I'm a novice here, with little better than an average high school student's knowledge in mathematics. That said, I'm an eager learner and don't shy away from difficult tasks. I could use a nod in the right direction. For some reason, this week I wanted to see how many decimal digits of pi I could come up with.
I first read up on an ancient technique of computing the area of regular polygons with a fixed radius but ever-increasing number of sides. I constructed a simple spreadsheet (I used Google Sheets) to allow me to experiment with the number of sides. By dividing the area I computed by the square of the radius I was able to get increasingly-accurate approximations of pi. As I expected, as the number of sides increased, the area approached what I would expect for a circle of the same radius. It took 12 sides to get 3 as the first digit before the decimal point. At 120 sides, I had 3.14. By 1000 sides, 3.1415. And when I got to the ridiculous point of 50 million sides, I got 3.14159265358979 as a result, which looked pretty good to me.
Then, I wanted greater precision. I formatted the output cell to show more decimal places, but this only yielded me answers like 3.14159265358979000000000, which undoubtedly tells me I ran up against the maximum precision of the software.
Given my limited math education (algebra, geometry, some trig, no calculus), what could I try next to get me, say, 25 digits of pi? I have MS Office, Google. I'm not afraid to dip my toes in programming, but I don't know of any full-featured programming tools that don't cost money. (I'm not prepared to invest money in this little project.)
How can I get greater precision? It seems like I'm bounded by the software only having a certain amount of bits available in the cell.
If you’re willing to do some tedious addition and subtraction you can calculate pi using the following:
$\pi=\dfrac41-\dfrac43+\dfrac45-\dfrac47+\dfrac49…$