I have seen this question asked countless times online, but almost every time it is misunderstood. What property of a circle causes the ratio between the circumference and 2 times the radius (pi) to equal 3.14159265358979...? What it is it about a unit circle that makes this number somewhere between 3 and 4, and not 4 and 5, or 2 and 3. In other words, I am looking for an informal mathematical proof of the numerical value of pi.
I am not asking: a) For the definition of pi b) For a tutorial in how to measure the ratio of pi with physical instruments c) For an explanation on why we approximate it 3.14, and not 3.14159, or 22/7 d) For someone to remind me that "pi is irrational and 3.14 is not the real value" or "the length of the circumference is different in every circle. Only the ratio is the same."
I understand the explanation is probably not simple. Explanations of the properties irrational numbers usually include infinite fractions or calculus. I just want an explanation of why it is such as seemingly arbitrary value. Thank you in advance!
Note: thanks to your search function I realized that a similar question had already been answered on this site. Why is $\pi $ equal to $3.14159...$? This answer referred to Archimedes's approximation of pi. The details of the explanation were a little confusing though, at least for me. If you want, a description or explanation of the algebraic formula he used to approach the value of pi will answer my question. Thank you!