$\begingroup$Archimedes famously determined that $223/71 < \pi < 22/7$ using the 96-gons circumscribed by and circumscribing a circle of unit diameter. But I haven't found a reference that explains the final step, making the rational approximation. For example, the exact perimeter of the circumscribed 96-gon is:
$$ 48\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}} $$
How did Archimedes come up with 223/71 from that?
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