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How many circles of radius $r$ can be placed around a circle of radius $R$ (close to it)? $r$ can be bigger, equal or smaller than $R$.

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up vote 5 down vote accepted

Draw the two lines through the center of the central circle that are tangent to one of the touching circles. Call the angle between them $\theta$. The question then is how many times will $\theta$ go into the full circle? You need $\left\lfloor\dfrac{2\pi}{\theta}\right\rfloor$, or if you're using degrees, $\left\lfloor\dfrac{360}{\theta}\right\rfloor$.

Now draw the line through the center of the two circles. The angle between that line and one of the lines you drew earlier is $\theta/2$. The distance between the centers is $R+r$. Now draw the segment from the center of the touching circle to the tangent line. Its length is $r$. Now you have a right triangle in which the hypotenuse has length $R+r$ and one of the legs has length $r$ and the angle opposite that leg is $\theta/2$. Therefore $$ \sin\frac\theta2 = \frac{r}{R+r}. $$ Take arcsines and multiply by $2$ and you've got $\theta$.

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Thank you for useful answer! – Ivan Bunin Jan 15 '13 at 11:40

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