Unused Cycles

May 26, 2008

Circumference of a Circle

Filed under: Mathematics — Kevin @ 2:06 am

So I was thinking about my previous area problem and the thought occurred to me that the same method could also be used to derive the circumference of the circle.

Using the same setup as before, the perimeter of the polygon is just Nb. So again if we let N\rightarrow\infty we should get the circumference of a circle.

Recall that I found the formula for b in terms of the number of sides and the radius: it was

\displaystyle b=\frac{r\sin(2\pi/N)}{\sin(\pi/2-\pi/N)}

The circumference of a circle is then

\begin{array}{rcl}\displaystyle C&\displaystyle=&\displaystyle \lim_{N\rightarrow \infty}N\frac{r\sin(2\pi/N)}{\sin(\pi/2-\pi/N)}\vspace{0.3 cm}\\&\displaystyle=&\displaystyle \lim_{N\rightarrow \infty}\frac{r\sin(2\pi/N)}{N^{-1}\sin(\pi/2-\pi/N)}\vspace{0.3 cm}\\&\displaystyle=&\displaystyle \lim_{N\rightarrow\infty}\frac{r\cos(2\pi/N)(-2\pi N^{-2})}{-N^{-2}\sin(\pi/2-\pi/N)+\pi N^{-3}\cos(\pi/2-\pi/N)}\vspace{0.3 cm}\\&\displaystyle=&\displaystyle \lim_{N\rightarrow\infty}\frac{r\cos(2\pi/N)(-2\pi)}{-\sin(\pi/2-\pi/N)+\pi N^{-1}\cos(\pi/2-\pi/N)}\vspace{0.3 cm}\\&\displaystyle=&\displaystyle 2\pi r,\end{array}

which is again exactly as expected.

And now with that out of the way, I can get some sleep. Good night!


1 Comment »

  1. Excellent site!!!!
    Provides with exactly what a Student Wants!
    But,I want to ask a question-
    i.e. Where will I find the exact Derivations of the Formulae for the Surface Area and Volume of Cylinder, Cone, Sphere, Hemiphere, Cuboid and Cube?

    Comment by Bakshinder — July 8, 2008 @ 2:53 am

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