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I'm not really sure how to word this question so feel free to edit it to make more sense:

I have a big circle of radius 32, and I want to divide it into smaller segments by both drawing smaller circle(s) with the same midpoint and drawing sectors (?) in such a way that each segment will fit into a circle of radius 9.5

I brought pictures!:

Imagine that if I were to divide the big circle into 8 slices and make 2 sub-circles of 20 and 10 radii, then you can see that it wont fit (the biege circle is the 9.5 radius one):

8 slices not enough doesn't fit

But if I make 16 slices, it will fit and I'll only need one smaller circle with a raidus of 18:

This will work fits!

So my question is: is there any optimal way to figure out the least amount of cuts and sub-circles that are needed to make this work?

The cuts don't need to go all the way through the from the edge of the big circle to the center, it can start or stop at any smaller circle.

By the way, I used Blender to draw the pictures so it may not be exact.

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    Interesting problem! If we draw a regular $n$-gon on the periphery, the main problem comes from the distance between adjacent vertices. A bit of trig (the angle whose sine is $9.5/32$) shows $n=10$ is not good enough, while $n=11$ looks OK. Perhaps, partly for simplicity, we might go to $n=12$. The rest should not be hard, but some experimentation s needed.2012-07-09
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    @AndréNicolas Can you explain how you got to that number. Also What is the radius of the inner circle and how many segments does the inner circle need?2012-07-09
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    Have not done the necessary calculations, so cannot tell you radius of inner circle. Draw the $n$-gon, and join its vertices to the centre. Then we get angles of $360/n$. bisect one of these angles, extend bisector to the side of the $n$-gon. Have angle of $180/n$. Call this $\theta$. We need $32\sin\theta\le 9.5$. That gives $\theta$ about $17.27$ degrees. So $180/n \le 17.27$, giving $n$ at least $11$.2012-07-09

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