Caroline!!!

I am curious about the derivation of pi and the formula for deriving a polygon of degree... can you help?

While I am more of an analyst than a mathematician, I will try to explain.

Historically, most people have used the perimeter of a polygon instead of its area to approximate Pi. To do this, you would use the formulaCircumference = Pi * Diameter to approximate the digits of Pi. To do
this, they inscribe a regular polygon in a circle of diameter 1. Then as the number of sides of the polygon gets big, its perimeter will approach Pi.

If you want to use the formula Pi*r^2 to find an approximation toPi, you'd use a circle of radius 1, and inscribe a regular polygon in it. It's WAY helpful to draw a picture of what's going on. You can divide up the polygon into pie-wedges, and then you find the
area of each one and multiply by n. So let's find the area of one of the pieces.

We'll divide the triangle into 2 triangles by constructing the segment from the center of the circle to the midpoint of the side of the polygon.
Then we've made a right triangle. One of the 2 non-right angles sticks out from the center of the circle; call this angle v. Now the lengths of the 2 legs of the triangle will be Sin[v] and Cos[v]. If you don't see why, write us back (remember, the circle we've inscribe
this triangle in has radius 1).

So the area of the smaller triangle we made is Sin[t] * Cos[t] /2, so the area of the larger triangle is Sin[t] * Cos[t]. Now, what is t in terms ofn? It's 360/2n, i.e. 180/n. Again, if you don't see why, ask us. So the
area of this triangle is Sin[180/n] * Cos[180/n], and there are n such triangles, so the total area of the polygon is n * Sin[180/n] * Cos[180/n]. If you take the limit of this term as n gets big, you should get Pi.

In fact, you could even simplify this term some. If you just look at the Cos[180/n] term, notice that 180/n is going to zero, and Cos[0] is 1. So this won't contribute to the limit in the end, it'll just multiply it by 1.
So you could just use n*Sin[180/n] if you wanted to.

I hope this helps!

OMG
this is another language to me..

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Posted by carolinebr



So the area of the smaller triangle we made is Sin[t] * Cos[t] /2, so the area of the larger triangle is Sin[t] * Cos[t]. Now, what is t in terms ofn? It's 360/2n, i.e. 180/n. Again, if you don't see why, ask us. So the
area of this triangle is Sin[180/n] * Cos

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the cos?

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Posted by DeannaNDave07

OMG
this is another language to me..

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this is the EXACT reason why I'm an artist.

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Posted by DeannaNDave07

OMG
this is another language to me..

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aaand this is EXACTLY why it took me 4times to pass my math regents....

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Posted by jtgarsik

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Posted by DeannaNDave07

OMG
this is another language to me..

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aaand this is EXACTLY why it took my 4times to pass my math regents....

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I failed my road test 5 times.

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Posted by carolinep24

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Posted by DeannaNDave07

OMG
this is another language to me..

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this is the EXACT reason why I'm an artist.

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LOL.. i dont blame you.. i cant even imagine figuring that out..

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Posted by carolinep24

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Posted by jtgarsik

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Posted by DeannaNDave07

OMG
this is another language to me..

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aaand this is EXACTLY why it took my 4times to pass my math regents....

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I failed my road test 5 times.

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you both crack me up !

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Posted by Colette

Holy G*d I think caroline(br) wrote out that entire thing, its not a cut and paste job. That's as freaky as the manboobs too me....

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If she did, then she is officially my new favorite person.

(it's not really an honor though... at one point wilfred brimely was my favorite person)

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Posted by carolinep24

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Posted by Colette

Holy G*d I think caroline(br) wrote out that entire thing, its not a cut and paste job. That's as freaky as the manboobs too me....

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If she did, then she is officially my new favorite person.

(it's not really an honor though... at one point wilfred brimely was my favorite person)

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I am about to pee my pants:

Wlfred





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