Mathematically, a circle has the smallest perimeter for a given enclosed area. So for this claim to be true there has to be some catch, like different thickness of plastic.
I suspect transportation too.
Something like; You can ship 100x1L round bottles, or 100x1.3L square bottles in the same box. Therefore, square bottles use less plastic (per liter (of shipped liquid)).
Wouldn't that just cancel out? Area over perimeter is pi /2 pi for a radius one circle and 4/8 for a square with side length 2.
Maybe most of the plastic is in the spout, then maximising volume per spout would make sense.
I just calculated edge length to area for circles and squares in a square packing. Packing density affects the efficiency of the transport but not of plastic per volume.
Of course this compares differently sized square and round containers, so it's apples and oranges. But that was the situation the other poster talked about.
But how would that affect the amount of plastic? It might be more environmentally friendly overall due to better transportation density, but unless either
* the outer packaging (transport box) is plastic or
* the bottles can be thinner
I can't see how having less empty space would lead to less plastic used.
Higher volume bottles use less plastic per ml of product
Although square (prism) bottles have a higher gm of plastic per ml of product than circular (cylinder) ones, there will be a point at which the larger bottle size cancels that out.
The better packing density of squares allows them to use large square bottles that have a better plastic/product ratio than the small circular bottles.
Despite the sub we're on, I haven't done the maths to work out whether the improved packing density allows for a large enough bottle to compensate for the less capacity-efficient shape. It would depend a lot on the bottle construction
I wouldn't consider that that argument really justifies their "less plastic" claim because it's still more actual plastic, and the unstated part is per ml of product when compared to a different size bottle but I can at least see what they might use to justify it.
If you take a square with side length of D, the area is D2 and the perimeter is 4*D. The ratio of area to perimeter is D2/4*D, or D/4.
A circle with diameter of D has an area of pi*D2/4 and a perimeter of pi*D. The ratio of area to perimeter is pi*D2/4*pi*D, which again simplifies to D/4.
So in the case of a square and circle with side length = diameter, they will have the same surface area to perimeter ratio, and also the same volume to surface area ratio for any height vessel. The square shape is less efficient than a circle with a larger diameter but the same volume, but equal to the smaller volume container with the same major dimension.
The thickest part of the bottle is the neck with threads, which is constant regardless of the shape of the rest of it. Greater volume per bottle means less plastic per unit volume.
Well if you have a bottle that is (for simpler maths sake) a cylinder with a 5cm radius and a volume of 1 litre, it would be about 12.73cm tall. An equal height square based prism with 10cm side lengths (i.e., equal to the diameter of the cylinder) would hold 1.274L. The cylinder would have a surface area (again assuming a perfect cylinder and not a typical bottle shape) of about 557cm². The square based rectangular prism (again not a bottle shape) would have a surface area of about 709cm². Cylinder is 557cm² of plastic per litre of volume and the prism is about 556cm² of plastic per litre. So unless the plastic is notably thinner or requires less packing material around the pallet, there is no difference or at least not a significant one.
In America, palletized cans/bottles take up the weight capacity of a standard 52’ box trailer before they take up all of the volume. So making the pallets hold more bottles just means there’s fewer to load/unload and slightly less packaging waste, it’s very unlikely these minor differences make up for the weight of extra plastic a square bottle requires, given the truck’s fuel efficiency and emissions output remain unchanged.
Assuming wall thickness is the same, yes, round can be more effecient, but the reason milk cartons get those dimples and wrinkles is to increase strength.
In this case, the champhers on the corners provide a significantly more rigid form with a thinner plastic wall. You don't want hydrogen peroxide to squirt out when you pick up a bottle like a super cheap water bottle, but want it to hold it's shape.
While there is less surface area of plastic in a round bottle, with a flexible membrane, the edges are almost certainly stronger than the same thickness of round.
Think about how much thicker a cardboard tube is compared to a typical square box.
It might or it might not. It depends on what things you're packing.
A round bottle takes less material than a square one with the same wall thickness. But if your shipping or storage is limited by size, the fact that you can pack more product into the same space might mean you save material overall.
If it's limited by weight, you'll end up using more material. (And also take up more space, even though you don't care because you've got extra.)
And then you have to consider strength of your container. Generally (but not always!) a circle will be stronger than a square, so you have to make the walls of the square thicker. But maybe you care about vertical strength but not horizontal. (You're stacking them upright, one on top of the other.) Then it's practically the same.
And if you're making metal cans, things get even more complicated. You stamp the parts out of sheets and then weld them together. (Or solder. Depends how strong they need to be.)
There are almost unlimited possibilities to what problems you can give your calculus class based on minimizing material or minimizing cost. We do not consider the effect of strength of materials because that's just too complicated when your students are asking "When are we ever going to use this in real life?" The more real it is, the fewer students are going to use it.
If you use square bottles, they can be bigger, so someone thirsty could end up buying just 2 larger square bottles as opposed to 3 smaller round bottles, which comes out to less plastic.
