how many circle of radius r can be placed inside on the border of another circle

Question

Suppose Radius R of a big circle is given. and I want to place some little circles of radius r inside on the border of that big circle. Like the picture:

But how to find how many small circle can be placed on the border?

Answer 1

I don't have a solution, I suspect someone will soon, but just thinking aloud here, if you want to have $n$ circles of radius $r$ then perhaps in the limit $n \rightarrow \infty$, $$ 2\pi(R-r) \approx 2nr, $$ $$ n \sim \frac{\pi}{r}(R-r), \qquad r \rightarrow 0. $$

Answer 2

As seen from the center of the large circle, a small circle subtends an angle $2\phi$, where $\sin\phi={r\over R-r}$. It follows that the quantities $n$, $r$, and $R$ are related by the equation $$n\cdot\arcsin{r\over R-r}=\pi\ .$$

Answer 3

If the circles have their maximal radius $r$, then draw the triangle between the center $A$ of one of the small circles, the point $B$ where it touches its neighbour, and the center $O$ of the big circle. This is a right triangle with one leg $r$ and hypotenuse $R - r$. The angle at $O$ is given by trigonometry: $$ \sin \angle OBA = \frac{r}{R - r} $$ But since this angle spans $\frac{1}{2n}$ of the whole circle, we must have $$ \sin \frac{360^\circ}{2n} = \frac{r}{R-r} $$ Now, $n$ and $R$ are numbers you know, so you just put them in and solve for $r$.