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

by josephlindenberg » Wed Jul 06, 2011 2:53 am

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

by xander » Wed Jul 06, 2011 8:33 am

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):

[tex]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)[/tex]

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.

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

by josephlindenberg » Wed Jul 06, 2011 8:42 am

Thanks!
PI is the SEMICIRCLE constant, not the circle constant . . . http://sites.google.com/site/taubeforeitwascool
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Re: Area of a regular n-gon inscribed in a unit circle

by SpikedMath » Wed Jul 06, 2011 1:15 pm

Depends if you prefer to divide by 2 or not 8-)
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Re: Area of a regular n-gon inscribed in a unit circle

by josephlindenberg » Wed Jul 06, 2011 9:08 pm

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. :D

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

by 1=0 » Sun Mar 19, 2017 11:13 am

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.
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