Smart Shapes: The Geometry Behind Packaging

Every time we buy a bottle, a box of cereal or a carton of milk, we are surrounded by geometric decisions. The shape of a package is not only aesthetically pleasing, but is the result of a mathematical optimisation process that combines efficiency, strength, functionality and transport. Geometry, in this case, literally involves our consumption decisions.

Imagine a supermarket aisle: cylindrical bottles in a perfect row, rectangular packs stacked on shelves, cans with circular lids fitted together like puzzle pieces. Every package you see has been designed to maximise content, minimise material, facilitate transport and withstand use. All of this is intimately linked to the choice of a precise geometric figure.

Packaging uses geometric shapes that respond to specific needs. Here are some examples:

• Cylinders (bottles, cans): they optimise the inner volume and allow a smooth emptying of the contents. They have a curved surface that resists internal pressure well (as in carbonated soft drinks) and can be easily mass-produced.
• Rectangular prisms (boxes, bricks): are easy to stack and store space efficiently. Their cubic or parallelepiped shape is ideal for logistical transport, as there are no gaps between units.
• Tetrahedrons (Tetra Pak): minimise material usage and facilitate complete emptying of the liquid by directing the flow towards the apex when pouring.
• Spheres (less common): although not widely used due to their instability, they offer the highest volume/surface area ratio, which is efficient in terms of content, but not in storage.

These decisions are based on geometric calculations of area, volume and surface to content ratio. For example, a package that reduces its surface area while maintaining the same volume uses less material (such as plastic or cardboard), which makes it more economical and environmentally friendly.

Geometry in packaging has implications beyond design:

• Ecological: packaging with optimised geometry generates less waste and requires less energy to manufacture and transport.
• Logistical: a well-designed shape allows more products to be loaded in less space, reducing the number of trips and emissions.
• Ergonomic: certain angles or curvatures make it easier for consumers to grip and use, especially in packaging for children or the elderly.
• Commercial: shape also communicates, differentiating products through recognisable geometry (a perfume bottle, a triangular box, etc.).

Even in the design of e-commerce packages, the exact space is calculated to protect the contents without creating unnecessary emptiness. An efficient geometric decision can translate into economic savings, reduced environmental impact and a better user experience.

References:

Delgado, M., & Recio, J. (2019). Geometría y diseño industrial: formas eficientes para el embalaje. Revista Tecnología y Sociedad, 17(3), 35–48.

López, V. (2021). Packaging inteligente: forma, función y sostenibilidad. Barcelona: Editorial Diseño Industrial.

Navarro, A. (2018). Matemáticas del envasado: volumen, área y proporción. Cuadernos de Didáctica, 29, 22–29.