## Area of a regular n-gon inscribed in a unit circle

An enlightening discussion about pi and tau.

### Area of a regular n-gon inscribed in a unit circle

In section 5, the area of a regular n-gon inscribed in a unit circle:

A = n * sin pi/n * cos pi/n = (n/2) sin 2pi/n = (n/2) sin tau/n

Isn't that simpler with tau?

(Sorry I haven't better mastered posting equations here yet.)
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: Area of a regular n-gon inscribed in a unit circle

josephlindenberg wrote:In section 5, the area of a regular n-gon inscribed in a unit circle:

A = n * sin pi/n * cos pi/n = (n/2) sin 2pi/n = (n/2) sin tau/n

Isn't that simpler with tau?

(Sorry I haven't better mastered posting equations here yet.)

Here it is for you (you can quote the post to see the syntax):

$$A = n\sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right) = \left(\frac{n}{2}\right)\sin\left(\frac{2\pi}{n}\right) = \left(\frac{n}{2}\right)\sin\left(\frac{\tau}{n}\right)$$

More details on what is possible can be found here, though do note that the forum software does not support everything that LaTeX (or even Plain TeX) can do.

xander

xander
University

Posts: 154
Joined: Fri Feb 11, 2011 12:14 am
Location: Sparks, NV, USA

### Re: Area of a regular n-gon inscribed in a unit circle

Thanks!
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: Area of a regular n-gon inscribed in a unit circle

Depends if you prefer to divide by 2 or not
Math - It's in you to give.

SpikedMath

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

### Re: Area of a regular n-gon inscribed in a unit circle

I'd rather divide by 2 than have to calculate a whole second trig function and then have to multiply that by the first trig function.

The TAU version takes fewer calculator button presses, fewer CPU cycles to calculate, less room on the paper, less writing, less pencil lead, and less remembering. But other than that, Mrs. Lincoln liked the play just fine.

Your Pi Manifesto paper is really well done, but this equation doesn't belong in it. It belongs in Hartl's Tau Manifesto.
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: Area of a regular n-gon inscribed in a unit circle

The proof of 1/2*n*sin(tau/n) also doesn't require bisecting the angles of the isosceles triangles that you split the polygon into, while n*sin(pi/n)cos(pi/n) does, which is why half tau shows up in that version. So tau is both more natural and simpler by every possible definition of simple in this formula.
τ>π
1=0
Mathlete

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