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 surface 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 is the original surface area and is the new surface area. where is the original volume, is the new volume, is the original length and is the new length. For example, a cube with a side length of 1 meter has a surface area of 6 m2 and a volume of 1 m3. If the sides of the cube were multiplied by 2, its surface area would be multiplied by the square of 2 and become 24 m2. Its volume would be multiplied by the cube of 2 and become 8 m3. The original cube (1 m sides) has a surface area to volume ratio of 6:1. The larger (2 m sides) cube has a surface area to volume ratio of (24/8) 3:1. As the dimensions increase, the volume will continue to grow faster than the surface area. Thus the square–cube law. This principle applies to all solids. When a physical object maintains the same density and is scaled up, its volume and mass are increased by the cube of the multiplier while its surface area increases only by the square of the same multiplier.