Editing Ellipsoid (section)

==Surface area==
{{see also|Area of a geodesic polygon}}

The [[surface area]] of a general (triaxial) ellipsoid is<ref>F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, editors, 2010, ''NIST Handbook of Mathematical Functions'' ([[Cambridge University Press]]), Section 19.33 {{cite web |url=http://dlmf.nist.gov/19.33 |title=Triaxial Ellipsoids |access-date=2012-01-08 }}</ref> 
:<math>S = 2\pi c^2 + \frac{2\pi ab}{\sin(\varphi)}\left(E(\varphi, k)\,\sin^2(\varphi) + F(\varphi, k)\,\cos^2(\varphi)\right),</math>

where
:<math> 
  \cos(\varphi) = \frac{c}{a},\qquad
  k^2 = \frac{a^2\left(b^2 - c^2\right)}{b^2\left(a^2 - c^2\right)},\qquad
  a \ge b \ge c,
</math>

and where {{math|''F''(''φ'', ''k'')}} and {{math|''E''(''φ'', ''k'')}} are incomplete [[elliptic integral]]s of the first and second kind respectively.<ref>{{Cite web|url=http://dlmf.nist.gov/19.2|title = DLMF: 19.2 Definitions}}</ref> 

The surface area of this general ellipsoid can also be expressed in terms of {{tmath|R_G}}, one of the [[Carlson symmetric form]]s of elliptic integrals:<ref>{{Cite web |title=Surface Area of an Ellipsoid |url=https://analyticphysics.com/Mathematical%20Methods/Surface%20Area%20of%20an%20Ellipsoid.htm |access-date=2024-07-23 |website=analyticphysics.com}}</ref>
:<math>S = 4\pi bc R_{G} \left( \frac{a^2}{b^2} , \frac{a^2}{c^2} , 1\right).</math>

Simplifying the above formula using properties of {{math|''R''<sub>''G''</sub>}},<ref>{{Cite web |title=DLMF: §19.20 Special Cases ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals |url=https://dlmf.nist.gov/19.20#ii |access-date=2024-07-23 |website=dlmf.nist.gov}}</ref> this can also be expressed in terms of the volume of the ellipsoid {{math|''V''}}:
:<math>S = 3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right).</math>

Unlike the expression with {{math|''F''(''φ'', ''k'')}} and {{math|''E''(''φ'', ''k'')}}, the equations in terms of {{math|''R''<sub>''G''</sub>}} do not depend on the choice of an order on {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. 

The surface area of an ellipsoid of revolution (or spheroid) may be expressed in terms of [[elementary function]]s:
:<math>
   S_\text{oblate} = 2\pi a^2\left(1 + \frac{c^2}{ea^2} \operatorname{artanh}e\right),
                   \qquad\text{where }e^2 = 1 - \frac{c^2}{a^2}\text{ and }(c < a), </math>
or
:<math>
 S_\text{oblate} = 2\pi a^2\left(1 + \frac{1 - e^2}{e} \operatorname{artanh}e\right)</math>

or
:<math>
S_\text{oblate} = 2\pi a^2\ + \frac{\pi c^2}{e}\ln\frac{1+e}{1-e}</math>

and 
:<math>
  S_\text{prolate} = 2\pi a^2\left(1 + \frac{c}{ae}  \arcsin e\right)
                   \qquad\text{where } e^2 = 1 - \frac{a^2}{c^2}\text{ and } (c > a),
</math>

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for {{math|''S''<sub>oblate</sub>}} can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases {{mvar|e}} may again be identified as the [[eccentricity (mathematics)|eccentricity]] of the ellipse formed by the cross section through the symmetry axis. (See [[ellipse]]). Derivations of these results may be found in standard sources, for example [[Mathworld]].<ref>{{cite web |url=http://mathworld.wolfram.com/ProlateSpheroid.html |title=Prolate Spheroid |first=Eric |last=Weisstein. |website=Wolfram MathWorld (Wolfram Research) |access-date=25 March 2018 |url-status=live |archive-url=https://web.archive.org/web/20170803085757/http://mathworld.wolfram.com/ProlateSpheroid.html |archive-date=3 August 2017}}</ref>

=== Approximate formula ===
: <math>S \approx 4\pi \sqrt[p]{\frac{a^p b^p + a^p c^p + b^p c^p}{3}}.\,\!</math>

Here {{math|''p'' ≈ 1.6075}} yields a relative error of at most 1.061%;<ref>[http://www.numericana.com/answer/ellipsoid.htm#thomsen Final answers] {{webarchive |url=https://web.archive.org/web/20110930084035/http://www.numericana.com/answer/ellipsoid.htm |date=2011-09-30}} by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments.</ref> a value of {{math|1=''p'' = {{sfrac|8|5}} = 1.6}} is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.

In the "flat" limit of {{mvar|c}} much smaller than {{mvar|a}} and {{mvar|b}}, the area is approximately {{math|2π''ab''}}, equivalent to {{math|1=''p'' = log<sub>2</sub>3 ≈ 1.5849625007}}.