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Math Help - Radius of equally sized small circle of a big Circle?!

  1. #1
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    Radius of equally sized small circle of a big Circle?!

    Radius of equally sized small circle of a big Circle?!-c.png

    Let say R= Radius of the big circle.
    r= radius of the small circles
    n= number of small circles in the big circle

    What would be the formula to figure out the 'r' for n number of small circles within that R radius big circle.
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    Re: Radius of equally sized small circle of a big Circle?!

    Quote Originally Posted by ameerulislam View Post
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    Let say R= Radius of the big circle.
    r= radius of the small circles
    n= number of small circles in the big circle

    What would be the formula to figure out the 'r' for n number of small circles within that R radius big circle.
    1. The midpoinst of the small circles form a regular polylateral with the side-length 2r. You have n isosceles triangles.

    2. The central angle of one of the isosceles triangles is 2\alpha, that means 2\alpha must be a divisor of 360.

    3. Each isosceles triangle can be split into 2 right triangles with

    \sin(\alpha)=\frac{r}{R-r}~\implies~r=\frac{R \cdot \sin(\alpha)}{\sin(\alpha) + 1}

    4. Keep in mind that n = \frac{360^\circ}{2 \alpha}
    Attached Thumbnails Attached Thumbnails Radius of equally sized small circle of a big Circle?!-klkrsingrkrs.png  
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    Re: Radius of equally sized small circle of a big Circle?!

    Quote Originally Posted by earboth View Post
    1. The midpoinst of the small circles form a regular polylateral with the side-length 2r. You have n isosceles triangles.

    2. The central angle of one of the isosceles triangles is 2\alpha, that means 2\alpha must be a divisor of 360.

    3. Each isosceles triangle can be split into 2 right triangles with

    \sin(\alpha)=\frac{r}{R-r}~\implies~r=\frac{R \cdot \sin(\alpha)}{\sin(\alpha) + 1}

    4. Keep in mind that n = \frac{360^\circ}{2 \alpha}
    Hei Thanks, I'm still trying to understand your solution though. not sure what {sin (\alpha)} means.

    and how to integrate n there. Its actually a programming problem that I'm solving but this is math first. !
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    Re: Radius of equally sized small circle of a big Circle?!

    Quote Originally Posted by ameerulislam View Post
    Hei Thanks, I'm still trying to understand your solution though. not sure what {sin (\alpha)} means.

    Have a look here: Sine - Wikipedia, the free encyclopedia

    and how to integrate n there. Its actually a programming problem that I'm solving but this is math first. !
    An example: You know (from my previous post) that n = \frac{360^\circ}{2 \alpha} = \frac{180^\circ}{\alpha}

    Now choose a value for \alpha which must be a divisor of 180: \tfrac14^\circ, \tfrac13^\circ, \tfrac12^\circ, 1^\circ, 2^\circ, 3^\circ, 4^\circ, 5^\circ, 6^\circ, 9^\circ ..... 90

    To each value of \alpha belongs the number of small circles: 720, 540, 360, 180, 90, 60, 45, 36, 30, 20, .... 2

    Of course you can choose the number of circles first and determine the value of \alpha afterwards:

    \alpha = \frac{180^\circ}n~, n \in \mathbb{N}~\wedge~n\ge2

    For instance: You want to draw 7 small circles then \alpha = \tfrac{180}7 ^\circ

    But in both cases you have to use the Sine function.
    Last edited by earboth; August 19th 2012 at 08:21 AM.
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