Editing Spheroid (section)

===Area===
An oblate spheroid with {{math|''c'' < ''a''}} has [[surface area]]
:<math>S_\text{oblate} = 2\pi a^2\left(1+\frac{1-e^2}{e}\operatorname{arctanh}e\right)=2\pi a^2+\pi \frac{c^2}{e}\ln \left( \frac{1+e}{1-e}\right) \qquad \mbox{where} \quad e^2=1-\frac{c^2}{a^2}. </math>
The oblate spheroid is generated by rotation about the {{mvar|z}}-axis of an ellipse with semi-major axis {{mvar|a}} and semi-minor axis {{mvar|c}}, therefore {{mvar|e}} may be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]].)<ref>A derivation of this result may be found at {{cite web|url=http://mathworld.wolfram.com/OblateSpheroid.html |title=Oblate Spheroid |publisher=Wolfram MathWorld |access-date=24 June 2014}}</ref>

A prolate spheroid with {{math|''c'' > ''a''}} has surface area
:<math>S_\text{prolate} = 2\pi a^2\left(1+\frac{c}{ae}\arcsin \, e\right) \qquad \mbox{where} \quad e^2=1-\frac{a^2}{c^2}. </math>
The prolate spheroid is generated by rotation about the {{mvar|z}}-axis of an ellipse with semi-major axis {{mvar|c}} and semi-minor axis {{mvar|a}}; therefore, {{mvar|e}} may again be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]].) <ref>A derivation of this result may be found at {{cite web|url=http://mathworld.wolfram.com/ProlateSpheroid.html |title=Prolate Spheroid |publisher=Wolfram MathWorld |date=7 October 2003 |access-date=24 June 2014}}</ref>

These formulas are identical in the sense that the formula for {{math|''S''{{sub|oblate}}}} can be used to calculate the surface area of a prolate spheroid and vice versa. However, {{mvar|e}} then becomes [[Imaginary number|imaginary]] and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.