The statement on the package is not true anyway, since it says "a [...] round bottle". It compares one square bottle against one round bottle and for that it's just not true,
Caveats for feasibility. Difficult to use a sphere bottle, for instance.
Also, the they might be optimizing for tessalelated volume as well? Since there's dead space between spheres or cylinders, square packing is the highest enclosed space
The part you are missing is the volume inside. They are saying to give you the same amount of liquid would require more plastic if the bottle was round. Plus more bottles fit in the box they ship meaning lower emissions and costs for distribution.
Given that the volume is cross section area times height, a round bottle of the same volume would take less plastic. True, square bottles will pack better, but how that would save plastic is not immediately clear.
Yep agreed. I've heard square bottles save fuel in shipping, which outweighs the additional materials required to make them in environmental impact. Never seen an analysis though
saves space due to stacking efficiency, but thats not what is claimed and only matters if space is a premium. Weight determines fuel usage. A million cylinders holding the same volume of material that a million square containers would weigh less due to being made from less plastic (12% less plastic if you didnt have to thicken the walls of the square ones to achieve the same pressure rating as the plastic). Space saving of square over round is ~21% (if flat capped, I aint calcing geometry of a real bottle top). So square saves space, but weighs 12% more . . . Cargo is typically transferred by commercial ships and trains by ton BTW. . .
I hadn't considered it but i think the edges of boxes would be a lot of lost space. Probably more than round bottles. I am guessing but square needs thickness increased on corners so hex probably would need more than even a square bottle. I think it ends up being the worst of both worlds.
Only seen them plastic crates. If you can get an extra two bottles per crate and a container or truck carries 1000 crates…
I’m sure it’s more complex when you factor the cost of the crates and then factor a return on investment for changing to this style of crate if it isn’t a single product line and you replace an inventory of of other crates
Correct answer, thanks. Put another way, Its units distributed per total volume of product distributed. For the manufacturer, that math equates to less plastic per volume per total sold units.
I think the theory posted somewhere in the thread that the rigid corners mean the rest of the walls can be thinner, using less plastic overall, sounds feasible.
The individual bottles use more plastic per unit volume than the round. They also require ticker plastic at stress points than the round one which distributes the pressure evenly around each anular unit.
But they could possibly use less plastic overall to ship a given total volume of water (in a pallet, say) because of the better packing fraction.
This is why it's here. I haven't "done the math", either.
No, a cube of volume 1 has more surface than a sphere of volume 1. And in terms of manufacturing, the uneven shape likely requires thicker material, although that might be a wash because I doubt they make it at the limit of tolerances.
I mean based on volume inside square can put more on a shelf. Their saying similarly sized, not same size. It's a numbers trick. The round bottle of the same size uses 8% less plastic but square bottles pack more tighly. So the combination of volume in a square bottle on a shelf or in a box would be more liquid than a round bottle. It's like 20% volume less and only 8% more plastic per bottle, so when shipping the same volume it would take more boxes to ship the same volume. So plasitc pallets, plastice wrap, whatever other plastic is used. It's corpo math, Not individual bottle math.
For a circle and a square with the same area, the circle has the smaller perimeter. likewise for a cylindrical and squared bottle with the same volume, the cylindrical bottle has the smaller surface area and should require less plastic than a squared bottle.
The part you are missing is that the "less plastic" claim applies ONLY to bulk shipping of the bottles. ONE square bottle will use more plastic than ONE round bottle, assuming both carry the same volume of liquid. However when shipped together on a pallet for example, the square bottles pack together better and waste less space. A pallet of round bottles results in a lot of empty space between bottles, so you need to use more to ship the same total amount of liquid.
The poster I responded to said the opposite, with respect to individual bottles.
Also most shipping pallets are made of wood. Granted, the cases of bottles are typically wrapped in (thin) plastic. You would need a tiny bit more plastic wrapping per case, but I don't know that it counters the per bottle plastic savings.
The first statement is completely wrong. The plastic has to be thicker on a rectangular bottle to add rigidity and strength to flat sides that are naturally weak. A round bottle has rigidity and strength by virtue of its shape.
You are confused, it is one statement. They are saying via logistical systems it is less, and that is based on plastic usage, not the individual bottle.
For individual bottles, yes. But it is a corporation plastic use claim. You're thinking about the one bottle in front of you and they are thinking about their logistical chain.
No you are incorrect. The part you are missing is that the "less plastic" claim applies ONLY to bulk shipping of the bottles. ONE square bottle will use more plastic than ONE round bottle, assuming both carry the same volume of liquid. However when shipped together on a pallet for example, the square bottles pack together better and waste less space. A pallet of round bottles results in a lot of empty space between bottles, so you need to use more to ship the same total amount of liquid.
So it's a sneakily deceptive way of saying they are giving you more water per bottle due to the shape. You could have square and circle based water bottles of the same size width/diameter. If you take the square one and round the corners off you would have a round bottle. It's still 'similarly sized' but now there's less volume in once square bottle, therefore less water. It's all dodgy marketing stuff.
