What is the mathematical proof behind the value of Pi?

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The discussion centers on the mathematical understanding of Pi, particularly its value and proofs related to it. Pi is defined as the ratio of a circle's circumference to its diameter, approximately 3.14159, but it is an irrational number, meaning it cannot be expressed as a simple fraction or a finite decimal. Various methods, such as Archimedes' polygon approximation and integral calculus, provide ways to derive Pi's value, but there is no singular proof of why Pi equals 3.14. The conversation also touches on the relationship between Pi and the volume and area of spheres, emphasizing that these concepts are often derived through calculus. Ultimately, the participants agree that while there are many ways to approximate Pi, its definition and properties are established rather than proven in a conventional sense.
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[Edit] OTHERS HAVE GIVEN THIS ANSWER, BUT I CAN'T DELETE IT.
There are trig functions that are known to equal π. Their series expansions are used to approximate π to the accuracy desired, but they also prove that the value of π is 3.14159...

For instance, knowing that π/2 = arcsin( 1 ) means that you can use the Taylor series of arcsin to prove the value of π to any accuracy you want. I don't know if this example is practical, but there are many other similar ways. See https://en.wikipedia.org/wiki/Pi#Infinite_series
 

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