Editing Earth radius

{{for multi|its historical development|Spherical Earth|its determination|Arc measurement}}
{{short description|Distance from the Earth surface to a point near its center}} 
{{Infobox quantity
| name = Earth radius
| othernames = terrestrial radius
| width =
| background =
| image = WGS84_mean_Earth_radius.svg
| caption = Equatorial (''a''), polar (''b'') and arithmetic mean Earth radii as defined in the 1984 [[World Geodetic System]] revision (not to scale)
| unit = meters
| otherunits =
| symbols   = ''R''<sub>🜨</sub>, ''R''<sub>E</sub>, ''a'', ''b'', ''a''<sub>E</sub>, ''b''<sub>E</sub>, ''R''<sub>''e''E</sub>, ''R''<sub>''p''E</sub>
| baseunits = [[Metre|m]]
| dimension = <math>\mathsf{L}</math>
| extensive = 
| intensive =
| conserved =
| transformsas = scalar
| derivations =
| value = ''Equatorial radius'': {{mvar|a}} = ({{val|6378137.0|u=m}}) <br/> ''Polar radius'': {{mvar|b}} = ({{val|6356752.3|u=m}})
}}
{{Infobox unit
| name = Nominal Earth radius
| image = EarthPieSlice.png
| caption = Cross section of Earth's Interior
| standard     = [[astronomy]], [[geophysics]]
| quantity     = [[distance]]
| symbol       = <math>\mathcal{R}^\mathrm N_\mathrm{E}</math>
| symbol2      = <math>\mathcal{R}^\mathrm N_{e\mathrm{E}}</math>, <math>\mathcal{R}^\mathrm N_{p\mathrm{E}}</math>
| units1       = [[SI base unit]]
| inunits1     = {{val|6.3781|e=6|u=m}}<ref name = "IAU XXIX">{{cite arXiv |eprint=1510.07674|last1=Mamajek|first1=E. E|title=IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties| last2=Prsa|first2=A|last3=Torres|first3=G|last4=Harmanec|first4=P|last5=Asplund|first5=M|last6=Bennett|first6=P. D|last7=Capitaine|first7=N|last8=Christensen-Dalsgaard|first8=J|last9=Depagne|first9=E|last10=Folkner|first10=W. M|last11=Haberreiter|first11=M|last12=Hekker|first12=S|last13=Hilton|first13=J. L|last14=Kostov|first14=V|last15=Kurtz|first15=D. W|last16=Laskar|first16=J|last17=Mason|first17=B. D|last18=Milone|first18=E. F|last19=Montgomery|first19=M. M|last20=Richards|first20=M. T|last21=Schou|first21=J|last22=Stewart|first22=S. G|class=astro-ph.SR|year=2015|display-authors= 3}}</ref>
| units2       = [[Metric system]]
| inunits2     = {{val|6357|to|6378|u=km|fmt=commas}}
| units3       = [[English units]]
| inunits3     = {{val|3950|to|3963|u=mi|fmt=commas}}
}}
{{Geodesy}}

'''Earth radius''' (denoted as ''R''<sub>🜨</sub> or ''R''<sub>E</sub>) is the distance from the center of [[Earth]] to a point on or near its surface. Approximating the [[figure of Earth]] by an [[Earth spheroid]] (an [[oblate ellipsoid]]), the radius ranges from a maximum ('''equatorial radius''', denoted ''a'') of about {{cvt|6378|km|mi}} to a minimum ('''polar radius''', denoted ''b'') of nearly {{cvt|6357|km|mi}}. 

A globally-average value is usually considered to be {{convert|6371|km|mi}} with a 0.3% variability (±10 km) for the following reasons.
The [[International Union of Geodesy and Geophysics]] (IUGG) provides three reference values: the ''mean radius'' (''R''{{sub|1}}) of three radii measured at two equator points and a pole; the ''authalic radius'', which is the radius of a sphere with the same surface area (''R''{{sub|2}}); and the ''volumetric radius'', which is the radius of a sphere having the same volume as the ellipsoid (''R''{{sub|3}}).<ref name="Moritz" /> All three values are about {{convert|6371|km|mi}}.

Other ways to define and measure the Earth's radius involve either the spheroid's [[radius of curvature]] or the actual [[topography]]. A few definitions yield values outside the range between the [[geographical pole|polar]] radius and [[equator]]ial radius because they account for localized effects.

A ''nominal Earth radius'' (denoted <math>\mathcal{R}^\mathrm N_\mathrm{E}</math>) is sometimes used as a [[unit of measurement]] in [[astronomy]] and [[geophysics]], a [[conversion factor]] used when expressing planetary properties as multiples or fractions of a constant terrestrial radius; if the choice between equatorial or polar radii is not explicit, the equatorial radius is to be assumed, as recommended by the [[International Astronomical Union]] (IAU).<ref name="IAU XXIX" />

==Introduction==
[[File:Earth_oblateness_to_scale.svg|thumb|300px|A scale diagram of the [[Flattening|oblateness]] of the 2003 [[IERS]] [[reference ellipsoid]], with north at the top. The light blue region is a circle.  The outer edge of the dark blue line is an [[ellipse]] with the same [[minor axis]] as the circle and the same [[Eccentricity (mathematics)#Ellipses|eccentricity]] as the Earth. The red line represents the [[Karman line]] {{cvt|100|km|mi}} above [[sea level]], while the yellow area denotes the [[apsis|altitude]] range of the [[International Space Station|ISS]] in [[low Earth orbit]].]]
{{main|Figure of the Earth|Earth ellipsoid|Reference ellipsoid}}

[[Earth's rotation]], internal density variations, and external [[tidal force]]s cause its shape to deviate systematically from a perfect sphere.<ref group=lower-alpha>For details see [[figure of the Earth]], [[geoid]], and [[Earth tide]].</ref> Local [[topography]] increases the variance, resulting in a surface of profound complexity. Our descriptions of Earth's surface must be simpler than reality in order to be tractable. Hence, we create models to approximate characteristics of Earth's surface, generally relying on the simplest model that suits the need.

