Maths and Biology
As the radius of a sphere increases, the volume increases much faster than the corresponding surface area.
You can plug some numbers on the sphere calculator on the right in order to check that for yourself.
Imagine a unicelular creature, such as a bacteria. It must get all its oxygen and food through its outer surface. Such membrane acts as an open gut, and also as lungs. In addition, all the waste products must get out via this same membrane.
Image courtesy of the National Institutes of Health image bank
Let's imagine some spherical bacteria (cocci),like the ones on the left. If their size increased a lot, they wouldn't be able to survive. Geometry explains: because the volume would increase much more than the surface area, so that the external membrane wouldn't be able to sustain the flux of nutrients and oxygen that would be needed to feed the much increased internal machinery, that makes up the volume of the bacterium.
There are implications to large animals too. A rat is a warm blooded animal, like all mammals. Because it is relatively small for a mammal, it has a relatively large surface area, for its body volume. That is a problem because it will be losing heat through the skin, and the metabolism will have to produce a lot of heat, all the time. Because of this reason, a rat eats about a quarter of its weight every day, and most of this food is used to maintain body temperature.
That is also the reason why you don't see rats or other small mammals in the cold regions of the planet.
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