I'm new to this forum (and all forums outside of the world of r/c helicopters) and I went through the listed topics looking for a discussion about circles and their diameters and did not find one. If there is such a thread please point me in that direction.

I have been troubled for a long time about several things in math but particularly about circles and the math used to describe them. Pi is used to represent the relationship between a circumference and its diameter. But in spite of the fact that there are circles all around us with finite circumferences and diameters it doesn't appear that we can compute finite results without approximations.

Is this a failure of our system of mathematics that will be left to be corrected in the future with a new system that will be able to describe a circle more in line with our familiarity with them?

Note: I am not a mathematician. I work in manufacturing management. My degrees are in philosophy and psychology but I did study math through calculus.