There are some advantages also for taking Pi/2 as the circle constant.

The choice between 2xPi and Pi/2 is equivalent with the choice between the cycle and the right angle as the unit of plane angle (the right angle corresponds with the "imaginary" unit, i.e. any point on the unit circle corresponds to i^x, with x in right angles). When extended to 3 dimensions, the corresponding constants for the solid angle become 4xPi and Pi/2, so Pi/2 is more consistent (it corresponds to taking the orthogonal trihedron as the unit of solid angle). The consistency of Pi/2 remains true for higher-dimensional spaces, but that has little practical importance.

"Pi" is by far the worst choice, in all computational applications either 2xPi or Pi/2 are needed, not Pi.

The only case where Pi appears naturally is in measurements, because both the circumference and the diameter are much easier to measure than the radius, and Pi is the relationship between these 2 practical measurements, allowing the conversion of one to the other.

Nowadays, the computational uses of Pi are many orders of magnitude more frequent than the conversions between the practical measurements of diameter and circumference, so the use of Pi is undesirable inside computer programs or in symbolic processing of mathematical formulae.

In general, it is much better to implement trigonometric functions where the argument is either x*Pi/2 or x*2*Pi, instead of traditional trigonometric functions, because the argument reductions are fast and exact. Sadly, the floating-point arithmetic standard defines useless functions of Pi*x, like sinPi, cosPi etc., instead of using any of the 2 better choices.

The only possible advantage of traditional trigonometric functions is at integration or differentiation, where they could save a multiplication, but in almost all applications of this kind the argument of the function is not x, but an expression with at least one multiplication, and the additional multiplication with a constant caused by using x*Pi/2 or x*2*Pi can frequently be done at compile time, or at run time, but only once, not at every computation.

What I don't understand is why pi is two taus stuck together but tau is 2*pi rather than the other way around.

Tau replacing pi? Sure, just start using it. Publish with it. If we want better, we have to make it happen!

I use it in my code at least.

Area of a circle?

Yes, even that one gets more beautiful, too.

Look at the usual equation: A = πr². Why is there no "2" there?

Let's derive it, and in particular, let's derive it from the onion proof, which is that a circle's area is composed of many small circles, arranged concentrically, like a 2D onion:

A = ∫_0^r 2πt dt

There's that blasted 2 again. The tau form is more beautiful:

A = ∫_0^r τt dt

Integrate it, and you'll get A = τr²/2, the constant being a result of the integral.

That is, to me, the usual equation is more properly A = 2πr²/2, the two 2s being different in their origins, and we just usually use & memorize the simplified form.

Unfortunately the ancients didn't invent calculus. Pi had been in use a long time when Liebniz and Newton came along.

Another way to look at that would have been visible to non-calculus bearing ancients:

  A = πr²
  C = πD

… why do we arbitrarily use r in one equation, and D in the other? (… because we're using the wrong constant, and it bugs us, and we're sweeping that under the mathematical rug.)