Does any extant Greek text prove that the area of a regular polygon inscribed in a fixed circle increases with the number of sides in the polygon?

I can't find such a proposition in Euclid, but the Greeks must surely have known of it, and had a proof, especially as a proof needn't even use Eudoxus's theory of proportion.

(I think this is valid, for example. It's something the Greeks could easily have come up with. I may even have overcomplicated it.)


Note on the edited title, Did Greeks know that the area of inscribed regular polygons increases with the number of sides?:

As stated in the body of the question, I personally don't doubt that the Greeks "knew" the result, both in the weak sense of finding it intuitively obvious, and in the strong sense of having at least one proof (which, in the absence of evidence, can only be guessed at). Of course, this is only a personal opinion, not even an educated one. I can't prove it; but on the other hand, I don't want to ask a question about it! Also, the edited title raises an epistemological question (what does it mean to "know" a result?), which I would rather avoid. Not only that, but it could lead to a complex historical debate, with no clear conclusion. I prefer my original concrete and narrowly-focussed wording. If it is still necessary to reword the question, please say why.


As a similar proposition about perimeters is being discussed in an answer and its comments, it seems worth reproducing here a comment in which I derive that result as a corollary:

Let $a_n$ be the area and $p_n$ the perimeter of a regular $n$-gon inscribed in a circle of radius $r$. If the vertices of the $(2n)$-gon are $P_0P_1P_2\cdots$, then $P_0P_2 \perp OP_1$, therefore the area of $\triangle OP_1P_2$ is $\frac{r}{2}\cdot\frac{p_n}{2n}$; but the same area is also equal to $\frac{a_{2n}}{2n}$; therefore $p_n = \frac{2a_{2n}}{r}$; and $a_{2n}$ increases with $n$, therefore so does $p_n$.

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    You can see Archimedes' The Quadrature of the Parabola. – Mauro ALLEGRANZA Nov 5 at 16:06
  • @MauroALLEGRANZA Thank you for reminding me to order a copy of The Works of Archimedes! I've seen that proof - at least in the special case treated by Stillwell in Mathematics and Its History (second edition 2002), but I can't recall reading anywhere a proof, by synthetic Euclidean methods, of the proposition I was asking about. Just to be clear: is it one of the propositions proved by Archimedes in the work you refer to? – Calum Gilhooley Nov 5 at 16:28
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    Just to be sure: you mean inscribed in what? – Francois Ziegler Nov 6 at 22:58
  • @FrancoisZiegler Sorry about that. I was trying to keep the title down to a manageable length, but that's no excuse for perpetuating the same ambiguity in the body of the question. I hope I've fixed it now, but I'm still not altogether happy with the wording! – Calum Gilhooley Nov 7 at 0:02
  • @CalumGilhooley Thank you. (What goes without saying sometimes goes even better by saying it.) As a side note, “taking limits” in your added fact ($p_n=2a_{2n}/r$) lets one prove $\smash{p_\infty=2\pi r}$ and $\smash{a_\infty=\pi r^2}$ from each other — i.e., “the same $\pi$ works for areas as for lengths”. So whoever established that, likely had something like your proof. – Francois Ziegler Nov 8 at 21:04

It is obvious geometrically when one duplicates the number of sides, but not in general. Archimedes considered up to 96-gons for his approximation of the ratio of the circumference to diameter, but he gets to 96 by successively duplicating sides of the regular triangle.

A similar (and somewhat stronger) proposition, "for regular polygons with the same perimeter, more sides imply greater area", is (presumably) proved by Zenodorus in On Isoperimetric Figures, the earliest known treatise on the isoperimetric problem. Unfortunately, it is not extant, and we only know about it from Theon’s commentary on Ptolemy’s Almagest, and Book V of Pappus’s Collection where some ideas of the proof are given. See Blasjo's Isoperimetric Problem for a reconstruction. Knorr in The Ancient Tradition of Geometric Problems, p.273 speculates that the proposition was already known to Archimedes, ingredients of a possible proof do occur in the Sand Reckoner.

To get the OP proposition about the inscribed regular polygons one would need to show that their perimeters increase with the number of sides in addition to the Zenodorus's lemma. This can also be proved, in principle, using the inequalities alluded to in the Sand Reckoner.

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