How are surface area to volume ratios important to animal functions?
For animals to function, they must absorb oxygen and nutrients while emitting waste products such as urea and carbon dioxide. For this to occur, these molecules must defuse across cell membranes. The greater the surface area is, the faster this diffusion can occur. On the other hand, the rates of metabolism and other processes are dependent on cell volume.
In general, as a cell increases in size, the volume grows at a faster rate than its surface area. Similarly, as an organism increases in volume, the available surface area for diffusion increases at a lower rate. This decreased surface area affects the ability for molecules to enter and exit cells, and affects metabolic rate, in turn. This is why larger animals have slower metabolisms than smaller animals, in general.