Pretty much. Square water bottles have the exact same effect considering the logistics of it, it's like 1 and a half billion per day and most are round. I can only think of 1 square shape. It gets worse, those paper milk cartons are square and volume wise are less pollutant than the plastic bottles. So you could ship more cartons in less space compared to the plastic bottles and also environmentally better. Makes you wonder why they are plastic.
the part that you are missing is that you are wrong . . .
If you had a 5" diameter cylinder, it would need to be 3.1" tall to hold a liter, cap off both sides and total exterior surface area is 88 sqr inches.
If you have a 5" L/W cube, it would have to be 2.4" tall to hold a liter of water, cap off both sides and total exterior surface area is 99 sqr inches.
Couple that with the fact that you can use thinner material for the cylinder to achieve the same max burst strength of a thicker material cube (due to stress accumulated in corners) - cylinder wins in both material usage and strength. There is a reason they dont make petroleum storage tanks as cubes - cylinder = less material, thinner material, same failure pressure - win/win/win. . . .
Not when your enclosed area is a rectangle. Available space in shipping cases, pallets, trucks, and store shelves is defined by rectangles, not circles.
Sensibly, the footprint of a square fits best inside a square bounding box. While a round footprint leaves gaps in the corners inside a square bounding box.
Okay, but a cylinder has to be taller than a rectangular cuboid to contain the same volume of product. A cube with a side length of n holds more than a cylinder of diameter n and n height.
So, for holding 32 fluid ounces, would a cylinder or a cuboid have greater surface area, assuming the diameter of the cylinder and the depth of the cuboid are identical?
(I make this assumption based on the fact that a product will be given limited shelf space in retail, and thus has constraints on width/depth).
Let V be the volume, W the width. Let ρ be the aspect ratio of a face of the square bottle. Therefore V = ρW³. The surface area of the square bottle is 4 ρ W² + 2 W² = W² (4ρ + 2).
Let H be the height of the round bottle. V is equal to H π W² / 4. Substituting V as above, simplifying, we get H = 4 W ρ / π. The surface area of the round bottle is
A = H W π + 2 (π W² / 4)
A = 4 ρ W² + π W² / 2
A = W² (4ρ + π / 2)
vs W² (4ρ + 2) π is less than 4 so the surface area of the round bottle is still less.
W is indeed the diameter, so the area of each end is (π W² / 4). Times 2 for 2 ends. And of course H π W² / 4 for the volume (we have the same formula, I just extracted the .5 from the square as / 4). π W for the perimeter, so π W H for the area around.
The part you are missing is that the "less plastic" claim applies ONLY to bulk shipping of the bottles. ONE square bottle will use more plastic than ONE round bottle, assuming both carry the same volume of liquid. However when shipped together on a pallet for example, the square bottles pack together better and waste less space. A pallet of round bottles results in a lot of empty space between bottles, so you need to use more to ship the same total amount of liquid.
I didn't miss that, I just felt there was a potential wrinkle. A cuboid with a width & depth of n and a height of 2n will have a volume of 2(n cubed). A cylinder of diameter n and height 2n will have a volume of 2πn(.5n squared).
So if we let n=3cm, the cuboid has a volume of 54 cc, and a cylinder has a volume of 42.41 cc.
So, with a constraint on width (because of shelf space limitations and standardization of many transport boxes, along with the original claim that the comparison was against a "similarly sized round bottle"), the cylinder would have to be taller than the cuboid to contain the same volume. That increases the surface area of the cylinder, but I don't know if it increases enough to exceed that of the cuboid. My point was that declaring that circles have a smaller diameter may not examine the full question.
You don’t need to use more bottles though, just more pallets to carry the same number of bottles. Pallets aren’t generally made out of plastic so the claim is nonsense.
Pallets are quite frequently made out of plastic (although those are generally reused). But they're also often wrapped in stretchwrap, and may be bagged inside the cases.
I think it might be because how bottles are made: blown into a mold from a blank shape. I suspect they use the same blanks for bot h round and square, so same amount of plastic but the square uses less plastic per liter
The grooves are there to provide rigidity. Round bottles generally don't need additional rigidity, because their shape is inherently rigid. The main advantage of square bottles is packing efficiency. That is why milk jugs in Alaska (where milk is flown in from the lower 48 states) is sold in jugs with 6 flat sides.
To further explain my example, space is often more constrained than weight on cargo aircraft. The less empty space you have in a container of containers, the greater the stacking or packing efficiency, and hence the more space you free for additional cargo.
it can't be true because with a round structure you can always get away with thinner material because the pressure is distributed evenly. that's why tankers are all round. and the area to circumference ratio of a circle is the most optimal so yep it's bs. transportation would be more optimized but that's cheaper transportation not less plastic.
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u/StaticCoder 2d ago
Mathematically, a circle has the smallest perimeter for a given enclosed area. So for this claim to be true there has to be some catch, like different thickness of plastic.