Volume of cylindrical vs. cube-shaped containers

Factual Questions
1

Way back in school, I had an algebra teacher that also taught physics and a couple of other “smart kid” courses. Occasionally, he would let us in on some of the more interesting topics his other classes were covering.

Two of the examples I remember him showing seemed odd. I didn’t voice my concerns at the time since it was a first period class, and I wasn’t completely awake.

He told us that the reason cans of soda, vegetables, and everything else in the store was cylindrical rather than cube-shaped, was that the cylinders held more liquid. This didn’t make sense to me. It’s probably easier to draw a picture than explain, so I whipped this up in a paint program (and compressed it before posting; it’s only 2kb)

http://internettrash.com/users/puffweed@hotmail.com/circlesquare.jpg

If the cylindrical objects are stacked or lined up, there is a large amount of empty space, represented by the red color in the graphic.

Well, if you made square containers, that space wouldn’t be empty anymore, would it? You’re using all of the available space, so there must be more liquid contained in a shelf full of square containers. Is there something I’m missing here?

Question #2 : In another demonstration, which was meant for his physics class, he showed us a little wind-up toy train. In the part of the engine where the smoke would come out in a real steam engine, he had rigged a little device that propelled a ball straight up in the air. The ball was supposed to fly straight up (while the train was moving) and land back in the little hole where it came from. He couldn’t get it to work, though, which he blamed on an error in the angle of the trajectory.

Even if the ball did go straight up, though, I don’t get this one. Once the ball is shot upwards, there is nothing propelling it forward other than the momentum of the train, which should be decreasing anyway, since the ball is no longer on the train.

The train, on the other hand, is still being propelled forward at the same rate of speed it had been travelling the whole time. So, shouldn’t the ball’s forward momentum decrease enough in the time that it is in the air that it wouldn’t land in the same little hole it had been shot from?

2

Actually, that image is only 773 bytes. Correcting myself before someone else does :slight_smile:

3

I agree on the bottle question…

Squares pack together more tightly than circles, ditto cubes and spheres…

Triangles/pyramids would work as well…

Dunno if there are any objects with more than six identical sides which fill space perfectly, like that. An eight-sided object might, as might a twelve-sided, if you ignore edge conditions. I don’t feel like digging out my AD&D dice to check.

I suspect bottle shape is dictated much more by tradition and convenience than space… Weight would probably be more important while shipping, as a truck full of water, which soda pretty well is, would have to be sturdier than you usual coca-cola truck.

On the subject of the train…

Without atmosphere, the ball wouldn’t lose forward speed, and would land, if shot straight up, right where it came from.

When you figure in air resistance, you need to shoot the ball forward to get it to work. When the train turns, this would be a problem, as when it was accelerating, because the angle the ball was shot at would need to depend on the train speed.

4

Re octagons: I dimly recall that hexagons are the figures with the largest number of faces that you could tessellate with no gaps: cf. a honeycomb.

5

Oh! I just realized what your teacher was getting at.

Circles (and by extension, cylinders) have the largest possible volume per unit perimeter of all figures. So say you have a circle with a perimeter of 10. Its volume is (((10/pi)/2)^2)pi, or about 7,9577.

A square with the same perimeter - that is, made out of the same amount of stuff - has a volume of (10/4)^2, or 6,25.

So your teacher was right, and either you misremembered the question (can’t blame you there - math class isn’t the most entertaining one in the world) or he misposed it.

6

My bad. Where I said “volume”, I meant of course “area”. Volume would be the area multiplied by the length of the can.

7

A sphere has the greatest volume to surface ration (ie it would contain the most liquid for the least amount of aluminum) however spherical cans would be impractical to make and use. Assuming you want a can with a flat top and bottom (ie a prism) a cylinder has the highest volume to surface ration.

As for packing a number of cans together, triangular, square, or hexagonal cans can be designed to fit completely together. All other cans will have some space between them. (I’m assuming the can cross sections are regular polygons.)

The pattern mentioned in the OP is not the tightest packing you can do with cylinders. Take a penny and arrange six other pennies around it. This will show you how you can pack more cylinders in a given volume by offsetting them. There will however still be some unused space.

8

Interesting corollary – how come soda cans are round while larger beverage containers such as orange juice actually are square?

I’m guessing that because OJ packages are paper, it’s actually easier to manufacture them square, and the larger size means that efficiency of packing is at a premium.

Making a square aluminum can is not impossible, but it’s probably a lot more timeconsuming than just wrapping aluminum sheet around a cylindrical mold and soldering it (or whatever they do).

