In living things its for interacting with the environment.
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A surface is where things happen between what is on one side of the surface and the other. Thus in general if you want more of these happenings (eg gas exchange) you need the biggest possible surface area. If however you want to minimise these happenings (eg heat loss) you try and reduce the surface area.
The function of air sacs (alveoli) is to provide a large surface area for gas exchange
The bunsen burner is used to heat items for experiments using a controllable temperature flame and surface area at which the heat is applied to the vessel.
Yes. There are ways to do that, but it depends on what you wish to change the surface area of. Changing the surface area of something like, say, a shotput is difficult, but changing the surface area of a balloon can be accomplished by merely blowing it up (more) or deflating it.
The rate of dissolution of a solid is dependent on the surface area reaction. By crushing the cube, the surface area is greatly increased, thus increasing the rate of dissolution.