- #1
ehj
- 79
- 0
I was wondering if the formulas for the volume of, for instance, a cube and a cylinder are definitions or if they can be proved. Does anybody know :)?
The volume of a cube and a cylinder.
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Discussion Overview
The discussion revolves around the nature of the formulas for the volume of a cube and a cylinder, specifically whether these formulas are definitions or can be derived through proofs. The scope includes theoretical aspects of geometry and mathematical reasoning related to volume calculations.
Discussion Character
- Exploratory, Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- Some participants question whether the volume formulas for a cube and a cylinder are definitions or if they can be proved.
- One participant suggests that volumes of various 3D shapes can be derived using volume integrals with appropriate limits.
- Another participant mentions that the proof for the volume formulas is based on the volume of a cylinder, although this claim is challenged by another participant.
- A participant proposes that the area of a unit square is defined as 1, but speculates that it could have been defined as any other number, indicating a belief that definitions simplify calculations.
- It is noted that the volume of a "unit" cube is defined as 1, but proving the volume of an n by n cube requires a proof, and proving the volume for non-integer dimensions is more complex.
- A request for elaboration on the concept of "x by x cubes" where x is not an integer is made, indicating interest in further exploration of the topic.
Areas of Agreement / Disagreement
Participants express differing views on whether the volume formulas are definitions or can be derived, with some supporting the idea of definitions while others emphasize the need for proofs. The discussion remains unresolved with multiple competing perspectives.
Contextual Notes
Participants reference various methods for deriving volume formulas, including integrals and the principle of Cavalieri, but do not reach a consensus on the foundational nature of the volume definitions.
Who May Find This Useful
This discussion may be of interest to those studying geometry, mathematical proofs, or anyone curious about the foundational aspects of volume calculations in mathematics.
- #2
Hootenanny
Staff Emeritus
Science Advisor
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One can derive the volumes of various 3D shapes using volume integrals with appropriate limits.ehj said:
- #3
ehj
- 79
- 0
The proof for the formula to derive volumes of those various 3D shapes is based on the volume of a cylinder, atleast the one I learned..
- #4
Pere Callahan
- 582
- 1
The volume of a cube based on the volume of cylinder?? Are you sure it wasn't the other way round?
Anyway, as Hottenanny pointed out you can just calculate these kind of volume using integrals or (essentially the same) http://fr.wikipedia.org/wiki/Principe_de_Cavalieri" (sorry, only available in French and German
)
Anyway, as Hottenanny pointed out you can just calculate these kind of volume using integrals or (essentially the same) http://fr.wikipedia.org/wiki/Principe_de_Cavalieri" (sorry, only available in French and German
)
Last edited by a moderator:
- #5
esbo
- 86
- 0
For simplicity consider a unit square.
We define the area of a unit square as 1. (at least I think we do!)
Seems to say so here:-
http://mathforum.org/library/drmath/view/60392.html
I think however we could have defined the area of a unit square a 7 or 12.738 or 1/4
or even -0.0009300203.
It would just make the maths a bit harder it we did!
So it seems to me it is a definition so proving it is trivial, for example an exam question
might be:-
a) Given the area of a unit square is 1, show the area of a unit square is 1.
I don't think that will be worth too many marks!
Going on to volume, I think we define a unit volune as one, but it could have been
defined as any number, one just makes the numbers easier to work with.
We define the area of a unit square as 1. (at least I think we do!)
Seems to say so here:-
http://mathforum.org/library/drmath/view/60392.html
I think however we could have defined the area of a unit square a 7 or 12.738 or 1/4
or even -0.0009300203.
It would just make the maths a bit harder it we did!
So it seems to me it is a definition so proving it is trivial, for example an exam question
might be:-
a) Given the area of a unit square is 1, show the area of a unit square is 1.
I don't think that will be worth too many marks!
Going on to volume, I think we define a unit volune as one, but it could have been
defined as any number, one just makes the numbers easier to work with.
Last edited:
- #6
HallsofIvy
Science Advisor
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Yes, a volume of "1" is defined as the volume of a "unit" cube- that is, the volume of a cube of length "1" on each side.
The fact that the volume of an n by n cube, for n an integer, is n3 does require a proof and the fact that the volume of an x by x cube, with x any real number, is x3 requires a significantly harder proof.
Once we are "given" the area of a circle, the proof that the volume of a right circular cylinder is [itex]\pi r^2 h[/itex] is fairly simple.
The fact that the volume of an n by n cube, for n an integer, is n3 does require a proof and the fact that the volume of an x by x cube, with x any real number, is x3 requires a significantly harder proof.
Once we are "given" the area of a circle, the proof that the volume of a right circular cylinder is [itex]\pi r^2 h[/itex] is fairly simple.
- #7
Pere Callahan
- 582
- 1
Would you mind elaborating on "x by x cubes" where x is not an integer ...? Or do you know a link/book where I could read about such things?
Thanks.
Thanks.
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