A few years ago, a lot of mathematicians started discussing an interesting proposed circle constant which has been named τ ("tau", or "tough" if you wanna use the Greek pronunciation) which has caused a lot of controversy, and the idea behind that constant is that it is based on the radius rather than the diameter, so that τ = C/r (as opposed to π = C/d).
The motivation behind using this constant is that it seems to be more intuitive in a lot of ways, and also make more sense.
It's not just about trivial arithmetic or about replacing 2·π with τ, it's about the belief that τ might actually be easier to understand for beginners and that it is probably more descriptive.
One major simplification that I can see right away is that τ much more directly tells you how many revolutions you have gone around the unit circle, because the number of revolutions is directly proportional to the number of τ.
For example, if an angle has the value (2/7)·τ, then you can see right away that you have gone 2/7 of one revolution, so that would be about 103°.
It also appears that τ would eliminate the need to memorise exact angles  you don't need to "learn" that 45° is equal to (1/4)·π radians, because if you express this in terms of τ then you will get (1/8)·τ;
in other words, exactly 1/8 of one period, which you can see from inspection!
The same is also true for 60°, which is of course 1/6 of a period, so that would be (1/6)·τ radians.
And to be honest, I still often confuse myself when I write angles in terms of π, even though I started trigonometry almost 15 years ago in high school in 2003.
It just seems more natural to me to use τ instead, since that constant much more directly corresponds to the number of periods.
It also seems to fit better with the general formula for the area of a circle sector;
that formula is A(θ) = (1/2)·θ·r² (basically, the closed integral of a circle's arc length in terms of radians), and you can see clearly that the special case where θ = τ gives you A(θ = τ) = (1/2)·τ·r², which does indeed give you the area for a full circle, and this might give new students some further insight.
I really like this idea, and I think that the only times when I use π nowadays are when I need to simplify an expression as much as possible and thus prefer to write π instead of τ/2, although I always think of angles in terms of τ, since it just seems simpler to me.
But I am also interested to know of your opinions as well.