Each of the models in common use involve some notion of the geometric [[radius]]. Strictly speaking, spheres are the only solids to have radii, but broader uses of the term ''radius'' are common in many fields, including those dealing with models of Earth. The following is a partial list of models of Earth's surface, ordered from exact to more approximate:
* The actual surface of Earth
* The [[geoid]], defined by [[mean sea level]] at each point on the real surface<ref group="lower-alpha">There is no single center to the geoid; it varies according to local [[Geodetic system|geodetic]] conditions. Where there is land, the geoid is generally below ground; it represents where the sea level would be if water could reach it from the ocean via an imaginary canal.</ref>
* A [[spheroid]], also called an [[ellipsoid]] of revolution, [[Geodetic system#Geodetic versus geocentric latitude|geocentric]] to model the entire Earth, or else [[Geodetic system#Geodetic versus geocentric latitude|geodetic]] for regional work<ref group=lower-alpha>In a geocentric ellipsoid, the center of the ellipsoid coincides with some computed center of Earth, and best models the earth as a whole. Geodetic ellipsoids are better suited to regional idiosyncrasies of the geoid. A partial surface of an ellipsoid gets fitted to the region, in which case the center and orientation of the ellipsoid generally do not coincide with the earth's center of mass or axis of rotation.</ref>
* A [[sphere]]

In the case of the geoid and ellipsoids, the fixed distance from any point on the model to the specified center is called ''"a radius of the Earth"'' or ''"the radius of the Earth at that point"''.<ref group=lower-alpha>The value of the radius is completely dependent upon the latitude in the case of an ellipsoid model, and nearly so on the geoid.</ref> It is also common to refer to any ''[[#Mean radii|mean radius]]'' of a spherical model as ''"the radius of the earth"''. When considering the Earth's real surface, on the other hand, it is uncommon to refer to a "radius", since there is generally no practical need. Rather, elevation above or below sea level is useful.

Regardless of the model, any of these ''geocentric'' radii falls between the polar minimum of about 6,357&nbsp;km and the equatorial maximum of about 6,378&nbsp;km (3,950 to 3,963&nbsp;mi). Hence, the Earth deviates from a perfect sphere by only a third of a percent, which supports the spherical model in most contexts and justifies the term "radius of the Earth". While specific values differ, the concepts in this article generalize to any major [[planet]].

===Physics of Earth's deformation===
{{further|Equatorial bulge}}

Rotation of a planet causes it to approximate an ''[[spheroid|oblate ellipsoid]]/spheroid'' with a bulge at the [[equator]] and flattening at the [[North Pole|North]] and [[South Pole]]s, so that the ''equatorial radius'' {{mvar|a}} is larger than the ''polar radius'' {{mvar|b}} by approximately {{mvar|aq}}. The ''oblateness constant'' {{mvar|q}} is given by
:<math>q=\frac{a^3 \omega^2}{GM},</math>
where {{mvar|ω}} is the [[angular frequency]], {{mvar|G}} is the [[gravitational constant]], and {{mvar|M}} is the mass of the planet.{{refn|This follows from the [[International Astronomical Union]] [[2006 definition of planet|definition]] rule (2): a planet assumes a shape due to [[hydrostatic equilibrium]] where [[gravity]] and [[centrifugal force]]s are nearly balanced.<ref>[http://www.iau2006.org/mirror/www.iau.org/iau0603/index.html IAU 2006 General Assembly: Result of the IAU Resolution votes] {{webarchive|url=https://web.archive.org/web/20061107022302/http://www.iau2006.org/mirror/www.iau.org/iau0603/index.html |date=2006-11-07 }}</ref>|group=lower-alpha}} For the Earth {{math|{{sfrac|1|''q''}} ≈ 289}}, which is close to the measured inverse [[flattening]] {{math|{{sfrac|1|''f''}} ≈ 298.257}}. Additionally, the bulge at the equator shows slow variations. The bulge had been decreasing, but since 1998 the bulge has increased, possibly due to redistribution of ocean mass via currents.<ref>[https://web.archive.org/web/20020810195620/http://www.gsfc.nasa.gov/topstory/20020801gravityfield.html Satellites Reveal A Mystery Of Large Change In Earth's Gravity Field ], Aug. 1, 2002, [[Goddard Space Flight Center]]. </ref>
[[Image:Lowresgeoidheight.jpg|400px|right]]
The variation in [[density]] and [[Crust (geology)|crustal]] thickness causes gravity to vary across the surface and in time, so that the mean sea level differs from the ellipsoid. This difference is the ''[[geoid]] height'', positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under {{convert|110|m|ft|abbr=on}} on Earth. The geoid height can change abruptly due to earthquakes (such as the [[2004 Indian Ocean earthquake|Sumatra-Andaman earthquake]]) or reduction in ice masses (such as [[Greenland]]).<ref>[https://archive.today/20120529015755/http://www.spaceref.com/news/viewpr.html?pid=18567 NASA's Grace Finds Greenland Melting Faster, 'Sees' Sumatra Quake], December 20, 2005, [[Goddard Space Flight Center]].</ref>

Not all deformations originate within the Earth. Gravitational attraction from the Moon or Sun can cause the Earth's surface at a given point to vary by tenths of a meter over a nearly 12-hour period (see [[Earth tide]]).

