The Greek letter 'pi', describes the mathematical relationship between the radius and circumference of a circle, but how is it actually calculated?

If you were to take a string the length of the radius of a circle (of any size) and wrap it around the outside of the same circle, you would need 3.1415926...... pieces of the string to go halfway around the circle. This is where the equation circumference=2(pi)r comes from.

This graphic really helped me to understand it:
https://en.wikipedia.org/wiki/Radian#/media/File:Circle_radians.gif

Since the usual definition is based on measurements, which are only as accurate as the measuring device used, it doesn't provide an answer for the value of pi to a zillion decimal places. (Since it goes on forever without developing a repeating pattern, you can insert any desired number in pace of the zillion.) Several such methods have been developed - but they are not straightforward, and involve an awful lot of number crunching.

https://www.mathscareers.org.uk/calculating-pi/

The calculation of pi and what pi actually is are two totally different things. Starting with Archimedes they use what was called the method of exhaustion. Take a square put it outside and inside the circle. Check the perimeter of both squares. And now you know that pi is in between those two values. Now just crank up the number of sides of each polygon, and you get a better and better approximation. The largest number of sides to a polygon that was used is astronomical. Here I'm going to copy and paste from a source.

Ludolph van Ceulen completed his calculations using the 2^62-sided polygon in the early 1600s, shortly before his death in 1610.

http://docmadhattan.fieldofscience.com/2016/03/ludolph-van-ceulen-in-searching-of-pi.html

Then along came Isaac Newton. I'm only going to throw out the titles of stuff so that it doesn't go down too big a mathematical rabbit hole. But the tools are surprisingly basic, at least they're all very visual. He started with the binomial theorem then worked his way into the graph of a circle. Combine that with the formula for the area of circle which contained pi, he ended up doing a little bit of basic algebra to isolate pie and came up with a very short sweet formula to calculate pi. Instead of having to do all of the crazy calculations with a billion sided polygon, he could get 16 places of accuracy with about five or six steps. No one messed with polygons after that. My guess would be that the supercomputers that are running today trying to figure out even more decimal places are using some form of Newton's method.

I know this is all very hand wavy, but it's not a bad thumbnail description of how to calculate pi. The concept of what pi is and how it is treated in mathematics is a totally different story.