How the Golden Ratio Optimizes Milk Carton Design for Cost Savings

Design of Milk Packaging Box

The figure shows the net content of a milk packaging box. When the volume (capacity) is fixed, how can we optimize the length, width, and height proportions to reduce packaging cost? This is an economic efficiency problem that can be solved using mathematics.

(1) Build a mathematical model: transform the real problem into a mathematical one—given a rectangular box with fixed volume, choose appropriate length, width, and height to minimize its surface area.

(2) Introduce parameters: let the volume be V, surface area be S, and the length, width, height be x, y, z respectively.

(3) Set the problem: with V fixed, find the relationship among x, y, z that makes S minimal.

(4) Analysis: because S = 2(xy + yz + zx), the minimum occurs when x = y = z, i.e., the box would be a cube. However, the actual packaging is a rectangular prism, not a cube. Why? The flattened box shape shown in the figure leads to a different optimal proportion.

From measurements, the actual box dimensions are ... (values omitted). The calculated optimal ratio is ... but the measured ratio does not match, suggesting other design considerations.

Nevertheless, the measured length‑width‑height ratios are close to the golden ratio. Both the front view and the top view of the milk carton approximate a golden rectangle, illustrating the harmonious beauty of mathematics in design.

Renowned astronomer Kepler said, “Geometry has two treasure houses: the Pythagorean theorem and the golden ratio.” The golden ratio, originating from ancient Greece, has long been associated with harmony and aesthetic appeal in human creations.

Reference: Shen Wenxuan, Yang Qingtiao, “Mathematical Modeling Attempts”.