===Radius and local conditions===
[[File:Abu Reyhan Biruni-Earth Circumference.svg|thumb|[[Al-Biruni#Geodesy and geography|Al-Biruni]]'s (973 – {{c.|1050}}) method for calculation of the Earth's radius simplified measuring the circumference compared to taking measurements from two locations distant from each other.]]
Given local and transient influences on surface height, the values defined below are based on a "general purpose" model, refined as globally precisely as possible within {{convert|5|m|ft|abbr=on}} of reference ellipsoid height, and to within {{convert|100|m|ft|abbr=on}} of mean sea level (neglecting geoid height).

Additionally, the radius can be estimated from the curvature of the Earth at a point. Like a [[torus]], the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The corresponding [[Radius of curvature (applications)|radius of curvature]] depends on the location and direction of measurement from that point. A consequence is that a distance to the [[horizon|true horizon]] at the equator is slightly shorter in the north–south direction than in the east–west direction.

In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by [[Eratosthenes]], many models have been created. Historically, these models were based on regional topography, giving the best [[Figure of the Earth#Historical Earth ellipsoids|reference ellipsoid]] for the area under survey. As satellite [[remote sensing]] and especially the [[Global Positioning System]] gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole.

==Extrema: equatorial and polar radii{{anchor|Fixed radius|Equatorial radius|Polar radius}}==
The following radii are derived from the [[World Geodetic System]] 1984 ([[WGS-84]]) [[reference ellipsoid]].<ref name=tr8350_2>{{cite web|url=https://nsgreg.nga.mil/doc/view?i=4085 |title=Department of Defense World Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems |access-date=2018-10-17}}</ref> It is an idealized surface, and the Earth measurements used to calculate it have an uncertainty of ±2&nbsp;m in both the equatorial and polar dimensions.<ref name="NGA">{{cite web |url=http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf |title=Info |website=earth-info.nga.mil |access-date=2008-12-31 |archive-date=2020-08-04 |archive-url=https://web.archive.org/web/20200804185855/https://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf |url-status=dead }}</ref> Additional discrepancies caused by topographical variation at specific locations can be significant. When identifying the position of an observable location, the use of more precise values for WGS-84 radii may not yield a corresponding improvement in [[accuracy]].{{unclear-inline|date=April 2021}}

The value for the equatorial radius is defined to the nearest 0.1&nbsp;m in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1&nbsp;m, which is expected to be adequate for most uses. Refer to the WGS-84 ellipsoid if a more precise value for its polar radius is needed.

* The Earth's ''equatorial radius'' {{mvar|a}}, or [[semi-major axis]],<ref name="Snyder manual"/>{{rp|11}} is the distance from its center to the [[equator]] and equals {{convert|6378.1370|km|mi|sigfig=8|abbr=on}}.<ref>{{cite web|url=http://maia.usno.navy.mil/NSFA/NSFA_cbe.html#EarthRadius2009|title=Equatorial Radius of the Earth|date=2012|website=Numerical Standards for Fundamental Astronomy: Astronomical Constants : Current Best Estimates (CBEs)|publisher=IAU Division I Working Group|access-date=2016-08-10|url-status=dead|archive-url=https://web.archive.org/web/20160826200953/http://maia.usno.navy.mil/NSFA/NSFA_cbe.html#EarthRadius2009|archive-date=2016-08-26}}</ref> The equatorial radius is often used to compare Earth with other [[Planet#Attributes|planets]].
* The Earth's ''polar radius'' {{mvar|b}}, or [[semi-minor axis]]<ref name="Snyder manual"/>{{rp|11}} is the distance from its center to the North and South Poles, and equals {{convert|6356.7523|km|mi|sigfig=8|abbr=on}}.

==Location-dependent radii{{anchor|Radii with location dependence}}==

[[Image:EarthEllipRadii.svg|thumb|Three different radii as a function of Earth's latitude. {{mvar|R}} is the geocentric radius; {{mvar|M}} is the meridional radius of curvature; and {{mvar|N}} is the prime vertical radius of curvature.]]

===Geocentric radius{{anchor|Geocentric}}===
{{distinguish|Geocentric distance}}

The ''geocentric radius'' is the distance from the Earth's center to a point on the spheroid surface at [[geodetic latitude]] {{mvar|φ}}, given by the formula
<ref>{{cite book |title=Global Navigation Satellite Systems, Inertial Navigation, and Integration |author=Mohinder S. Grewal |author2= Angus P. Andrews |author3= Chris G. Bartone |page=512 |publisher=John Wiley & Sons |year=2020 |isbn=978-1-119-54783-9 |url=https://books.google.com/books?id=ppjDDwAAQBAJ&dq=%22geocentric+radius%22&pg=PA512 |edition=4}}</ref>
:<math>R(\varphi)=\sqrt{\frac{(a^2\cos\varphi)^2+(b^2\sin\varphi)^2}{(a\cos\varphi)^2+(b\sin\varphi)^2}},</math>
where {{mvar|a}} and {{mvar|b}} are, respectively, the equatorial radius and the polar radius.

The extrema geocentric radii on the ellipsoid coincide with the equatorial and polar radii.
They are [[Vertex (curve)|vertices]] of the ellipse and also coincide with minimum and maximum radius of curvature.

===Radii of curvature{{anchor|Radius of curvature|Curvature}}===
{{see also|Spheroid#Curvature}}

====Principal radii of curvature====

There are two [[Principal curvature|principal radii of curvature]]: along the meridional and prime-vertical [[normal section]]s.  

