MOST ECONOMIC SIZE RATIOS FOR VARIOUS DESIGNS
The formulas below were calculated using partial differentiation. Naturally,
it is impossible to include bend allowances or manufacturing joints in
such formulas, as they bear no direct relationship to board area in terms
of length, width and depth.
L = length of pack.
W = width of pack.
D = depth of pack.
V = volume of pack.
C = the proportion of the width which is to cover by the outer flaps, expressed
as a fraction of the width.
Naturally, these formula will be of most help when deciding on the arrangement
of retail packs in an outer case. However, they should be kept in mind
at all times.
A size calculator is available in Boxcomp Help
MATHEMATICAL PROOF OF AN 0201's MOST ECONOMIC SIZE RATIOS BY PARTIAL
W/2 | | | | | Board area = A
| | | | | A = 2DL + 2DW + 2WL + 2W2
D | | | | |
| L | W | L | W | Volume = V = LWD
| | | | | Therefore - D = V
W/2 |_______|_______|_______|_______| LW
By substitution:- A = 2V + 2V + 2WL +2W2
By Partial differentiation:-
-- = -- + 2W
-- = -- + 2L + 4W
For max or min value of A:- dA = 0 and dA = 0
-2V + 2W = 0 -2V + 2L + 4W = 0
2W = 2V 2L + 4W = 2V
2WL2 = 2V 2LW2 + 4W3 = 2V
L2 = V 2LW2 + 4W3 = 2WL2
LW + 2W2 = L2
Since V = LWD Therefore:- L2 - LW - 2W2 = 0
L2 = LWD (L - 2W)(L + W) = 0
Therefore:- L = 2W or L = -W
L2 = LD
Therefore:- L = D
Therefore, for maximum board economy L = D = 2W