It's τ/2. The area of a circle is τr²/2. You may be familiar with the idea of x²/2 from calculus: it's an integral, which can be used to compute areas.

Yes, that certainly is the integral of x * dx. So voila, there's your 1/2. And tau relates the triangle to the circle. Nice trick.

Mind you, I don't have a dog in this hunt so I'm not all up to speed on it. All I know for sure is that tau = 2 * pi, so I won't be terribly upset if I see either usage. I generally favor the use of notations which better reveal an underlying concept, but I don't like it when people get all high and mighty about things.

As the tau manifesto discusses, "half tau" is the more meaningful answer: the area of a circle is equal to the area of a triangle whose base is the circumference and whose height is the radius.

It's the integral of tau * r * dr from 0 to 1, which is tau/2.

personally, i'd answer "half tau".

half tau r^2 is a fine equation for an area.

:) Still, two syllables instead of one.

I suppose if I found myself writing "2 * pi" or "4 * pi^2" a lot, I might throw in a dash of tau here and there for brevity. Otherwise it's all the same to me.

Though in fairness to tau, one can always choose a pithy question whose answer is shorter either way, to wit:

What is the area of a unit circle? Pi.

How many radians are in a circle? Tau.