=====Meridional=====
In particular, the ''Earth's [[meridian (geography)|meridional]] radius of curvature'' (in the north–south direction) at {{mvar|φ}} is<ref name="Jekeli">{{Cite book|author=Christopher Jekeli|title=Geometric Reference Systems in Geodesy|url=https://kb.osu.edu/bitstream/handle/1811/77986/Geom_Ref_Sys_Geodesy_2016.pdf|access-date=13 May 2023|year=2016|publisher=Ohio State University, Columbus, Ohio}}</ref>
:<math>M(\varphi)=\frac{(ab)^2}{\big((a\cos\varphi)^2+(b\sin\varphi)^2\big)^\frac32}
=\frac{a(1-e^2)}{(1-e^2\sin^2\varphi)^\frac32}
=\frac{1-e^2}{a^2} N(\varphi)^3,</math>
where <math>e</math> is the [[Eccentricity (mathematics)|eccentricity]] of the earth.  This is the radius that [[Eratosthenes#Measurement of the Earth|Eratosthenes]] measured in his [[arc measurement]].

=====Prime vertical=====
[[Image:Geodetic latitude and the length of Normal.svg|thumb|The length PQ, called the ''prime vertical radius'', is <math>N(\phi).</math> The length IQ is equal to <math>e^2 N(\phi).</math> <math>R = (X, Y, Z).</math>]]

If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.<ref group=lower-alpha name=curvprim>East–west directions can be misleading. Point B, which appears due east from A, will be closer to the equator than A. Thus the curvature found this way is smaller than the curvature of a circle of constant latitude, except at the equator. West can be exchanged for east in this discussion.</ref>

This ''Earth's [[prime vertical|prime-vertical]] radius of curvature'', also called the ''Earth's transverse radius of curvature'', is defined perpendicular ([[orthogonal]]) to {{mvar|M}} at geodetic latitude {{mvar|φ}}<ref group=lower-alpha>{{mvar|N}} is defined as the radius of curvature in the plane that is normal to both the surface of the ellipsoid at, and the meridian passing through, the specific point of interest.</ref> and is<ref name="Jekeli"/>
:<math>N(\varphi)=\frac{a^2}{\sqrt{(a\cos\varphi)^2+(b\sin\varphi)^2}}
=\frac{a}{\sqrt{1-e^2\sin^2\varphi}}.</math>

''N'' can also be interpreted geometrically as the [[normal distance]] from the ellipsoid surface to the polar axis.<ref name="Bowring">{{cite journal
|title=Notes on the curvature in the prime vertical section
|first=B. R.| last=Bowring |journal=Survey Review |volume=29
|issue=226 |date=October 1987 |pages=195–196|doi=10.1179/sre.1987.29.226.195|bibcode=1987SurRv..29..195B }}
<!--https://www.tandfonline.com/doi/abs/10.1179/sre.1987.29.226.195--></ref>
The radius of a [[parallel of latitude]] is given by <math>p=N\cos(\varphi)</math>.<ref>{{cite book |last=Bomford |first=G. |title=Geodesy |title-link=Geodesy (book) |publisher=Oxford University Press |year=1952 |author-link=Guy Bomford}}</ref><ref>{{Cite book|author=Christopher Jekeli|title=Geometric Reference Systems in Geodesy|url=https://kb.osu.edu/bitstream/handle/1811/77986/Geom_Ref_Sys_Geodesy_2016.pdf|access-date=13 May 2023|year=2016|publisher=Ohio State University, Columbus, Ohio}}</ref>

=====Polar and equatorial radius of curvature=====
The ''Earth's meridional radius of curvature at the equator'' equals the meridian's [[semi-latus rectum]]:

:<math>M(0\text{°})=\frac{b^2}{a}=6,335.439 \text{ km.}</math>

The ''Earth's prime-vertical radius of curvature at the equator'' equals the equatorial radius, <math>N(0\text{°})=a</math> 

The ''Earth's polar radius of curvature'' (either meridional or prime-vertical) is <math>M(90\text{°})=N(90\text{°})=\frac{a^2}{b}=6,399.594 \text{ km.}</math>

=====Derivation=====
{{collapse top}}
The principal curvatures are the roots of Equation (125) in:<ref name="Lass">{{cite book
|last=Lass |first=Harry |title=Vector and Tensor Analysis |url=https://archive.org/details/vectortensoranal00hlas |url-access=limited |date=1950
|publisher=McGraw Hill Book Company, Inc. |pages=[https://archive.org/details/vectortensoranal00hlas/page/n83 71]–77|isbn=9780070365209 }}</ref>

:<math>(E G - F^2) \kappa^2 - (e G + g E - 2 f F) \kappa + (e g - f^2) = 0
= \det(A - \kappa B),</math>

where in the [[first fundamental form]] for a surface (Equation (112) in<ref name="Lass"></ref>):

:<math>ds^2 = \sum_{ij} a_{ij} dw^i dw^j = E \,d\varphi^2 + 2 F \,d\varphi \,d\lambda + G \,d\lambda^2,</math>

''E'', ''F'', and ''G'' are elements of the [[metric tensor]]:

:<math> A = a_{ij} = \sum_\nu \frac{\partial r^\nu}{\partial w^i} \frac{\partial r^\nu}{\partial w^j}
= \begin{bmatrix} E & F \\ F & G \end{bmatrix},</math>

<math>r = [r^1, r^2, r^3]^T = [x, y, z]^T</math>, <math>w^1 = \varphi</math>, <math>w^2 = \lambda,</math>

in the [[second fundamental form]] for a surface (Equation (123) in<ref name="Lass"></ref>):

:<math> 2 D = \sum_{ij} b_{ij} dw^i dw^j =  e \,d\varphi^2 + 2 f \,d\varphi \,d\lambda + g \,d\lambda^2,</math>

''e'', ''f'', and ''g'' are elements of the shape tensor:

:<math>B = b_{ij} = \sum_\nu n^\nu \frac{\partial ^2 r^\nu}{\partial w^i \partial w^j}
= \begin{bmatrix} e & f \\ f & g \end{bmatrix},</math>

<math>n = \frac{N}{|N|}</math> is the unit normal to the surface at <math>r</math>, and because
<math>\frac{\partial r}{\partial \varphi}</math> and <math>\frac{\partial r}{\partial \lambda}</math> are tangents to the surface,

:<math>N = \frac{\partial r}{\partial \varphi} \times \frac{\partial r}{\partial \lambda}</math>
is normal to the surface at <math>r</math>.

