Square-cube law

The square-cube law (or cube-square law) is a mathematical principle, applied in a variety of scientific fields, which describes the relationship between the volume and the area as a shape's size increases or decreases. It was first described in 1638 by Galileo Galilei in his Two New Sciences as the "...ratio of two volumes is greater than the ratio of their surfaces."

This principle states that, as a shape grows in size, its volume grows faster than its surface area. When applied to the real world this principle has many implications which are important in fields ranging from mechanical engineering to biomechanics. It helps explain phenomena including why large mammals like elephants have a harder time cooling themselves than small ones like mice, and why building taller and taller skyscrapers is increasingly difficult.

The square-cube law can be stated as follows:

When an object undergoes a proportional increase in size, its new surface area is proportional to the square of the multiplier and its new volume is proportional to the cube of the multiplier.

Represented mathematically:

where ${\displaystyle A_{1}}$ is the original surface area and ${\displaystyle A_{2}}$ is the new surface area.

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