arbitrary sets see volume element , which is used to integrate.

Furthermore the boundary $\partial M$ of an $n$-dimensional manifold $M^n$ is generally $n-1$ dimensional.