With <math>F = f = 0</math> for an oblate spheroid, the curvatures are

:<math>\kappa_1 = \frac{g}{G}</math> and <math>\kappa_2 = \frac{e}{E},</math>

and the principal radii of curvature are

:<math>R_1 = \frac{1}{\kappa_1}</math> and <math>R_2 = \frac{1}{\kappa_2}.</math>

The first and second radii of curvature correspond, respectively, to the Earth's meridional and prime-vertical radii of curvature.

Geometrically, the second fundamental form gives the distance from <math>r + dr</math> to the plane tangent at <math>r</math>.
{{collapse bottom}}

====Combined radii of curvature====

=====Azimuthal{{anchor|Directional}}=====
The Earth's ''azimuthal radius of curvature'', along an [[Earth normal section]] at an [[azimuth]] (measured clockwise from north) {{mvar|α}} and at latitude {{mvar|φ}}, is derived from [[Euler's theorem (differential geometry)|Euler's curvature formula]] as follows:<ref name=Torge/>{{rp|97}}
:<math>R_\mathrm{c}=\frac{1}{\dfrac{\cos^2\alpha}{M}+\dfrac{\sin^2\alpha}{N}}.</math>

=====Non-directional=====
It is possible to combine the principal radii of curvature above in a non-directional manner.

{{anchor|Gaussian}}{{anchor|Gaussian radius of curvature}}The ''Earth's [[Gaussian curvature|Gaussian radius of curvature]]'' at latitude {{mvar|φ}} is<ref name=Torge>{{cite book|url=https://books.google.com/books?id=pFO6VB_czRYC&pg=PA98 |title=Geodesy|isbn=9783110170726|last1=Torge|first1=Wolfgang|year=2001|publisher=Walter de Gruyter }}</ref>
:<math>R_\text{a}(\varphi) = \frac{1}{\sqrt{K}} = \frac{1}{2\pi}\int_{0}^{2\pi}R_\text{c}(\alpha)\,d\alpha = \sqrt{MN} = \frac{a^2b}{(a\cos\varphi)^2+(b\sin\varphi)^2}
= \frac{a\sqrt{1-e^2}}{1-e^2\sin^2\varphi},</math>
where ''K'' is the ''Gaussian curvature'', <math>K = \kappa_1\,\kappa_2 = \frac{\det B}{\det A}.</math>

{{anchor|Mean radius of curvature}}The ''Earth's [[Mean curvature|mean radius of curvature]]'' at latitude {{mvar|φ}} is<ref name=Torge />{{rp|97}}
:<math>R_\text{m} = \frac{2}{\dfrac{1}{M} + \dfrac{1}{N}}.</math>

==Global radii{{anchor|Mean radii}}==
The Earth can be modeled as a sphere in many ways. This section describes the common ways. The various radii derived here use the notation and dimensions noted above for the Earth as derived from the [[WGS-84]] ellipsoid;<ref name=tr8350_2 /> namely,

:''Equatorial radius'': {{mvar|a}} = ({{val|6378.1370|u=km}})
:''Polar radius'': {{mvar|b}} = ({{val|6356.7523|u=km}})

A sphere being a gross approximation of the spheroid, which itself is an approximation of the geoid, units are given here in kilometers rather than the millimeter resolution appropriate for geodesy.

===Arithmetic mean radius===
[[File:WGS84_mean_Earth_radius.svg|thumb|Equatorial (''a''), polar (''b'') and arithmetic mean Earth radii as defined in the 1984 [[World Geodetic System]] revision (not to scale)]]

In geophysics, the [[International Union of Geodesy and Geophysics]] (IUGG) defines the ''Earth's [[arithmetic mean]] radius'' (denoted {{math|''R''<sub>1</sub>}}) to be<ref name="Moritz">Moritz, H. (1980). [https://geodesy.geology.ohio-state.edu/course/refpapers/00740128.pdf ''Geodetic Reference System 1980''] {{Webarchive|url=https://web.archive.org/web/20160220054607/https://geodesy.geology.ohio-state.edu/course/refpapers/00740128.pdf |date=2016-02-20 }}, by resolution of the XVII General Assembly of the IUGG in Canberra.</ref>
:<math>R_1 = \frac{2a+b}{3}.</math>
The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid.<ref name="Moritz2000">{{cite journal |last=Moritz |first=H. |date=March 2000 |title=Geodetic Reference System 1980 |journal=Journal of Geodesy |volume=74 |issue=1 |pages=128–133 |doi=10.1007/s001900050278 |bibcode = 2000JGeod..74..128. |s2cid=195290884 }}</ref>
For Earth, the arithmetic mean radius is published by IUGG and [[National Geospatial-Intelligence Agency|NGA]] as {{convert|6371.0087714|km|mi|abbr=on}}.<ref name="IAU XXIX"/><ref name="NGA"/>

===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>

===Volumetric radius===
Another spherical model is defined by the ''Earth's volumetric radius'', which is the radius of a sphere of volume equal to the ellipsoid. The [[IUGG]] denotes the volumetric radius as {{math|''R''<sub>3</sub>}}.<ref name="Moritz"/>
:<math>R_3 = \sqrt[3]{a^2b}.</math>
For Earth, the volumetric radius equals {{convert|6,371.0008|km|mi|abbr=on}}.<ref name=Moritz2000/>

===Rectifying radius===
{{see also|Quarter meridian|Rectifying latitude}}
Another global radius is the ''Earth's rectifying radius'', giving a sphere with circumference equal to the [[circumference|perimeter]] of the ellipse described by any polar cross section of the ellipsoid. This requires an [[Circumference#Ellipse|elliptic integral]] to find, given the polar and equatorial radii:
:<math>M_\text{r} = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \sqrt{a^2 \cos^2\varphi + b^2 \sin^2\varphi} \,d\varphi.</math>

The rectifying radius is equivalent to the meridional mean, which is defined as the average value of {{mvar|M}}:<ref name="Snyder manual"/>
:<math>M_\text{r} = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} M(\varphi) \,d\varphi.</math>

For integration limits of [0,{{sfrac|{{pi}}|2}}], the integrals for rectifying radius and mean radius evaluate to the same result, which, for Earth, amounts to {{convert|6,367.4491|km|mi|abbr=on}}.

