Volume of a Cube: Definition & Explanation

[McFluffy]

Suppose if we have a cube:

The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see that the above image is a cube. But how I see it is that, if you have a square of length $a$ and stack $a$ amount of "layers" above/beside/behind/whatever direction such that it is perpendicular the square, you'll form a cube. My question is that, is this a wrong way to look at volumes in general? If it is, why is it wrong and what is the correct definition? I searched the internet but I can't find any satisfying answer. I see definitions that state that volume is defined as how much space can a closed surface contain but that's just a bit vague for me.

 

[DrClaude]

Volume is basically the amount of space enclosed by a closed surface. Mathematically, I would say that it is best defined by a triple integral:

https://en.wikipedia.org/wiki/Volume#Volume_in_calculus

 

[McFluffy]

Volume is basically the amount of space enclosed by a closed surface. Mathematically, I would say that it is best defined by a triple integral:

https://en.wikipedia.org/wiki/Volume#Volume_in_calculus

Then is the way I looked at volume the wrong way?

 

[DrClaude]

Then is the way I looked at volume the wrong way?

No, it is basically a good visualization of the integration process. You can see the volume arising from summing a stack of surfaces. (you have to make each surface infinitely thin and sum and infinite number of them).

And you don't necessarily need to stack identical squares: you could start from one corner of the cube and add progressively bigger surfaces then progressively smaller surfaces as you move to the opposite corner.

 

[McFluffy]

Thank you for the answer. I was having doubts about how I defined volume and feared that this interpretation of stacking layers upon layers of the surfaces would break down once you start introducing 3d shapes that doesn't work well with it.

 

[FactChecker]

You can think of volume all those ways. You can stack them upward, sideways, forward, backward, in a circle, any way you want. You can divide a weird-shaped solid into tiny cubes, stack up each tiny cube, then add them up. All those ways should give you the same answer. Just make sure that you are dividing the entire shape into disjoint parts that are added up.

 

[Leo Authersh]

You might imagine this in this way:
Length is the total space occupied one dimensionally.
Area is the total space occupied two dimensionally.
Volume is the total space occupied three dimensionally.

Is this comprehensible? If not, feel free to ask further questions.

 

[Ssnow]

Hi, there are a lot of definitions and intuitive way to think it, for example you can think the volume in a "dynamic'' way as the result of a movement of a surface (2D dimensional) along a direction (1D dimensional). This is the Newton point of view, moving a point you have a line, moving a line a surface, and so on ...
Ssnow