Editing Area (section)

==Optimization==
Given a wire contour, the surface of least area spanning ("filling") it is a [[minimal surface]].  Familiar examples include [[soap bubble]]s.

The question of the [[filling area conjecture|filling area]] of the [[Riemannian circle]] remains open.<ref>{{citation
 |last        = Gromov
 |first       = Mikhael
 |issue       = 1
 |journal     = Journal of Differential Geometry
 |mr          = 697984
 |pages       = 1–147
 |title       = Filling Riemannian manifolds
 |url         = http://projecteuclid.org/euclid.jdg/1214509283
 |volume      = 18
 |year        = 1983
 |doi = 10.4310/jdg/1214509283
 |citeseerx   = 10.1.1.400.9154
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20140408110006/http://projecteuclid.org/euclid.jdg/1214509283
 |archive-date = 2014-04-08
}}</ref>

The circle has the largest area of any two-dimensional object having the same perimeter.

A [[cyclic polygon]] (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths.

A version of the [[isoperimetric inequality]] for triangles states that the triangle of greatest area among all those with a given perimeter is [[equilateral]].<ref name=Chakerian/>

The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.<ref>Dorrie, Heinrich (1965), ''100 Great Problems of Elementary Mathematics'', Dover Publ., pp. 379–380.</ref>

The ratio of the area of the incircle to the area of an equilateral triangle, <math>\frac{\pi}{3\sqrt{3}}</math>, is larger than that of any non-equilateral triangle.<ref>{{cite journal|author1=Minda, D.|author2=Phelps, S.|title=Triangles, ellipses, and cubic polynomials|journal=[[American Mathematical Monthly]]|volume=115|issue=8|date=October 2008|pages=679–689: Theorem 4.1|doi=10.1080/00029890.2008.11920581|url=https://www.researchgate.net/publication/228698127|jstor=27642581|s2cid=15049234|url-status=live|archive-url=https://web.archive.org/web/20161104141707/https://www.researchgate.net/publication/228698127_Triangles_ellipses_and_cubic_polynomials|archive-date=2016-11-04}}</ref>

The ratio of the area to the square of the perimeter of an equilateral triangle, <math>\frac{1}{12\sqrt{3}},</math> is larger than that for any other triangle.<ref name=Chakerian>Chakerian, G.D. (1979) "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums''. R. Honsberger (ed.). Washington, DC: Mathematical Association of America, p. 147.</ref>