Wikipedia talk:WikiProject Mathematics/Archive/2024/Nov

Area of a circle

Please see recent edit history at Area of a circle where some new editor insists that Archimedes proof needs to be labeled as "a logic proof" and that a calculation of the areas of some isosceles triangles needs to be replaced by subdividing the triangles into right triangles and summing their areas instead, in not-well-written English. —David Eppstein (talk) 06:35, 3 November 2024 (UTC)

I agree that these edits are not good. However I hope that someone can improve the readability of this section.

I think the 'not greater' argument can be described in a clear way almost entirely without symbols. It has two parts: (1) any inscribed regular polygon has smaller area than the right triangle and (2) there exist inscribed regular polygons with area arbitrarily close to the circle area. So if the circle area is greater than the triangle area, by (2) there is an inscribed regular polygon with area larger than the triangle area, but this contradicts (1).

The argument for (1) is that the polygon perimeter is less than the circle circumference (as follows from the fact that lines minimize distance between two points) and the polygon's inner radius is less than the circle radius. Since polygon area is one half the perimeter times the inner radius and triangle area is one half the circumference times the circle radius, (1) follows immediately. Fact (2) is extremely intuitive, and could even be acceptable here as self-evident. Archimedes' construction of iterated bisection is a good illustration but probably not a proper proof. Is it clear without doing some extra calculation that the 'gap area' eventually becomes arbitrarily small?

I think it's a really marvelous proof (or almost-proof) but I found its wiki-description rather hard to read. For me a description of the above kind is much easier.

(And if nothing else, symbol $c$ is presently referred to multiple times but not defined!) Gumshoe2 (talk) 09:00, 3 November 2024 (UTC)