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Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...
to a point on or near its surface. Approximating the figure of Earth by an
Earth spheroid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
, the radius ranges from a maximum of nearly (equatorial radius, denoted ''a'') to a minimum of nearly (polar radius, denoted ''b''). A ''nominal Earth radius'' is sometimes used as a
unit of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a mult ...
in
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and geophysics, which is recommended by the
International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreach ...
to be the equatorial value. A globally-average value is usually considered to be 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) 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); and the ''volumetric radius'', which is the radius of a sphere having the same volume as the ellipsoid (R). All three values are about . Other ways to define and measure the Earth radius involve the radius of curvature. A few definitions yield values outside the range between polar radius and
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
ial radius because they include local or geoidal
topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary s ...
or because they depend on abstract geometrical considerations.


Introduction

Earth's rotation, internal density variations, and external tidal forces cause its shape to deviate systematically from a perfect sphere.For details see
figure of the Earth Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...
, geoid, and Earth tide.
Local
topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary s ...
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 In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
. 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 There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...
at each point on the real surfaceThere is no single center to the geoid; it varies according to local geodetic conditions. * A spheroid, also called an ellipsoid of revolution, geocentric to model the entire Earth, or else geodetic for regional workIn 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. * A
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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"''.The value of the radius is completely dependent upon the latitude in the case of an ellipsoid model, and nearly so on the geoid. It is also common to refer to any '' 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 radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (3,950 to 3,963 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 A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a ...
.


Physics of Earth's deformation

Rotation of a planet causes it to approximate an '' oblate ellipsoid/spheroid'' with a bulge at the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
and flattening at the
North North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating direction or geography. Etymology The word ''no ...
and
South Pole The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
s, so that the ''equatorial radius'' is larger than the ''polar radius'' by approximately . The ''oblateness constant'' is given by :q=\frac\,, where is the angular frequency, is the gravitational constant, and is the mass of the planet. For the Earth , which is close to the measured inverse flattening . 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. The variation in
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
and 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 on Earth. The geoid height can change abruptly due to earthquakes (such as the
Sumatra-Andaman earthquake An earthquake and a tsunami, known as the Boxing Day Tsunami and, by the scientific community, the Sumatra–Andaman earthquake, occurred at 07:58:53 local time ( UTC+7) on 26 December 2004, with an epicentre off the west coast of northern Su ...
) or reduction in ice masses (such as Greenland). 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

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 of reference ellipsoid height, and to within 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 depends on the location and direction of measurement from that point. A consequence is that a distance to the 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 reference ellipsoid for the area under survey. As satellite
remote sensing Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ or on-site observation. The term is applied especially to acquiring information about Ear ...
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

The following radii are derived from the
World Geodetic System The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also describ ...
1984 ( WGS-84) reference ellipsoid. It is an idealized surface, and the Earth measurements used to calculate it have an uncertainty of ±2 m in both the equatorial and polar dimensions. 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. The value for the equatorial radius is defined to the nearest 0.1 m in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1 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'' , or semi-major axis, is the distance from its center to the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
and equals . The equatorial radius is often used to compare Earth with other planets. * The Earth's ''polar radius'' , or semi-minor axis, is the distance from its center to the North and South Poles, and equals .


Location-dependent radii


Geocentric radius

The ''geocentric radius'' is the distance from the Earth's center to a point on the spheroid surface at
geodetic latitude Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''. They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as g ...
: :R(\varphi)=\sqrt where and 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 vertices of the ellipse and also coincide with minimum and maximum radius of curvature.


Radii of curvature


Principal radii of curvature

There are two principal radii of curvature: along the meridional and prime-vertical normal sections.


=Meridional

= In particular, the ''Earth's meridional radius of curvature'' (in the north–south direction) at is: :M(\varphi)=\frac =\frac =\frac N(\varphi)^3\,. where e is the eccentricity of the earth. This is the radius that Eratosthenes measured in his arc measurement.


=Prime vertical

= If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.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. This ''Earth's prime-vertical radius of curvature'', also called the ''Earth's transverse radius of curvature'', is defined perpendicular (
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
) to at geodetic latitude is: 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. :N(\varphi)=\frac =\frac\,. ''N'' can also be interpreted geometrically as the normal distance from the ellipsoid surface to the polar axis. The radius of a parallel of latitude is given by p=N\cos(\varphi).


