I've written a lot about Archimedes and the approximation of pi over the years (appropriated from the web, actually).
Most of that material is in my Analytic Geometry book. I am in the process of re-doing the Github repos for the books, so they're not currently up-to-date, but that will change over the next week or so. You can find me on Github here.
In the last couple of days, I wrote a summary of the math of Archimedes with respect to pi, including how his formulas connect with others based on perimeter and area that are due to Gregory. There is also extensive discussion of methods to approximate the square root of 3, essential to Archimedes work.
That pdf is on Dropbox here. Enjoy, and as always, comments would be welcome.
