Finding pi

**Chokfull** · August 26th 2010, 11:41 AM

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?

**Archie Meade** · August 26th 2010, 12:02 PM

Originally Posted by Chokfull

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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,

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

$=\displaystyle\lim_{N\rightarrow\infty}\pi\,\frac{ Sin\left(\frac{2\pi}{N}\right)}{\left(\frac{2\pi}{ N}\right)}=\pi$

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