Editing Circumference (section)

== Ellipse ==
[[File:Ellipses same circumference.png|thumb|Circle, and ellipses with the same circumference]]
{{Main|Ellipse#Circumference}}
Some authors use circumference to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the [[semi-major and semi-minor axes]] of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the [[canonical form|canonical]] ellipse, 
<math display=block>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
is 
<math display=block>C_{\rm{ellipse}} \sim \pi \sqrt{2\left(a^2 + b^2\right)}.</math>
Some lower and upper bounds on the circumference of the canonical ellipse with <math>a\geq b</math> are:<ref>{{cite journal|last1=Jameson|first1=G.J.O.|title=Inequalities for the perimeter of an ellipse| journal= Mathematical Gazette|volume= 98 |issue=499|year=2014|pages=227–234|doi=10.2307/3621497|jstor=3621497|s2cid=126427943 }}</ref>
<math display=block>2\pi b \leq C \leq 2\pi a,</math>
<math display=block>\pi (a+b) \leq C \leq 4(a+b),</math>
<math display=block>4\sqrt{a^2+b^2} \leq C \leq \pi \sqrt{2\left(a^2+b^2\right)}.</math>

Here the upper bound <math>2\pi a</math> is the circumference of a [[Circumscribed circle|circumscribed]] [[concentric circle]] passing through the endpoints of the ellipse's major axis, and the lower bound <math>4\sqrt{a^2+b^2}</math> is the [[perimeter]] of an [[Inscribed figure|inscribed]] [[rhombus]] with [[Vertex (geometry)|vertices]] at the endpoints of the major and minor axes.

The circumference of an ellipse can be expressed exactly in terms of the [[complete elliptic integral of the second kind]].<ref>{{citation|first1=Gert|last1=Almkvist|first2=Bruce|last2=Berndt|s2cid=119810884|title=Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, {{pi}}, and the Ladies Diary|journal=American Mathematical Monthly|year=1988|pages=585–608|volume=95|issue=7|mr=966232|doi=10.2307/2323302|jstor=2323302}}</ref> More precisely,
<math display=block>C_{\rm{ellipse}} = 4a \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2\theta}\ d\theta,</math>
where <math>a</math> is the length of the semi-major axis and <math>e</math> is the eccentricity <math>\sqrt{1 - b^2/a^2}.</math>