EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that $\pi \approx \frac{7920}{2521}$. And this is pretty close: the continued fraction expansion of $\pi$ is $(3; 7,15,1,292,\ldots)$, and the continued fraction for the approximation is $(3;7,16,4,2,2)$.

However, if you switched out 360 for dividing the circle into $r$ pieces, you are looking for approximations of $\pi$ of the form $rn / 2(rk+1)$ as $n$ and $k$ vary.

- Is this particular approximation for $\pi$ you get, based on 360 degrees, unusually close for the size ($n=44$, $k=7$) present?

I feel like I have to tell a story to justify this question, because otherwise it's a little weird -- please bear with me.

Some time (a long time) ago, I was in a programming class for grade-school kids and we were learning to do some basic graphics. We were drawing fireworks, and encountered the problem that the "bursts" seemed to have their points going off in random directions. You can guess why: the standard library functions for sin and cos were expecting radians instead of degrees.

Our teacher told us that there was some conversion factor, but didn't remember it. So I set the computer on a loop to draw lines out from the center of a circle, to see if I could figure out what the conversion factor was. It animated drawing lines at angles of 1 radian, 2 radians, 3 radians, ... 360 radians, and then cleared the screen and started over drawing lines at 2 radians, 4 radians, 6 radians ... and then again drawing at 3 radians, 6 radians, 9 radians, ...

Of course, this was kind of misguided, but it was interesting to watch for a few minutes. (If I was more motivated, I would try and animate it again, but those skills are very rusty.) And then, at the multiples of 44 radians, it suddenly worked: it visibly drew a nice, sweeping path all the way around the circle in one pass.

I tweaked around with it a little and found that while (cos(360*44),sin(360*44)) were pretty close to (1,0), a better approximation was had if you used about 43.99975 instead. I still remember this number all this time later.

Now I know radians a little better, and abstractly know that this is because $2\pi(7 + \tfrac{1}{360})$ is very close to the integer 44. Another way to say this is by trying to find approximations as I stated above. As I learned more math I understood this less and less, because the answer seems much closer than it deserves to be based on how small the numbers in question are.