Today, I discovered this Vi Hart video:
The video presents a remarkably sound idea that I had never considered; to paraphrase:
π is only half as fundamental a number as you think it is
Using the diameter to define the circle constant π was a bad idea. To make Maths a better and less confusing place, we should rather use τ=2π.
Some further reading, in a more serious tone than Vi's videos: The Tau Manifesto by Michael Hartl