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Math Help - Finding pi

  1. #1
    Member Chokfull's Avatar
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    Finding pi

    Finding pi-didddd.psd

    I know you can estimate the area of a circle by inscribing N-gons in the circle with higher and higher values on N. I tried this multiple times, but it didn't work out for me. Can anyone tell me what I'm doing wrong?

    I use a circle with radius 1. I start by making a triangle out of two radii and a side of the polygon. I then draw an altitude from the center of the circle to the side of the triangle, bisecting the angle. The new angles I name \theta. The altitude I call y, and I call the base z. The area of the triangle is A. So,

    \theta=\frac{180}{x}

    \cos \theta=y

    \sin \theta=z

    \frac{zy}{2}=A

    Now, since we can find A, the total area should be 2xA.

    Also, \pi=2xA since r=1.

    Therefore, the area of the circle should be the limit as x approaches infinity of

    2xA=xyz=x\cos\theta\sin\theta=x\cos\frac{180}{x}\s  in\frac{180}{x}

    However, this keeps giving me 180. If this were \pi, this would be right. But it's not, so it's not. And if it's wrong then I'm wrong. And if it's wrong I'm doing something wrong. So what am I doing wrong?
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  2. #2
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    Quote Originally Posted by Chokfull View Post
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    I know you can estimate the area of a circle by inscribing N-gons in the circle with higher and higher values on N. I tried this multiple times, but it didn't work out for me. Can anyone tell me what I'm doing wrong?

    I use a circle with radius 1. I start by making a triangle out of two radii and a side of the polygon. I then draw an altitude from the center of the circle to the side of the triangle, bisecting the angle. The new angles I name \theta. The altitude I call y, and I call the base z. The area of the triangle is A. So,

    \theta=\frac{180}{x}

    \cos \theta=y

    \sin \theta=z

    \frac{zy}{2}=A

    Now, since we can find A, the total area should be 2xA.

    Also, \pi=2xA since r=1.

    Therefore, the area of the circle should be the limit as x approaches infinity of

    2xA=xyz=x\cos\theta\sin\theta=x\cos\frac{180}{x}\s  in\frac{180}{x}

    However, this keeps giving me 180. If this were \pi, this would be right. But it's not, so it's not. And if it's wrong then I'm wrong. And if it's wrong I'm doing something wrong. So what am I doing wrong?
    First, don't evaluate any limit...

    Instead test your formula for increasing values of x.

    \displaystyle\ x\,Cos\left(\frac{180^o}{x}\right)\,Sin\left(\frac  {180^o}{x}\right)=3.14157198278

    for x=1000...

    so you are correct, but are working in degree mode!
    Hence you can evaluate the limit in radian mode.

    Alternatively,

    \displaystyle\lim_{N\rightarrow\infty}N\left(\frac  {1}{2}r^2\,Sin\theta\right)=\lim_{N\rightarrow\inf  ty}\pi\left(\frac{N}{2\pi}\right)\,Sin\left(\frac{  2\pi}{N}\right)

    =\displaystyle\lim_{N\rightarrow\infty}\pi\,\frac{  Sin\left(\frac{2\pi}{N}\right)}{\left(\frac{2\pi}{  N}\right)}=\pi
    Attached Thumbnails Attached Thumbnails Finding pi-pi.jpg  
    Last edited by Archie Meade; August 27th 2010 at 04:31 AM.
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