The cube-square law is the mathematical principle that describes how volume increases faster than surface area. This principle has implications across many fields and helps explain why many of the difficulties that come with a larger sized animals, buildings, microorganisms, and many others.

Volume scales with length faster than surface area does.

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

\[ \begin{equation} A_{2} = A_{1}(\frac{L_{2}}{L_{1}})^2 \end{equation} \] \[ \begin{equation} V_{2} = V_{1}(\frac{L_{2}}{L_{1}})^3 \end{equation} \] - A1 is the original surface area and A2 is the new surface area. - V1 is the original volume and V2 is the new volume. - L1 is the original length and L2 is the new length.

As the size of something increases, there is a higher ratio of volume to surface area. This can cause several issues for larger objects or organisms. Elephants have difficulties staying cool because they have so much volume, and heat is dissipated with surface area. Cells can only become so large because the cell wall (surface area) can only supply nutrients for a certian amount of volume.

ratio = volume/area
df <- data.frame(length, ratio)
ggplot(data = df, aes(x=length, y=ratio)) + geom_line() + labs(x = "Length", y = "Ratio of Volume to Surface Area")

## Advantages of Large Creatures and Objects

There are some advantages that come with increasing size and addition volume compared to surface area. Large planes have less air resistance per volume compared to smaller planes.

Large objects have to support more weight with their support system. Elephants have more pressure per amount of bone than mice do.

George Johnson (January 12, 1999). “Of Mice and Elephants: A Matter of Scale”. The New York Times. https://en.wikipedia.org/wiki/Square%E2%80%93cube_law