A unit radius circle is surrounded in the plane by $n$ equal radius circles, such that each of them externally touches the unit circle and the two adjacent circles in the "chain". Determine the radius of the circles as a function of $n$. Also, calculate the radii to four decimal places for the first four possible values of $n$.


Let the radius of the tangent circles be $r$, the distance between the centers of two adjacent tangent circles is $2r$, and this distance is seen from the center of the unit circle at an angle of $\frac{2 \pi}{n}$. From the right triangle $O O_{1} E$ as shown in the figure:

$$
\sin \frac{\pi}{n}=\frac{r}{1+r}, \quad \text { hence } \quad r=\frac{\sin \frac{\pi}{n}}{1-\sin \frac{\pi}{n}}
$$

The fraction is undefined if $\sin \frac{\pi}{n}=1$, which means $\frac{\pi}{n}=90^{\circ}$, or $n=2$. (We cannot draw two tangent circles such that they touch each other and the unit circle.)

The values of $r$ calculated to four decimal places:

For $n=3$, $r_{3}=6.4641$  
For $n=4$, $r_{4}=2.4142$  
For $n=5$, $r_{5}=1.4259$  
For $n=6$, $r_{6}=1$.

![](https://cdn.mathpix.com/cropped/2024_05_02_f0d4eee0e369c39d1c0eg-1.jpg?height=445&width=458&top_left_y=642&top_left_x=823)