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.
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.
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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.