Editing Earth radius (section)

===Authalic radius===
{{see also|Authalic latitude}}
''Earth's authalic radius'' (meaning [[equal-area projection|"equal area"]]) is the radius of a hypothetical perfect sphere that has the same surface area as the [[reference ellipsoid]]. The [[IUGG]] denotes the authalic radius as {{math|''R''<sub>2</sub>}}.<ref name="Moritz"/>
A closed-form solution exists for a spheroid:<ref name="Snyder manual">Snyder, J. P. (1987). ''[https://pubs.usgs.gov/pp/1395/report.pdf Map Projections – A Working Manual (US Geological Survey Professional Paper 1395)]'' p.&nbsp;16–17. Washington D.C: United States Government Printing Office.</ref>
:<math>R_2
=\sqrt{\frac12\left(a^2+\frac{b^2}{e}\ln{\frac{1+e}{b/a}} \right) }
=\sqrt{\frac{a^2}2+\frac{b^2}2\frac{\tanh^{-1}e}e} 
=\sqrt{\frac{A}{4\pi}},</math>
where {{tmath|1=\textstyle e = \sqrt{a^2 - b^2}\big/a }} is the eccentricity, and {{tmath|A}} is the surface area of the spheroid.

For the Earth, the authalic radius is {{convert|6,371.0072|km|mi|abbr=on}}.<ref name=Moritz2000/>

The authalic radius <math>R_2</math> also corresponds to the ''radius of (global) mean curvature'', obtained by averaging the Gaussian curvature, <math>K</math>, over the surface of the ellipsoid. Using the [[Gauss–Bonnet theorem]], this gives
:<math> \frac{\int K \,dA}{A} = \frac{4\pi}{A} = \frac{1}{R_2^2}.</math>