Number of the circles of radius $\frac{\pi}{4}$ in the circle of radius $\pi$ .

Question

QUESTION:

How many circles of radius $\frac{\pi}{4}$ can be drawn within a circle of radius $\pi$ such that they do not intersect one another?

I have seen a solution where it is stated to use the expression below :

$$\sin \frac{360}{2n}=\frac{\frac{\pi}{4}}{\pi-\frac{\pi}{4}}$$

where $n$=number of circles possible.

But I'm unable to understand the geometry behind it.

Answer 1

The answer is eleven (proved by Melissen) [thanks @Rushabh Mehta]

(And why is this problem posed with the radii in multiples of $\pi$ rather than just $1$?)

Answer 2

I previously answered this question, of course, I did not prove that 11 is maximal but I showed OP that at least you can make 11 circles inside and 9 circles touching the outer circle

but @rushabhamehta said that it is not possible to draw 9 circles touching inner part of the outer circle.

So I decided to plot them one by one and you can clearly see that drawing 9 circles touching the inner part of the outer circle is possible

however here I am not claiming that the circles themselves must touch they may not touch themselves.

Then I decided to draw 2 extra inner circles which also seems easy

and by looking at the picture it is clear that you can not draw an extra circle, however, I did not prove it.