I bought this brand of hydrogen peroxide today. On the bottle are the words, “Square bottle uses less plastic than a similarly sized round bottle.”

I don’t see how that could be true. Imagine a bottle that’s perfectly rectangular, with no tapering or round corners. Assume for the sake of argument that its sides have a length of 1, and it has a height of h. The volume will be 1 X 1 X h = h, and the surface area will be 4h + 2.

Now imagine a bottle that’s perfectly circular in cross-section, and that the surface area of the bottom is 1 and the height is h. It will have the same volume as the square bottle. Its surface area will be 2 plus 2h times the square root of pi, or about 2 + 3.54h. This is less than the surface area of the square bottle.

Am I missing something? Are my calculations wrong? Is it possible to use thinner plastic when making a square bottle? Is there less manufacturing waste with a square bottle? Something else?

Maybe they include the total plastic used to ship it there as square bottles pack together with less wasted space and no need for rings to hold them together and thus need less plastic in as part of the large containers of hundreds of individual bottles?

It may even be that the bottles being able to be packed together efficiently in shipping does require less thickness of the plastic in the bottle than cylindrical bottles with smaller points of contact between each bottle. Not sure about that.

Pretty sure though that it is related to how they pack together.

Jeff Lichtman, your calculations are correct. For a container with the same area base, a cylinder is always more efficient in surface area than a rectangle, even the most efficient rectangle, a square. However, they are selling you a bit of a straw man with their statement, since they use the words “similarly sized”. What they can claim is that a round bottle is "similarly sized if it has a diameter equivalent to the diagonal of the square. Now, the square is smaller. Mind you, of course, it also holds less fluid, but then, they didn’t say it had to do that.

Most of the time, when manufacturers choose rectangles over circles (that is, rectangular prisms over cylinders), they do so for packing reasons. It can often be the case that the same volume of liquid, contained in a rectangular prism, will cost less to package and ship than if contained in a cylinder. This is because rectangles are very efficient in packing schemes, and circles are not. One of my favorite assignments as a Geometry teacher was to design a container, along with both a packing box and a shipping carton, which had to be able to palletize easily on a standard 40x48 shipping pallet. The goal was to design a useful container that minimized the cost of materials. Fun times!

Yes. A single round bottle would use less material to hold a given volume than a single square bottle, but I would expect that in packaging a given large volume of product to be packed into rectangular boxes (so as not to be shipping a bunch of empty space) that a group of square bottles would use less material.

But then there’s the fact that for most materials (at least when holding liquid), the walls need to be thicker for square containers than round ones, so some savings gets eaten back up.

More fun and more educational to design multiple sizes of cartons that could be easily packed on a standard pallet with no empty space or hanging over the edge. From memory there are five.

It is not just for maximizing volume per pallet either. When pallets are well wrapped and standing together on a truck bed with no gaps, they are pretty stable. If any part of a carton overhangs the edge, it is more likely to be damaged.

No, **DSeid **has it. Whatever spin they put on it for the shelf buyer is marketing blither, but designing container to pack together closely and without additional support materials (trays, buffers, rings, whatever) is a reduction in material use.

No idea if this applies in this case but for non-pressurized liquids a thinner wall is needed than for something like carbonated drinks. That would still favor a cylindrical shape but you can see in the newer low cost water bottles that the thinner plastic makes the bottle less sturdy, it will easily collapse with little external pressure. So perhaps, the square shape is more rigid with thinner wall plastic and for something like peroxide that is considered necessary to prevent spills and spurting, and also provides a more stable base to keep the bottle upright. That’s all just speculation, denser packing seems to be a more obvious reason.

If you examine a soft drink bottle, you will see that it is not a perfect cylinder, but has quite a few ridges and deformations whose purpose is rigidity. It must be possible to hold a bottle without “pinching” it too much. The rigidity is obtained by a careful choice of material, geometry and thickness.

Hydrogen peroxide is a relatively dangerous substance, and you don’t want squirts of it all over your clothes which will be destroyed as a consequence. Therefore, the container must be pretty rigid. As it happens, structural angles (90 degree profile) are very rigid relative to the amount of material needed (like tent stakes). If you carefully look at the body of the peroxide container, you will also see small ridges that also increase the rigidity of the structure in the Z dimension.

It seems to me very possible that a square based bottle of the same volume and rigidity will consume less material than a round one.

I just cut through a rectangular section plastic cleaning fluid bottle to see for myself.

There was a considerable variation in the thickness of the walls versus the corners. The corners were thick right in the angle but rapidly tapered out so that the walls were very thin. They bowed in slightly at rest.

Compared it to another dissected plastic bottle [dishwashing liquid] of about the same cross-sectional area. As expected, walls were uniform, but these were ?doubleish the thickness of the square section vessel.

All else being equal, I would have to say that the OP’s hydrogen peroxide bottle is telling the truth!

Am now about to leave for work on a high. I feel like an actual Mythbuster.

No, no one ever tried that cute idea. I believe, as I recall, that the smaller box had to fit 12 “cans” and the bigger shipping container had to fit 4 to 6 boxes. So there were some parameters. But they DID come up with interesting “can” solutions sometimes.

Back when I was a much more prolific poster here, I wasn’t a math teacher; I was an attorney. But then one day, walking through WalMart, I saw a rack in the aisle with souls available half-price. When I bought one, that ended my ability to be an attorney.

I went back to college to get my teaching certificate and fell into one of my old loves, mathematics. Close to ten years of misleading high school students about mathematical principles has followed, with Geometry being my special joy.

Nitpick: Per your linked article, the bottles never left the prototype stage.

Cute idea, but a product that sucks at its primary function (to hold beer), and awesome at its secondary function (to build walls), is never going to succeed.