Why is the pan handle made of plastic?

The first and last answers simply are not correct, nor do they answer your question. The second, middle question is much better, but it does not fully answer your question either.

If a metal pan with a plastic handle is put in the oven at, let's say, 350 deg., then the temperature of the entire pan, metal and plastic, will eventually reach 350 deg. It is not true that the plastic won't get hot, actually it should heat up faster than the metal.

There are two properties in consideration: heat capacity and thermal conductivity. Metals in general have large heat capacities. One can think about the amount of heat a substance can store per unit mass, called the "Specific Heat," or one can consider the amount of heat a substance can store per unit volume. In the case of a pan and handle, it makes more sense to look at the question in terms of the heat stored per volume.

Most people know from everyday experience that metals stay hot longer than non-metals like wood and plastic. That is because metals have a much larger heat capacity per volume than most non-metals; that is they hold more heat.

What about thermal conductivity? In this case, we only need to consider the thermal conductivity of air.

Heat flows FROM a hotter object TO a cooler object. If a pan w/ plastic handle is put in a 350 deg. oven and allowed to heat up completely, when it is taken out the oven the pan and handle are at 350 deg. We only need to consider the heat transfer from the metal to the air and from the plastic to the air. Since the cooler material is the air, all that matters is the thermal conductivity of the air. You can look up thermal conductivities in a table, but air, and all gasses, have much lower thermal conductivities than almost any solid, and it is certainly lower than the pan and Bakelite® (the pan's handle). Since the handle does not contain as much heat as the pan, it will cool down to a touchable temperature faster.

The same applies if the pan is placed on a burner. The handle will take longer to heat up than the metal because 1) the metal is directly over the flame, and 2) the metal has a higher thermal conductivity than the plastic meaning that a lot more of the heat from the burner will be absorbed by the metal.

Answer

The reason the pan Handle is made of plastic is because Plastic is not a conductor of heat so it won't get hot. If the handle was metal it would get very hot because it is a conductor of heat.

Answer

Frying pans have plastic handles because plastic is a terrible conductor of heat that is coming from a stove. Therefore the handle is cool even when the metal of the frying pan is hot, when you need to lift it, or perhaps something else.

Answer

Safety. Plastic will not conduct either heat or electricity, so in order you can safely handle pots and pans, plastic is used for those reasons.

Conductivity of air plays practically no role in this. If the pan heats up air in contact, that air is immediately removed by convection and replaced by cool air.