I have read about how they calculate the value of $\pi$ using polygons.

A hexagon inside a circle gives an approximation of $\mathbf{\pi} = 3.00$, then doubling the number of the number of polygon sides gives a closer approximation of $3.10582$, and if we keep doubling the amount of sides the value for $\pi$ starts to plateau. At $1536$ sides, we get the first six digits of $\pi = 3.14159$.

calculating pi spreadsheet

Would it be fair to say that the value of $\pi$ that we use is just a very close approximation? To me it seems that no matter how many sides a polygon has, its perimeter will always be less than the circumference of a circle that encloses it.

Are there any other proofs for $\pi$ that are 'not too technical'?

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    $\begingroup$ any practical experiments done that prove the value of π What does that even mean? See perhaps Buffon's needle. $\endgroup$ – dxiv Jan 29 '18 at 0:18
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    $\begingroup$ Circles do not exist in the physical world, so what kind of practical experiments do you want? $\endgroup$ – Yuriy S Jan 29 '18 at 0:19
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    $\begingroup$ Look at this interesting site the world of PI for a history about the calculations / algorithms etc. on $\pi$ $\endgroup$ – G Cab Jan 29 '18 at 0:28
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    $\begingroup$ A practical experiment (in contrast to a calculation) always gives approximate values. You can never measure exactly $1$ second, $1$ meter or whatsoever because the precision is always limited. Moreover, the decimal expansion of $\pi$ does not have a period (We know that $\pi$ is irrational), so we can never write down the complete decimal expansion ( not even with marking a period like in the case of rational numbers). $\endgroup$ – Peter Jan 29 '18 at 0:42
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    $\begingroup$ We can also safely say that there is no practical expermient (or measurement) , that can approve the first $200$ digits of $\pi$ (Probably, this is already a big overshoot). And, if we have , for example , a circle or a ball, we cannot even determine whether it is actually an exact circle, ball or whatsoever by an experiment. The polygon-method never reaches $\pi$ , but it approaches $\pi$. Archimedes determined $\pi$ astonishing precisely this way. The accuracy appears poor for today standards, but with the tools Archimedes had it was a really good performance. $\endgroup$ – Peter Jan 29 '18 at 0:49

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