## Areas of sectors - an argument for tau

An enlightening discussion about pi and tau.

### Areas of sectors - an argument for tau

I know in the Pi Manifesto it is stressed that when looking at areas it is a clear win for $$\pi$$. But when looking at areas of sectors a bit more closely, it seems more natural to think of it in terms of $$\tau$$. In particular, the area of a circular sector with central angle $$\theta$$ is:

$$A = \frac{1}{2}\theta r^2$$

When $$\theta=2\pi$$, the 2's cancel giving the area of a circle. The area of a circle is just a special case of the above formula and so it seems there should be a $$1/2$$ in it.
Math - It's in you to give.

SpikedMath

Posts: 133
Joined: Mon Feb 07, 2011 1:31 am

### Re: Areas of sectors - an argument for tau

Hi everyone,

I would like to say two things: First, that I agree with Spiked Math that areas of sectors provide an argument for $$\tau$$.

But I would even go beyond what Spiked Math said when writing: "I know in the Pi Manifesto it is stressed that when looking at areas it is a clear win for $$\pi$$."

In my opinion, "tauists" are being too generous with our adversaries when they make such a concession regarding the area of a circle. I believe that when advocates for $$\pi$$ make such a claim, then advocates for $$\tau$$ are letting off the pro-$$\pi$$ people too easily.

First, let me say that I recently saw someone else make essentially the same argument that I am about to make, albeit with different words. I wish that I could give proper attribution, but I cannot remember where I saw the argument -- perhaps it was in this very forum.

In any event, pro-$$\pi$$ advocates will point out that the following formula looks so tidy:

$$\pi = \frac{A}{{{r^2}}}$$

So it does. But in writing that formula, those who advocate $$\p$$ fail to have the courage of their convictions. They claim that diameter is the fundamental circle constant, not the radius. When it suits them, however, as it does here in order to make a succinct formula, then they are quick to abandon diameter and utilize radius in its place.

I say that the advocates of $$\p$$ do not have the right to refer to radius in the context of explaining any of the parameters of a circle. To be consistent with their own argument, they are logically bound to always refer to diameter and never to radius when discussing circles. And hence, the formula they should use is the much more cluttered version of the circle's diameter, as such:

$$\pi = \frac{4A}{{{d^2}}}$$

In short, if pro-$$\pi$$ people don't like radius, if they don't recognize radius to be the fundamental circle constant, then they shouldn't use it.

Thanks to all of you who read my commentary.
metric man
Kindergarten

Posts: 8
Joined: Wed Dec 21, 2011 10:38 pm

### Re: Areas of sectors - an argument for tau

metric man wrote:First, let me say that I recently saw someone else make essentially the same argument that I am about to make, albeit with different words. I wish that I could give proper attribution, but I cannot remember where I saw the argument -- perhaps it was in this very forum.

It was probably Bob Palais' website where you saw that:
http://www.math.utah.edu/~palais/pi.html wrote:"It seems to me that you can't have it both ways on area $$A= \pi r^2$$ and circumference $$C= \pi d$$. If you believe diameter is fundamental, then it should be $$A= {\pi d^2}$$/$$4$$."
PI is the SEMICIRCLE constant, not the circle constant . . . http://sites.google.com/site/taubeforeitwascool
josephlindenberg
Elementary School

Posts: 18
Joined: Wed Jul 06, 2011 2:34 am

### Re: Areas of sectors - an argument for tau

For the Wikipedia article on tau, this is how I summarized why radius is more fundamental:
• A circle is defined as all points in a plane a certain distance — the radius — away from a center point.
• Standard circle formulas use radius:
$$\;r^2 = (x-a)^2 + (y-b)^2$$
$$\;x = a\,+\,r\,cos\,t \;\;y = b\,+\,r\,sin\,t$$
• The unit circle — note the word unit — has a radius of 1, not a diameter of 1.
• Angles are measured in radians.
PI is the SEMICIRCLE constant, not the circle constant . . . http://sites.google.com/site/taubeforeitwascool
josephlindenberg
Elementary School

Posts: 18
Joined: Wed Jul 06, 2011 2:34 am

### Re: Areas of sectors - an argument for tau

I agree with all four of josephlindenberg's excellent points.

Incidentally, could somebody please explain to me how to quote another posting. Thank you.
metric man
Kindergarten

Posts: 8
Joined: Wed Dec 21, 2011 10:38 pm

### Re: Areas of sectors - an argument for tau

metric man wrote:I agree with all four of josephlindenberg's excellent points.

Incidentally, could somebody please explain to me how to quote another posting. Thank you.

when signed in, the upper right corner has a box with four dots and a right arrow. Click that, and the quote will open.
bmonk
University

Posts: 140
Joined: Thu Feb 10, 2011 4:03 pm

### Re: Areas of sectors - an argument for tau

Thank you, bmonk.

bmonk wrote:when signed in, the upper right corner has a box with four dots and a right arrow. Click that, and the quote will open.
metric man
Kindergarten

Posts: 8
Joined: Wed Dec 21, 2011 10:38 pm