|WikiProject Mathematics||(Rated Start-class, Low-priority)|
A part of a circle or a disk?
In German and Russian the sector is a part of a disk (2-dimentional object) rather than merely a circular arc. I think that en.wp mistakes, as other languages which align to this article. Incnis Mrsi (talk) 10:21, 17 April 2010 (UTC)
Center of Mass
The current paragraph that replaced the incorrect expression is neither comprehendible nor referenced, and not very useful at any rate because it gives no actual mathematical expression. Unless we have something concrete to say about the center of mass of a circular sector, there's really no point in having this section. — Preceding unsigned comment added by 18.104.22.168 (talk) 08:41, 18 July 2011 (UTC)
Ehrenpreis used to point out this logical circle in most calculus texts. The Wikipedia article on the proof of the derivative of sine references this article. Now, the issue is, how does one prove that the area of a circular sector is proportional to the angle subtended? The answer is, you can't. Unless you use integral calculus, there is no proof using geometry alone....in fact, there is no way to "measure" an angle in Euclidean geometry, i.e., no way to associate a real number to it. One could develop a theory of proportion, perhaps, but I do not see any reference to a source on this.
So it seems to me that only the explanation using integrals here is valid. But in that case, it cannot be used for the other Wikipedia article that refers to this one.
- Interesting observation. I don't have a source, and it may be hard to find one because authors might tend to assume it's "obvious", on grounds of symmetry: If you have a sector, then rotate the circle by theta, then you have a sector congruent to the first one and hence with the same area. Together the two sectors have twice the angle and twice the area, and by symmetry they're in the same circle. Since the sector is infinitely divisible into sub-sectors, and all of them can be rotated the same way to be adjacent to the original sector, you get proportionality of the area to the angle subtended. Loraof (talk) 18:20, 17 January 2015 (UTC)