This (123-112)/123 = 1-0.91 = 9% reduction in diameter doesn't look much for a single roll:
|123mm v 112mm, to scale overlaid for comparison|
|the difference is clear with 40 rolls|
But we could do better, surely? I've shown the rolls on a square grid, because that's how they are packed:
But what about a hexagonal packing?
|smaller rolls, and now hexagonally packed|
|square boundary hexagonal boundary equilateral triangle|
What space does this take? The hexagon is made of 6 equilateral triangles, each of height r. If the side is of length h, then we have r = h sin 60 = h √3 / 2. The area of the triangle = 1/2 x base x perpendicular height = h/2 x r = r2 / √3. The area of the hexagon is 6 times this, or 2√3 r2 = 3.464 r2. The square, meanwhile, has area 4 r2. (And not coincidentally, 3.464 is a better approximation to π than is 4.) Hence hexagonal packing is (4 - 3.464)/4 = 0.134, or a little over 13%, better than square packing.
Doing the sums to combine the percentages properly [viz, 1-(1-17%)(1-13%)=0.28], this means hexagonal packing and smaller rolls combined gives about 28% improvement over the original large, square packed rolls.
So Sainsbury's could save a further 300-odd lorry trips, and a further 90 tonnes CO2, by packing the rolls hexagonally.
Maybe I should write them a letter?
UPDATE (19 Jan 2014)
A commenter queried the correctness of the QI calculations. I just so happen to have an old and new roll size cardboard tube. Here's the difference:
|the smaller tube has a diameter of ~35mm, the larger a diameter of ~50mm; also the cardboard is much flimsier in the smaller tube|