- #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 :)?
In summary, the conversation discussed the derivation and proof of volume formulas for various 3D shapes. The volume of a unit cube is defined as 1, but it could have been defined as any number. The volume of an n by n cube for an integer n is n^3 and the volume of an x by x cube for any real number x is x^3, requiring a more complex proof. The volume of a right circular cylinder is \pi r^2 h, which can be easily proven once the area of a circle is given.
- #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
- 586
- 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
Homework Helper
- 42,989
- 975
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
- 586
- 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.
1. What is the formula for calculating the volume of a cube?
The formula for calculating the volume of a cube is V = s^3, where s is the length of one side of the cube.
2. How do you find the volume of a cylinder?
To find the volume of a cylinder, you can use the formula V = πr^2h, where r is the radius of the base and h is the height of the cylinder.
3. Can the volume of a cube and a cylinder be the same?
No, the volume of a cube and a cylinder cannot be the same as they have different shapes and formulas for calculating their volume. However, they can have the same numerical value if the dimensions are equal.
4. How does changing the dimensions of a cube or cylinder affect its volume?
Changing the dimensions of a cube or cylinder will directly affect its volume. For a cube, increasing the length of one side will increase the volume by the same factor. For a cylinder, increasing the radius or height will increase the volume by different factors depending on the dimensions.
5. How is the volume of a cube or cylinder used in real life?
The volume of a cube or cylinder is used in various real-life applications such as determining the amount of liquid a container can hold, calculating the capacity of a storage space, and designing structures with specific dimensions. It is also used in fields such as engineering, architecture, and chemistry.
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