13 Mar '08 01:371 edit

Start with a 2D square with a side length of Z. What volume can be made from its surface area? Im not asking for a specific answer, partly because after some experimenting i see that this question is not explicitly answerable in this form.

I realized there are several ways to get several volumes and it all depends on the approach one takes

My first instinct:

divide Z^2 into 9 equal areas and use 5/9th's of this area to create a five sided cube. then take the remaining 4/9th's of the area and divide those 9ths and so on and so forth.

the end result for this surface area changed to volume is

n=0

4^n(Z^3/3^3)+ 4^(n+1)(Z^3/3^6)+ 4^(n+2)(Z^3/3^9)+...........

and I was fine with that until morning when i realized Z could also = 2*Pi*r and the volume = Pi *( Z/2*Pi)^2 * (2*Pi(Z/2*Pi))

after doing some numerical testing, the values i got were a good bit different.

then in hopes of finding a volume that is closer to my right circular cylinder value i decided to use 4/9 of Z^2 to make my cube

which would give me a volume of

2(Z/3)^3 + 2(Z/9)^3 + 2(Z/27)^3 + ..........

this produced a volume that was closer to my cirrcular cylinder , but not quite?

so how is volume measured? is in in little cubes or not?

are my methods accurate? what is the true max Volume?ðŸ˜•

oh and if at all possible keep this simple so i can undestand...thanks

I realized there are several ways to get several volumes and it all depends on the approach one takes

My first instinct:

divide Z^2 into 9 equal areas and use 5/9th's of this area to create a five sided cube. then take the remaining 4/9th's of the area and divide those 9ths and so on and so forth.

the end result for this surface area changed to volume is

n=0

4^n(Z^3/3^3)+ 4^(n+1)(Z^3/3^6)+ 4^(n+2)(Z^3/3^9)+...........

and I was fine with that until morning when i realized Z could also = 2*Pi*r and the volume = Pi *( Z/2*Pi)^2 * (2*Pi(Z/2*Pi))

after doing some numerical testing, the values i got were a good bit different.

then in hopes of finding a volume that is closer to my right circular cylinder value i decided to use 4/9 of Z^2 to make my cube

which would give me a volume of

2(Z/3)^3 + 2(Z/9)^3 + 2(Z/27)^3 + ..........

this produced a volume that was closer to my cirrcular cylinder , but not quite?

so how is volume measured? is in in little cubes or not?

are my methods accurate? what is the true max Volume?ðŸ˜•

oh and if at all possible keep this simple so i can undestand...thanks