Preface: This question is a result of personal interest, and has nothing to do with any assignment. Okay here's the deal; I've tutored trig in the past, and I've noticed one thing that a lot of folks have trouble with is the fact that there are two pi radians per circle. (It doesn't help that pi can't help but remind most of us of a deliciously full pie.) Anyway, that got me thinking: why are radians defined such that two pi fits in each pie? We could easily redefine radians such that pi = pie, which is much more intuitive. (Don't believe me? Radian angle = S/2R. It works.) So I'm hoping someone here is a math history buff, and can give me an answer. It'll probably be "Well this is how the first guy did it, so that's how we do it too," but I can hope for a more interesting answer. Thanks in advance. :)
Google "pi is wrong" and you will see many links about this subject. There are a few threads here on Physics Forum as well. Some people use the Greek letter tau to stand for 2*pi, and then there are exactly tau radians in a circle; half a circle is tau/2, etc. Much more sensible. Maybe this will catch on. My theory (which I have no confirmation for) is that historically, people invented geometry and trigonometry to measure the size of their fields (agricultural fields, not algebraic ones of course!) so that the diameter of a circle is the most important measurement. Circumference/diameter was what they studied, and that's how they got pi. However in modern times we understand circles mathematically in terms of the plane and the unit circle; and in that context, the radius is more important. And the circumference divided by the radius is unfortunately 2*pi, from which much student confusion follows.
It may catch on in education circles, but unlikely in academia. We use tau as a variable in too many fields. The reason why pi never seems to be used as a variable is because we use it as a constant and it's ingrained. Similarly, you see in high school texts: [itex]\overline{A}[/itex] to mean the complement of a set, not its closure the use of [itex]\mathbb{J}[/itex] to be the integers not [itex]\mathbb{Z}[/itex] the use of cis and so on. And students have to unlearn these conventions because no academic paper uses them.