Area and volume?

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?😕

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

Originally posted by joe shmo
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 ...[text shortened]... e true max Volume?😕

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

Originally posted by serigado
Are you asking which form should a volume have to get the minimal surface area to volume ratio?
That would be a sphere (I think)
As for your little squares, wouldn't you need 6/9 of the little squares to make a closed surface?

Originally posted by joe shmo
Ok so the spherical surface area brings the most volume....I had another person mention that to me when discussing the problem

As for the six surface area's need to make a cube, not sure

the way i look at it there has to be a minimum of 4/9 and a maximum of 6/9

that is what is strange to me for defining volume.....how can 3 different surface areas ...[text shortened]... cube

this all is strange to me....do not all of the gemotrical objects have the same volume?

Originally posted by Aetherael
lastly, as others have said before, the optimal shape for the greatest space enclosed within a minimal surface area (so, for instance, given 100 sq. feet of wrapping paper what's the biggest volume i could wrap up?) is indeed a sphere. 🙂

Originally posted by Aetherael
try to think of volume's relationship to surface area in 3D as similar to the relationship between area and perimeter in 2D

you can clearly make closed geometric objects with different values for area that have the same perimeter. for example, take a 4x4 square (perimeter 16, area 16) and a 2x6 rectangle (perimeter 16, area 12). this includes non-po ...[text shortened]... sq. feet of wrapping paper what's the biggest volume i could wrap up?) is indeed a sphere. 🙂

Originally posted by joe shmo
That is a very good question..

It seemed like 4 sides for my cube was the minimum, simply because I had never used a 3 sided cube in any application..there are open boxes and closed boxes and boxes without tops, so thats why i drew the line at 4

altough now I agree, what is really stopping me from taking another side off? its kind of a cool question, ...[text shortened]... e max volume it could contain with the least suface area is

V = 4/3*Pi*(Z/2*sqrt(Pi))^3

??