=Polar and equatorial radius of curvature

= The ''Earth's meridional radius of curvature at the equator'' equals the meridian's semi-latus rectum: :=6,335.439 km The ''Earth's prime-vertical radius of curvature at the equator'' equals the equatorial radius, . The ''Earth's polar radius of curvature'' (either meridional or prime-vertical) is: :=6,399.594 km


=Derivation

= The principal curvatures are the roots of Equation (125) in: :(E G - F^2)\, \kappa^2 - ( e G + g E - 2 f F )\, \kappa + ( e g - f^2 ) = 0 =\det(A - \kappa\,B), where in the first fundamental form for a surface (Equation (112) in ): :ds^2 = \sum_ a_ dw^i dw^j = E\, d\varphi^2 + 2 F\, d\varphi\, d\lambda + G\, d\lambda^2, E, F, and G are elements of the metric tensor: : A = a_ = \sum_ \frac \frac = \left \begin E & F \\ F & G \end \right r = ^1, r^2, r^3T = x, y, zT, w^1 = \varphi, w^2 = \lambda, in the second fundamental form for a surface (Equation (123) in ): : 2 D = \sum_ b_ dw^i dw^j = e\, d\varphi^2 + 2 f\, d \varphi\, d \lambda + g\, d \lambda^2, e, f, and g are elements of the shape tensor: :B = b_ = \sum_ n^ \frac = \left \begin e & f \\ f & g \end \right n = \frac is the unit normal to the surface at r, and because \frac and \frac are tangents to the surface, :N = \frac \times \frac is normal to the surface at r. With F = f = 0 for an oblate spheroid, the curvatures are :\kappa_1 = \frac and \kappa_2 = \frac\,, and the principal radii of curvature are :R_1 = \frac and R_2 = \frac. 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 r + dr to the plane tangent at r.


Combined radii of curvature


=Azimuthal

= The Earth's ''azimuthal radius of curvature'', along an Earth normal section at an
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...
(measured clockwise from north) and at latitude , is derived from Euler's curvature formula as follows: :R_\mathrm=\frac\,.


=Non-directional

= It is possible to combine the principal radii of curvature above in a non-directional manner. The ''Earth's Gaussian radius of curvature'' at latitude is: :R_\mathrm(\varphi)= \frac = \frac\int_^R_\mathrm(\alpha)\,d\alpha\,=\sqrt=\frac =\frac\,. Where ''K'' is the ''Gaussian curvature'', K = \kappa_1\,\kappa_2 = \frac. The ''Earth's mean radius of curvature'' at latitude is: :R_\mathrm=\frac\,\!


Global 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; namely, :''Equatorial radius'': = () :''Polar radius'': = () 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.


Nominal radius

In astronomy, the
International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreach ...
denotes the ''nominal equatorial Earth radius'' as \mathcal^\mathrm N_\mathrm, which is defined to be . The ''nominal polar Earth radius'' is defined as \mathcal^\mathrm N_\mathrm = . 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. The nominal radius serves as a unit of length for
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. (The notation is defined such that it can be easily generalized for other planets; e.g., \mathcal^\mathrm N_\mathrm for the nominal polar Jupiter radius.)


Arithmetic mean radius

In geophysics, the International Union of Geodesy and Geophysics (IUGG) defines the ''Earth's
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
radius'' (denoted ) to beMoritz, H. (1980)
''Geodetic Reference System 1980''
by resolution of the XVII General Assembly of the IUGG in Canberra.
:R_1 = \frac\,\! The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid. For Earth, the arithmetic mean radius is .


Authalic radius

''Earth's authalic radius'' (meaning "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 . A closed-form solution exists for a spheroid:Snyder, J.P. (1987). ''Map Projections – A Working Manual (US Geological Survey Professional Paper 1395)'' p. 16–17. Washington D.C: United States Government Printing Office. :R_2 =\sqrt =\sqrt =\sqrt\,, where and is the surface area of the spheroid. For the Earth, the authalic radius is . The authalic radius R_2 also corresponds to the ''radius of (global) mean curvature'', obtained by averaging the Gaussian curvature, K, over the surface of the ellipsoid. Using the Gauss-Bonnet theorem, this gives : \fracA = \fracA = \frac1.


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 . :R_3=\sqrt ,. For Earth, the volumetric radius equals .


