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 Jul 5 accepted Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” Thanks, that's an elegant rendition. Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” Thank you. Even though, as you point out, this proof is not complete, and therefore (rigorously speaking) it isn't rigorous, it contains the essentials of what I was after. It does a very good job of demarcating those bits that may really require non-elementary methods from those that don't. Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” @DavidC.Ullrich: I've been intentionally vague on the points you mention precisely because I don't know what's possible. For example, AFAIK, at least as of not too long, the only proof known of Fermat's Last Theorem relied on mathematics established well after Fermat's time. For all I know, something similar is true of the theorem in this post. In that case, I'm interested in a proof that requires the least additional mathematics than was know by the ancients who were already familiar with the theorem. Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” @ajotatxe: did you learn integration before you learned this result? Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” @MichaelHardy: I suppose that one can use either the area or the circumference of the unit circle as the basis for the definition of $\pi$. But then one would need to show that the same number of this definition reappears in the post's theorem's formula. Jul 5 revised Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” deleted 5 characters in body Jul 5 asked Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” May 28 revised What's the name of this tensor product? deleted 18 characters in body May 28 asked What's the name of this tensor product? May 26 accepted Looking for a “Guide for the Perplexed by Low-dimensional Topology” May 25 accepted Looking for student's guide to diagram chasing May 25 comment Looking for a “Guide for the Perplexed by Low-dimensional Topology” @GregoryGrant: At least part of the reason I find this stuff unsettling is that pretty much everywhere else (outside of areas devoted to these dimension-specific peculiarities) the sets $\mathbb{R}^n$ are treated as "basically the same" modulo a "trivial" difference in the dimensions. In particular, people routinely reason about "$\mathbb{R}^n$" (for some unspecified $n$) by analogy with their intuitions about $\mathbb{R}^d$ for $d \in \{1, 2, 3\}$. If every dimension has its own quirks, such thinking by analogy (where the target of the analogy is wholly unfamiliar) becomes highly suspect. May 25 asked Looking for a “Guide for the Perplexed by Low-dimensional Topology” May 23 awarded Popular Question May 19 revised On random rotational fluctuations in $\mathbb{R}^n$ removed a redundant term in second equation; fixed a couple of small typos May 16 revised Conditions for “equal cardinality $\Rightarrow$ equal (finite) measure”? deleted 3 characters in body May 16 comment Conditions for “equal cardinality $\Rightarrow$ equal (finite) measure”? @PhoemueX: I've edited my post in response to your comment. May 16 revised Conditions for “equal cardinality $\Rightarrow$ equal (finite) measure”? added 420 characters in body May 16 revised Conditions for “equal cardinality $\Rightarrow$ equal (finite) measure”? edited title