[capnrefsmmat]

There is actually statistical theory on the best methods for fitting a circle to data. Here's an open-access paper:

https://projecteuclid.org/euclid.ejs/1251119958

I believe this website is using what the paper calls the geometric fit, which is considered one of the better methods, but the paper develops an even better method.

(I had to use this, strangely, in a physics lab course. I was measuring the Zeeman effect by looking at the diameter of rings of light projected by an interferometer; I took webcam images of the ring and used the algorithm to fit circles to the ring and hence measure its diameter.)