- #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 :)?
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SUMMARY
The discussion centers on the derivation and proof of volume formulas for geometric shapes, specifically cubes and cylinders. It is established that the volume of a cube is defined as the cube of its side length (x³), while the volume of a cylinder is derived from the area of a circle (πr²) multiplied by its height (h). The conversation highlights that while the definitions of these volumes are straightforward, proving them, especially for non-integer values, requires more complex mathematical techniques such as volume integrals. The participants agree that the area of a unit square is defined as 1, which simplifies calculations.
PREREQUISITES- Understanding of basic geometric shapes and their properties
- Familiarity with volume integrals and calculus concepts
- Knowledge of the definition of π and its application in geometry
- Ability to work with real numbers and their properties in mathematical proofs
- Research volume integrals and their application in deriving volumes of 3D shapes
- Study the proof of the volume formula for non-integer cubes
- Learn about Cavalieri's Principle and its implications in geometry
- Explore advanced geometric proofs involving the volume of cylinders and other solids
Students of mathematics, educators teaching geometry, and anyone interested in the proofs and derivations of volume formulas for geometric shapes.
- #2
Hootenanny
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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
)
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- #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 \pi r^2 h 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 \pi r^2 h 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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