Rectifying radius

Another global radius is the ''Earth's rectifying radius'', giving a sphere with circumference equal to the perimeter of the ellipse described by any polar cross section of the ellipsoid. This requires an elliptic integral to find, given the polar and equatorial radii: :M_\mathrm=\frac\int_^\sqrt\,d\varphi\,. The rectifying radius is equivalent to the meridional mean, which is defined as the average value of : :M_\mathrm=\frac\int_^\!M(\varphi)\,d\varphi\,. For integration limits of , the integrals for rectifying radius and mean radius evaluate to the same result, which, for Earth, amounts to . The meridional mean is well approximated by the semicubic mean of the two axes, :M_\mathrm\approx\left(\frac\right)^\frac23\,, which differs from the exact result by less than ; the mean of the two axes, :M_\mathrm\approx\frac\,, about , can also be used.


Topographical radii

The mathematical expressions above apply over the surface of the ellipsoid. The cases below considers Earth's
topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary s ...
, above or below a reference ellipsoid. As such, they are ''topographical geocentric distances'', ''Rt'', which depends not only on latitude.


Topographical extremes

* Maximum ''Rt'': the summit of Chimborazo is from the Earth's center. * Minimum ''Rt'': the floor of the
Arctic Ocean The Arctic Ocean is the smallest and shallowest of the world's five major oceans. It spans an area of approximately and is known as the coldest of all the oceans. The International Hydrographic Organization (IHO) recognizes it as an ocean, ...
is approximately from the Earth's center.


Topographical global mean

The ''topographical mean geocentric distance'' averages elevations everywhere, resulting in a value larger than the IUGG mean radius, the
authalic radius Earth radius (denoted as ''R''🜨 or R_E) 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, the radius ranges from a maximum of nearly (equatorial radius, deno ...
, or the volumetric radius. This topographical average is with uncertainty of .


Derived quantities: diameter, circumference, arc-length, area, volume

Earth's
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
is simply twice Earth's radius; for example, ''equatorial diameter'' (2''a'') and ''polar diameter'' (2''b''). For the WGS84 ellipsoid, that's respectively: *, *. '' Earth's circumference'' equals the perimeter length. The ''equatorial circumference'' is simply the circle perimeter: ''Ce''=2''πa'', in terms of the equatorial radius, ''a''. The ''polar circumference'' equals ''Cp''=4''mp'', four times the quarter meridian ''mp''=''aE''(''e''), where the polar radius ''b'' enters via the eccentricity, ''e''=(1−''b''2/''a''2)0.5; see Ellipse#Circumference for details.
Arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services ...
of more general
surface curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
s, such as
meridian arc In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to ...
s and 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 The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodesi ...
. Earth's volume, or that of the reference ellipsoid, is . Using the parameters from WGS84 ellipsoid of revolution, and , .


Published values

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


History

The first published reference to the Earth's size appeared around 350 BC, when
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
reported in his book '' On the Heavens'' that mathematicians had guessed the circumference of the Earth to be 400,000 stadia. Scholars have interpreted Aristotle's figure to be anywhere from highly accurate to almost double the true value. The first known scientific measurement and calculation of the circumference of the Earth was performed by Eratosthenes in about 240 BC. Estimates of the accuracy of Eratosthenes's measurement range from 0.5% to 17%. For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant.


See also

* Earth's circumference * Earth mass *
Effective Earth radius Line-of-sight propagation is a characteristic of electromagnetic radiation or acoustic wave propagation which means waves travel in a direct path from the source to the receiver. Electromagnetic transmission includes light emissions traveli ...
*
Geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), Earth rotation, orientation in space, and Earth's gravity, gravity. The field also incorporates studies of how these properti ...
*
Geographical distance Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitud ...
* Osculating sphere *
History of geodesy The history of geodesy deals with the historical development of measurements and representations of the Earth. The corresponding scientific discipline, ''geodesy'' ( /dʒiːˈɒdɪsi/), began in pre-scientific antiquity and blossomed during the ...
* Template:Earth radius *
Planetary radius A planetary coordinate system is a generalization of the geographic coordinate system and the geocentric coordinate system for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the ''selen ...


Notes


References


External links

* {{DEFAULTSORT:Earth Radius
Radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
Planetary science Planetary geology