The meridional mean is well approximated by the semicubic mean of the two axes,{{cn|date=November 2020}}
:<math>M_\text{r} \approx \left(\frac{a^\frac32 + b^\frac32}{2}\right)^\frac23,</math>

which differs from the exact result by less than {{convert|1|um|sigfig=1|abbr=on}}; the mean of the two axes,
:<math>M_\text{r} \approx \frac{a + b}{2},</math>

about {{convert|6,367.445|km|mi|abbr=on}}, can also be used.

==Topographical radii==
{{see also|Earth#Size and shape}}

The mathematical expressions above apply over the surface of the ellipsoid.
The cases below considers Earth's [[topography]], above or below a [[reference ellipsoid]].
As such, they are ''topographical [[geocentric distance]]s'', ''R''<sub>t</sub>, which depends not only on latitude.

===Topographical extremes===
* Maximum ''R''<sub>t</sub>: the summit of [[Chimborazo]] is {{convert|6384.4|km|mi|abbr=on}} from the Earth's center.
* Minimum ''R''<sub>t</sub>: the floor of the [[Arctic Ocean]] is {{convert|6,352.8|km|mi|abbr=on}} from the Earth's center.<ref name=extrema>{{cite web |url=http://guam.discover-theworld.com/Country_Guide.aspx?id=96&entry=Mariana+Trench |title=Discover-TheWorld.com – Guam – POINTS OF INTEREST – Don't Miss – Mariana Trench |publisher=Guam.discover-theworld.com |date=1960-01-23 |access-date=2013-09-16 |url-status=dead |archive-url=https://web.archive.org/web/20120910021956/http://guam.discover-theworld.com/Country_Guide.aspx?id=96&entry=Mariana+Trench |archive-date=2012-09-10 }}</ref>

===Topographical global mean===
The ''topographical mean geocentric distance'' averages elevations everywhere, resulting in a value {{val|230|u=m}} larger than the [[#Arithmetic mean radius|IUGG mean radius]], the [[authalic radius]], or the [[#Volumetric radius|volumetric radius]]. This topographical average is {{convert|6371.230|km|mi|abbr=on}} with uncertainty of {{convert|10|m|ft|abbr=on}}.<ref name="chambat">{{cite journal
| title = Mean radius, mass, and inertia for reference Earth models
| author1 = Frédéric Chambat
| author2 = Bernard Valette
| journal = Physics of the Earth and Planetary Interiors
| date = 2001
| volume = 124
| issue = 3–4
| pages = 234–253
| doi = 10.1016/S0031-9201(01)00200-X
| bibcode = 2001PEPI..124..237C
| url = http://frederic.chambat.free.fr/geophy/inertie_pepi01/article.pdf
| access-date = 18 November 2017
| archive-date = 30 July 2020
| archive-url = https://web.archive.org/web/20200730215818/http://frederic.chambat.free.fr/geophy/inertie_pepi01/article.pdf
| url-status = dead
}}</ref>

==Derived quantities: diameter, circumference, arc-length, area, volume {{anchor|Derived quantities|Derived geometric quantities}}==

{{anchor|Diameter}}'''Earth's [[diameter]]''' is simply twice Earth's radius; for example, ''equatorial diameter'' (2''a'') and ''polar diameter'' (2''b''). For the WGS84 ellipsoid, that's respectively:
: {{math|1=2''a'' = {{convert|12,756.2740|km|mi|sigfig=8|abbr=on}}}},
: {{math|1=2''b'' = {{convert|12,713.5046|km|mi|sigfig=8|abbr=on}}}}.

''[[Earth's circumference]]'' equals the [[perimeter]] length. The ''equatorial circumference'' is simply the [[circle perimeter]]: ''C''<sub>e</sub> = 2''πa'', in terms of the equatorial radius ''a''. The ''polar circumference'' equals ''C''<sub>p</sub> = 4''m''<sub>p</sub>, four times the [[quarter meridian]] ''m''<sub>p</sub> = ''aE''(''e''), where the polar radius ''b'' enters via the eccentricity ''e'' = (1 − ''b''<sup>2</sup>/''a''<sup>2</sup>)<sup>0.5</sup>; see [[Ellipse#Circumference]] for details.

[[Arc length]] of more general [[surface curve]]s, such as [[meridian arc]]s and [[Earth geodesics|geodesics]], can also be derived from Earth's equatorial and polar radii.

Likewise for [[surface area]], either based on a [[map projection]] or a [[geodesic polygon]].

