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I came across this website (see also) where the author (a supposedly alternative mathematician) claims to have a better approximation to $\pi$.

$\pi\approx 3.1547…$

Can someone tell me where is the error in his reasoning? (Sorry for the pessimistic viewpoint)

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    If you look closely at any carefully crafted argument, it will usually make sense. Making sense and mathematical correctness are not correlated, at all.2012-10-05

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The triangle he refers to does not have one half of the area of the circle, as he claims. This is where the error comes from.

In the future, I might recommend to not bother reading sites that claim obviously false things, like $\pi=3.15...$.

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    @pion: For example [the proof that \pi<22/7.](https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80) Since it is easily verified that 22/7<3.15, this also proves that \pi<3.15.2012-09-18
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I'm not sure he is trying to say that his estimate of $\pi$ is better, just that this construction provides a reasonably good estimate. And it does, but we know this only because we know a good approximation of $\pi$ beforehand (that is, we can compare the calculated value to 3.1415926...)

His error estimate makes me wonder though, since it's way off. If he is trying to claim that his estimate of $\pi$ is better than the usual 3.1415926..., then he would need to better explain his error estimate, which comes out of nowhere.

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    @BenMillwood Didn't Archimedes get$22/7$by calculating the areas of inscribed and circumscribed 96-gons? That sounds like a lot of work to me... Unless I'm mistaken about the origin of 22/7. I'm honestly not willing to look too closely into the rest of this crackpot's website though to see what lengths he goes through and how much "work" it actually is to obtain his estimate of $\pi$.2012-09-18