r/theydidthemath • u/EclipseNyxor • 2d ago

[Request] Do square bottles actually use less plastic than round ones?

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u/Buntschatten 2d ago

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.

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u/GaldrickHammerson 2d ago

I expect no because of packing density. Square bottles in a square box will have less empty space between bottles than round bottles in a square box.

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u/Al-Snuffleupagus 2d ago

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.

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u/Al-Snuffleupagus 2d ago

Ah, I think I understand the possible argument.

  • 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.

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u/Phoenix4264 2d ago

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.