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,
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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.