{{anchor|Volume}}'''Earth's volume''', or that of the reference ellipsoid, is <math>V = \tfrac{4}{3}\pi a^2 b.</math> Using the parameters from [[WGS84]] ellipsoid of revolution, {{nowrap|1=''a'' = 6,378.137 km}} and {{nowrap|1=''b'' = {{val|6356.7523142|u=km}}}}, {{nowrap|1=''V'' = {{convert|1.08321E12|km3|mi3|abbr=on}}}}.<ref name="earth_fact_sheet">{{citation | last = Williams | first = David R. | date = 2004-09-01 | url = http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html | title = Earth Fact Sheet | publisher = [[NASA]] | access-date = 2007-03-17 }}</ref>

==Nominal radii{{anchor|Nominal|Nominal radius}}==
In astronomy, the [[International Astronomical Union]] denotes the ''nominal equatorial Earth radius'' as <math>\mathcal{R}^\text{N}_\text{eE}</math>, which is defined to be exactly {{convert|6378.1|km|mi|abbr=on}}.<ref name = "IAU XXIX"/>{{rp|3}} The ''nominal polar Earth radius'' is defined exactly as <math>\mathcal{R}^\text{N}_\text{pE}</math> = {{convert|6356.8|km|mi|abbr=on}}. These values correspond to the zero [[Earth tide]] convention. Equatorial radius is conventionally used as the nominal value unless the polar radius is explicitly required.<ref name = "IAU XXIX"/>{{rp|4}}
The nominal radius serves as a [[unit of length]] for [[Astronomical system of units|astronomy]].
(The notation is defined such that it can be easily generalized for other [[Planet#Size and shape|planets]]; e.g., <math>\mathcal{R}^\text{N}_\text{pJ}</math> for the nominal polar [[Jupiter radius]].)

==Published values==
This table summarizes the accepted values of the Earth's radius.

{| class ="wikitable sortable"
! Agency
! Description
! Value (in meters)
! Ref
|-
| [[International Astronomical Union|IAU]]
| nominal "zero tide" equatorial
| {{val|6378100}}
| {{r|IAU XXIX}}
|-
| IAU
| nominal "zero tide" polar
| {{val|6356800}}
| {{r|IAU XXIX}}
|-
| [[International Union of Geodesy and Geophysics|IUGG]]
| equatorial radius
| {{val|6378137}}
| {{r|Moritz}}
|-
| IUGG
| semiminor axis (''b'')
| {{val|6356752.3141}}
| {{r|Moritz}}
|-
| IUGG
| polar radius of curvature (''c'')
| {{val|6399593.6259}}
| {{r|Moritz}}
|-
| IUGG
| mean radius (''R<sub>1</sub>'')
| {{val|6371008.7714}}
| {{r|Moritz}}
|-
| IUGG
| radius of sphere of same surface (''R<sub>2</sub>'')
| {{val|6371007.1810}}
| {{r|Moritz}}
|-
| IUGG
| radius of sphere of same volume (''R<sub>3</sub>'')
| {{val|6371000.7900}}
| {{r|Moritz}}
|-
| [[National Geospatial-Intelligence Agency|NGA]]
| [[WGS-84]] ellipsoid, semi-major axis (''a'')
| {{val|6378137.0}}
| {{r|tr8350_2}}
|-
| NGA
| WGS-84 ellipsoid, semi-minor axis (''b'')
| {{val|6356752.3142}}
| {{r|tr8350_2}} 
|-
| NGA
| WGS-84 ellipsoid, polar radius of curvature (''c'')
| {{val|6399593.6258}}
| {{r|tr8350_2}} 
|-
| NGA
| WGS-84 ellipsoid, Mean radius of semi-axes (''R<sub>1</sub>'')
| {{val|6371008.7714}}
| {{r|tr8350_2}} 
|-
| NGA
| WGS-84 ellipsoid, Radius of Sphere of Equal Area (''R<sub>2</sub>'')
| {{val|6371007.1809}}
| {{r|tr8350_2}} 
|-
| NGA
| WGS-84 ellipsoid, Radius of Sphere of Equal Volume (''R<sub>3</sub>'')
| {{val|6371000.7900}}
| {{r|tr8350_2}} 
|-
|
| [[GRS 80]] semi-major axis (''a'')
| {{val|6378137.0}}
|
|-
|
| [[GRS 80]] semi-minor axis (''b'')
| {{val|6356752.314140|p=≈}}
|
|-
|
| Spherical Earth Approx. of Radius (''R<sub>E</sub>'')
| {{val|6366707.0195}}
| <ref name="Phillips">{{cite book |last1=Phillips |first1=Warren |title=Mechanics of Flight |date=2004 |publisher=John Wiley & Sons, Inc. |isbn=0471334588 |page=923}}</ref>
|-
|
| meridional radius of curvature at the equator
| {{val|6335439}}
|
|-
| 
| Maximum (the summit of Chimborazo)
| {{val|6384400}}
| {{r|extrema}}
|-
| 
| Minimum (the floor of the Arctic Ocean)
| {{val|6352800}} 
| {{r|extrema}}
|-
| 
| Average distance from center to surface
| {{val|6371230|10}}
| {{r|chambat}}
|}

==History==
{{See also|History of geodesy|Spherical Earth#History|Earth's circumference#History|Meridian arc#History}}

The first published reference to the Earth's size appeared around 350&nbsp;[[Anno Domini|BC]], when [[Aristotle]] reported in his book ''[[On the Heavens]]''<ref>{{cite book|last=Aristotle|author-link=Aristotle|title=On the Heavens|volume=Book II 298 B|url=http://classics.mit.edu/Aristotle/heavens.html|access-date=5 November 2017}}</ref> that mathematicians had guessed the circumference of the Earth to be 400,000 [[Stadia (length)|stadia]]. Scholars have interpreted Aristotle's figure to be anywhere from highly accurate<ref>
{{ cite journal
| journal = The Classical Journal
| title = On the Science of the Egyptians and Chaldeans, Part I
| volume = 16
| page = 159
| first = William
| last = Drummond
| year = 1817
}}</ref> to almost double the true value.<ref>
{{Cite EB1911|wstitle= Earth, Figure of the |volume= 8 |last1= Clarke |first1= Alexander Ross |author1-link= Alexander Ross Clarke |last2= Helmert |first2= Friedrich Robert |author2-link= Friedrich Robert Helmert | pages = 801&ndash;813 }}</ref> The first known scientific measurement and calculation of the circumference of the Earth was performed by [[Eratosthenes]] in about 240&nbsp;BC. Estimates of the error of Eratosthenes's measurement range from 0.5% to 17%.<ref>
{{cite encyclopedia
| encyclopedia = Britannica.com
| title = Eratosthenes, the Greek Scientist
| url = https://www.britannica.com/biography/Eratosthenes
| year = 2016
}}</ref> For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant.