9

Just to finish off the little limit problem…

The greatest volume/area ratio for a cylindrical can occurs when the can has a “square” profile; that is, where the diameter is equal to the height.

Most cans are not this shape for marketing reasons: given a normal soda can and a “square” profile can of the same volume, most people will say the tall skinny can looks like it contains more. They’re also slightly easier to hold, which is another consideration.

Your teacher was correct that the ball should have gone back down into the stack. Reasons for it not to do so would be that it didn’t go quite straight up to begin with, or (particularly if the ball was light, like a cork ball or ping-pong ball) air resistance slowed it down too much. The forward momentum of the ball wouldn’t change in the absence of an outside force (such as air resistance), so from the point of view of an observer on the train it should go straight up and come straight back down (to an observer in the room it would follow a parabolic trajectory).

10

While the various reponses to volume, area, etc., were accurate, I doubt that they had anything to do with the shape picked to convey liquids from place to place.

If you wanted to make a leak-proof container out of cheap metal (tin in the previous centuries), you would generally find that having a single “side” (circular) that only needed to be joined in a flat interface at the ends is the simplest shape. (Spheres may have a better surface/volume ratio, but they don’t stack well and they are truly miserable to manufacture by beating or rolling.) Containers (of all sorts) were made by the simple process of rolling a sheet of metal into a cylinder, then placing one or two ends on it. (A single end for household cups, watering cans, etc.)

On the other hand, packages where leaking liquids were not a problem could be made of paper (cardboard). Simply fold flat pieces together into a six-sided shape with rectangular surfaces. They stack closely together with no spaces between them. (Hexagonal bases give better volume/area ratios, but are harder to manufacture and lose their efficiency when placed on our rectangular shelves.)

Once the process of blowing glass into molds was made economical, glass containers began to be made in a wide variety of shapes. Since glass is rather more expensive than cardboard or tin, glass (in all its different shapes) was used to store more valuable products that did not need to be packed closely to justify being stacked. Square-bottomed milk bottles may be held up as a counter example, but milk in bottles was not generally stored for long-term storage or distant shipping and qualifies as a more expensive item.

Today we continue the same shapes (for some of the same reasons). With the introduction of plastics, some of the old rules (regarding cost and structural integrity) are not necessarily true for that medium.

Tom~

11

Another advantage of round containers is better resistance to internal pressure. Square or hexagonal containers bulge on the flat faces (take a look at a large milk carton), and have a lot of strain on the corners.

Bob the Random Expert
“If we don’t have the answer, we’ll make one up.”

12

torq, I like the theory about the “square profile” cylinder, but it doesn’t make sense to me. If you double the height of a cylinder, you double its volume. You also double the surface area of its sides, while the top and bottom remain constant. That implies that the taller a cylinder is, the larger the volume to surface area ratio. It would rise to an asymptote of (r^2)/2r I think.

To put it in less abstract terms, two coke cans stacked on top of each other contain twice as much coke, but have slightly lest than twice as much aluminum, that is if you assume you’ve sliced off the top of one and the bottom of the other. Thus your coke to aluminum ratio has gone up. The same would be true by continuing to double the stack to four cans and so on.

I betting there is something special about the square profile cylinder, although probably that means the perimeter of the top and bottom matches the height, not the diameter. It must maximize something, but offhand I’m not sure what.

13

Greg, I just worked through this again and I’m positive that for a given volume the minimum surface area solution for a cylindrical container is where the height is equal to the diameter.

I’m not sure if I can explain what’s wrong with your counter-example of the double-tall can, but if you think about scaling it down to the same height as an ordinary can you’ll see that the volume decreases to one-fourth that of the normal can, but the surface area only decreases to somewhere between one-half and one-fourth that of the normal can (depending on exactly what the height/width ratio is to start with).

14

I did a math test over in the testing forum, and evidently it’s possible to do this. So here goes:

<pre>
V = pi r[sup]2[/sup]h
so
h = V / pi r[sup]2[/sup]

A = pi r[sup]2[/sup] (end) + 2 pi rh (side) + pi r[sup]2[/sup] (other end)
or
A = 2 pi ( r[sup]2[/sup] + rh )

Substituting for h,

A = 2 pi ( r[sup]2[/sup] + V / pi r )

Therefore

dA/dr = 2 pi ( 2r - V / pi r[sup]2[/sup] )
or
dA/dr = 2 ( 2 pi r[sup]3[/sup] - V ) / r[sup]2[/sup]

The minimum occurs when dA/dr equals zero, or when

V = 2 pi r[sup]3[/sup]

but

V = pi r[sup]2[/sup]h
</pre>

so h = 2r is the minimum surface area for a given volume V.