O.k.  Math folks like to calculate things as precisely as possible.  However, there really is a limit on how precise one can measure something, even with good temperature controls.  So, what is the upper limit on the practical number of digits one needs for the approximation of pii?

Take the example where a disc is created whose radius is 10 meters, and is created out of a sheet of graphene.  The potential limit of precision, then, becomes bounded by the size of the carbon atom.

Now, in my world, I have had occasion to measure something I've machined whose diameter is 48", and I typically measure to 0.0001" (but not something that big to that level of precision). So, using those extremes, my limit for pi is only seven digits (to avoid any round-off errors).

My question for you is to justify a greater level of precision, and express how many digits that entails.  I'm going to go out on a limb, and say that anything beyond 12 decimal points is really a waste of time for practical applications.

4, 2 on each side of .

It depends on how big your circle is

JPL uses 15 decimal places for its most precise calculations: 3.141592653589793

I think partially you answered your own question, it depends on the application.  At DARPA, for most equations and algorithms, we always used Pi in code and then truncated the answer  for the equation to so many digits, again depending on the application. For actual calculations I memorized, 3.14156295 in high school and still use it when I try to impress my kids but I rarely ever use the actual number to calculate anything. I usually just use the equation and let others do the manual labor ;)

All I ever use pi for at work is 3.14.
Funny note though.  In 10th grade my sons math teacher gave out an assignment to memorize pi to the greatest number of digits to win a "Pizza Pie" lunch for the winners class. In 4 days my son had it out to something like 540 digits and won pizza for his class.  When he went to college and was in his first math class (He was a math major) pi came up and my son ripped off 300 or so digits on the spot to his professor who actually looked it up and was absolutely floored.  Talk about crowding a brain with useless info.....  IIRC there are people in this world that spend lifetimes memorizing it.  Records are in the 10,000 digit range in memory..

Depends.

For shop stuff or estimating the Pi button or 3.14, or even 3.1 is usually good enough.

For something more exacting if no Pi button is present, I use what I have memorized -> 3.1415926 <-

And the only reason I memorized that is that I am a mechanical engineerl