# What comes after length, area and volume? [duplicate]

The length of the unit is 1.
The area of the unit square is 1.
The volume of the unit cube is 1.
The $\color{red}{???}$ of the unit tesseract is 1.
The $\color{blue}{???}$ of the unit 5-cube is 1.

So this is a fill in the blanks-question, but I've been wondering if there's a common term for this? I'd imagine it'd be "n-volume", but I cannot seem to find any uses of that term, so I'm guessing there is another term?

• Related (duplicate?): "What is the general term for concepts like length, area and volume?". I recall looking through old-school geometry texts where "content" was the preferred general term (with "$n$-content" the specific $n$-dimensional term), so it's what I tend to use.
– Blue
Mar 15, 2016 at 9:17
• @Blue I thought there had to be similar question asked about this, but it seem like a more general question than mine. Mar 15, 2016 at 9:26
• I've heard the term תפיסה used to describe the space time occupied by an object: volume * timeInExistence. That might translate to "holding" or "capture" in English, but note that one of the dimensions was the time dimension, not another spatial dimension. Mar 15, 2016 at 12:47
• A common theme is called measure theory. In this language length of unit is the induced Lebesgue measure from the real line and so on.
– DBS
Mar 15, 2016 at 14:09

The term "hypervolume" is in use, but that doesn't tell you the dimension. I don't see any trouble using "$4$-volume" to describe the amount of space the unit tesseract occupies.