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Thread: 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 $\displaystyle \theta$. The altitude I call y, and I call the base z. The area of the triangle is A. So,

    $\displaystyle \theta=\frac{180}{x}$

    $\displaystyle \cos \theta=y$

    $\displaystyle \sin \theta=z$

    $\displaystyle \frac{zy}{2}=A$

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

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

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

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

    However, this keeps giving me 180. If this were $\displaystyle \pi$, 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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    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 $\displaystyle \theta$. The altitude I call y, and I call the base z. The area of the triangle is A. So,

    $\displaystyle \theta=\frac{180}{x}$

    $\displaystyle \cos \theta=y$

    $\displaystyle \sin \theta=z$

    $\displaystyle \frac{zy}{2}=A$

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

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

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

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

    However, this keeps giving me 180. If this were $\displaystyle \pi$, 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.

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

    for x=1000...

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

    Alternatively,

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

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