Radius of equally sized small circle of a big Circle?!

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Aug 18th 2012, 09:29 AM
ameerulislam

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Radius of equally sized small circle of a big Circle?!

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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.
Aug 18th 2012, 10:51 AM
earboth

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Re: Radius of equally sized small circle of a big Circle?!

Quote:

Originally Posted by ameerulislam

Attachment 24535

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 $n = \frac{360^\circ}{2 \alpha}$
Aug 19th 2012, 02:33 AM
ameerulislam

Re: Radius of equally sized small circle of a big Circle?!

Quote:

Originally Posted by earboth

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. !
Aug 19th 2012, 04:59 AM
earboth

Re: Radius of equally sized small circle of a big Circle?!

Quote:

Originally Posted by ameerulislam

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.