Why Does πr^2 Work?

Recently, I have been alluding to the pi vs tau argument, which is debating the proposal of replacing pi with tau (the equivalent of 2π). Since both sides have pretty convincing points, it is a fun thing to talk about.

Pi fans argue that by changing pi to tau, we would ruin the formula for the area of the circle. It would go from:

A = πr^2  –>   A = 1/2τr^2

As you can see, the formula looks a lot sloppier.

Tau-ists rebut this by saying that the proof of this formula requires one to multiply 1/2 by 2πr^2, thus proving the significance of 2π. I mention this a lot, but I have never actually proven it.

Rather than writing out the proof, I think it would make more sense to watch this YouTube video. It made it a lot more interesting for me.

I find this proof fascinating on its own. Also, you may have noticed when the 2π was present and was cancelled out by the 1/2. For the pi vs tau argument, this specific instance can fall to either side.