Around 100&nbsp;BC, [[Posidonius of Apamea]] recomputed Earth's radius, and found it to be close to that by Eratosthenes,<ref>Posidonius, [http://www.attalus.org/translate/poseidonius.html#202.K fragment 202]</ref> but later [[Strabo]] incorrectly attributed him a value about 3/4 of the actual size.<ref>Cleomedes ([http://adsabs.harvard.edu/abs/1975QJRAS..16..152F in Fragment 202]) stated that if the distance is measured by some other number the result will be different, and using 3,750 instead of 5,000 produces this estimation: 3,750 x 48 = 180,000; see Fischer I., (1975), ''Another Look at Eratosthenes' and Posidonius' Determinations of the Earth's Circumference'', Ql. J. of the Royal Astron. Soc., Vol. 16, p. 152.</ref> [[Claudius Ptolemy]] around 150&nbsp;[[Anno Domini|AD]] gave empirical evidence supporting a [[spherical Earth]],<ref>{{Cite book|title=Early astronomy|last=Thurston|first=Hugh|publisher=Springer-Verlag New York|year=1994|isbn=0-387-94107-X|location=New York|page=138}}</ref> but he accepted the lesser value attributed to Posidonius. His highly influential work, the ''[[Almagest]]'',<ref>{{Cite web |title=Almagest – Ptolemy (Elizabeth) |url=https://projects.iq.harvard.edu/predictionx/almagest-ptolemy-elizabeth |access-date=2022-11-05 |website=projects.iq.harvard.edu |language=en}}</ref> left no doubt among medieval scholars that Earth is spherical, but they were wrong about its size.

By 1490, [[Christopher Columbus]] believed that traveling 3,000 miles west from the west coast of the [[Iberian Peninsula]] would let him reach the eastern coasts of [[Asia]].<ref>[[John Freely]], [https://books.google.com/books?id=MfhjAAAAQBAJ ''Before Galileo: The Birth of Modern Science in Medieval Europe''] (2013), {{ISBN|978-1468308501}}</ref> However, the 1492 enactment of that voyage [[Voyages of Christopher Columbus| brought his fleet to the Americas]]. The [[Magellan expedition]] (1519–1522), which was the first [[circumnavigation]] of the World, soundly demonstrated the sphericity of the Earth,<ref>{{Cite book|author=Nancy Smiler Levinson|title=Magellan and the First Voyage Around the World|url=https://books.google.com/books?id=1PbBzjBuW8IC&pg=PA39|access-date=31 July 2010|year=2001|publisher=Houghton Mifflin Harcourt|isbn=978-0-395-98773-5}}</ref> and affirmed the original measurement of {{convert|40000|km|mi|abbr=on}} by Eratosthenes.

Around 1690, [[Isaac Newton]] and [[Christiaan Huygens]] argued that Earth was closer to an [[Spheroid#Oblate spheroids | oblate spheroid]] than to a sphere. However, around 1730, [[Jacques Cassini]] argued for a [[Spheroid#Prolate spheroids | prolate spheroid]] instead, due to different interpretations of the [[Newtonian mechanics]] involved.<ref>{{cite book |last=Cassini |first=Jacques |title=Méthode de déterminer si la terre est sphérique ou non |date=1738 |url=https://bibnum.obspm.fr/items/show/41044#?c=0&m=0&s=0&cv=0&z=-0.0556%2C-0.339%2C1.1111%2C1.9323 |language=fr |access-date=2023-02-09 |archive-date=2018-01-27 |archive-url=https://web.archive.org/web/20180127004440/https://bibnum.obspm.fr/items/show/41044#?c=0&m=0&s=0&cv=0&z=-0.0556%2C-0.339%2C1.1111%2C1.9323 |url-status=dead }}</ref> To settle the matter, the [[French Geodesic Mission]] (1735–1739) measured one degree of [[latitude]] at two locations, one near the [[Arctic Circle]] and the other near the [[equator]]. The expedition found that Newton's conjecture was correct:<ref>{{Cite web|url=https://gallica.bnf.fr/ark:/12148/bpt6k5470853s|title=La Vie des sciences|last=Levallois|first=Jean-Jacques|date=1986|website=Gallica|pages=277–284, 288|access-date=2019-05-22}}</ref> the Earth is flattened at the [[Geographical pole|poles]] due to rotation's [[centrifugal force]].

== See also ==
{{Div col|colwidth=24em}}
* [[Earth's circumference]]
* [[Earth mass]]
* [[Effective Earth radius]]
* [[Geodesy]]
* [[Geographical distance]]
* [[Osculating sphere]]
* [[History of geodesy]]
* [[Planetary radius]]
{{Div col end}}

==Notes==
{{notes}}

==References==
{{Reflist}}

==External links==
{{EB1911 Poster|Earth, Figure of the}}
* {{cite web|last=Merrifield|first=Michael R.|title=<math>R_\oplus</math> The Earth's Radius (and exoplanets)|url=http://www.sixtysymbols.com/videos/earthradius.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2010}}

{{Commons category|Earth radius}}
{{Units of length used in Astronomy}}

{{DEFAULTSORT:Earth Radius}}
[[Category:Earth|Radius]]
[[Category:Planetary science]]
[[Category:Planetary geology]]
[[Category:Radii]]