Carton boxes are a small cost on each pack, but we buy them in large numbers, so every square centimetre of cardboard counts. In 2010 we asked a simple question: what shape of carton holds 20 half-litre pouches using the least cardboard?
How a carton is cut
Most cartons are regular slotted cartons (RSC). The box is cut from one rectangular sheet: four side panels in a row, with flaps above and below that fold in to form the top and bottom.
If l is the length, b the breadth and h the height of the finished box, each flap is b/2 deep, so the sheet measures 2(l + b) by (h + b). Ignoring the small glue flap, the cardboard used is:
A = 2(l + b)(h + b)
That is 2b² more than the surface area of a closed box, 2(lb + bh + lh). The extra goes into the inner flaps, which add strength, and it keeps the cutting simple. It is also why a cube, the best shape for a closed box, is not the best shape here.
Finding the best shape
We want the smallest A for a fixed volume V = lbh.
- Replace the length using l = V/(bh). Halving A for simplicity:
A/2 = V/h + V/b + b² + bh - At the minimum, the slope in each direction is zero:
∂/∂h: −V/h² + b = 0 ⟹ V = bh²
∂/∂b: −V/b² + 2b + h = 0 - Put V = bh² into the second equation and multiply by b:
2b² + bh − h² = 0 ⟹ (2b − h)(b + h) = 0 ⟹ h = 2b - Then l = V/(bh) = bh²/(bh) = h. So:
l = h = 2b - Since V = 2b × b × 2b = 4b³:
b = ∛(V/4), l = h = ∛(2V), A = 18b²
The rule: the cheapest slotted carton is as long as it is tall, and half as wide.
Our carton
Twenty 500 ml pouches hold 10 litres. Pouches are soft and not perfectly regular, so we allow 25% extra space: 12.5 litres, or 12,500 cm³. The best carton for that is 29.24 cm long, 14.62 cm wide and 29.24 cm tall, using 3,847 cm² of cardboard.
For now we use one standard carton for both half-litre and one-litre pouches, because our order quantities are too small for two sizes. Here is how it compares with the best shape for the same volume:
| Box | Length | Breadth | Height | Volume | Cardboard |
|---|---|---|---|---|---|
| Our carton | 32 cm | 19 cm | 21 cm | 12,768 cm³ | 4,080 cm² |
| Best shape, same volume | 29.45 cm | 14.72 cm | 29.45 cm | 12,768 cm³ | 3,902 cm² |
The best shape uses about 4.4% less cardboard, 178 cm² on every box. It sounds small, but we use cartons in large numbers, and the saving adds up.
Try it yourself
Enter the volume you need to pack to see the best carton size.
At Cocoguru, we look closely at the cost of every item, to spend less without giving the customer less. Cutting costs is not about cutting corners; it is smart value engineering.