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nload as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from radius of the earth for its historical development see spherical earth for its determination see arc measurement distance from the earth surface to a point near its center earth radius equatorial a polar b and arithmetic mean earth radii as defined in the 1984 world geodetic system revision not to scale other names terrestrial radius common symbols r r e a b a e b e r e e r p e si unit meters in si base units m behaviour under coord transformation scalar dimension l displaystyle mathsf l value equatorial radius a 6 378 137 0 m polar radius b 6 356 752 3 m nominal earth radius cross section of earth s interior general information unit system astronomy geophysics unit of distance symbol r e n displaystyle mathcal r _ mathrm e mathrm n r e e n displaystyle mathcal r _ e mathrm e mathrm n r p e n displaystyle mathcal r _ p mathrm e mathrm n conversions 1 r e n displaystyle mathcal r _ mathrm e mathrm n in is equal to si base unit 6 3781 10 6 m 1 metric system 6357 to 6378 km english units 3950 to 3963 mi geodesy fundamentals geodesy geodynamics geomatics history concepts geographical distance geoid figure of the earth radius and circumference geodetic coordinates geodetic datum geodesic horizontal position representation latitude longitude map projection reference ellipsoid satellite geodesy spatial reference system spatial relations vertical positions technologies global nav sat systems gnsss global pos system gps glonass russia beidou bds china galileo europe navic india quasi zenith sat sys qzss japan discrete global grid and geocoding standards history ngvd 29 sea level datum 1929 osgb36 ordnance survey great britain 1936 sk 42 systema koordinat 1942 goda ed50 european datum 1950 sad69 south american datum 1969 grs 80 geodetic reference system 1980 iso 6709 geographic point coord 1983 nad 83 north american datum 1983 wgs 84 world geodetic system 1984 navd 88 n american vertical datum 1988 etrs89 european terrestrial ref sys 1989 gcj 02 chinese obfuscated datum 2002 geo uri internet link to a point 2010 international terrestrial reference system spatial reference system identifier srid universal transverse mercator utm v t e 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 an oblate ellipsoid the radius ranges from a maximum equatorial radius denoted a of about 6378 km 3963 mi to a minimum polar radius denoted b of nearly 6357 km 3950 mi the globally averaged value is 6371 kilometres 3959 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 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 2 and the volumetric radius which is the radius of a sphere having the same volume as the ellipsoid r 3 2 all three values are about 6371 kilometres 3959 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 polar radius and equatorial radius because they account for localized effects a nominal earth radius denoted r e n displaystyle mathcal r _ mathrm e mathrm n 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 1 introduction edit a scale diagram of the 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 as the earth the red line represents the karman line 100 km 62 mi above sea level while the yellow area denotes the altitude range of the iss in low earth orbit main articles figure of the earth earth ellipsoid and reference ellipsoid earth s rotation internal density variations and external tidal forces cause its shape to deviate systematically from a perfect sphere a 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 b a spheroid also called an ellipsoid of revolution geocentric to model the entire earth or else geodetic for regional work c 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 d 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 of these geocentric radii falls between the polar minimum of about 6357 km and the equatorial maximum of about 6378 km 3950 to 3963 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 edit further information equatorial bulge rotation of a planet causes it to approximate an oblate ellipsoid spheroid with a bulge at the equator and flattening at the north and south poles so that the equatorial radius a is larger than the polar radius b by approximately aq the oblateness constant q is given by q a 3 ω 2 g m displaystyle q frac a 3 omega 2 gm where ω is the angular frequency g is the gravitational constant and m is the mass of the planet e for the earth 1 q 289 which is close to the measured inverse flattening 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 4 the variation in density 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 110 m 360 ft on earth the geoid height can change abruptly due to earthquakes such as the sumatra andaman earthquake or reduction in ice masses such as greenland 5 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 edit 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 5 m 16 ft of reference ellipsoid height and to within 100 m 330 ft 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 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 edit the following radii are derived from the world geodetic system 1984 wgs 84 reference ellipsoid 6 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 7 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 clarification needed 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 a or semi major axis 8 11 is the distance from its center to the equator and equals 6 378 1370 km 3 963 1906 mi 9 the equatorial radius is often used to compare earth with other planets the earth s polar radius b or semi minor axis 8 11 is the distance from its center to the north and south poles and equals 6356 7523 km 3949 9028 mi location dependent radii edit various radii as a function of latitude geocentric radius edit not to be confused with geocentric distance the geocentric radius is the distance from the earth s center to a point on the spheroid surface at geodetic latitude φ given by the formula 10 r φ a 2 cos φ 2 b 2 sin φ 2 a cos φ 2 b sin φ 2 displaystyle r varphi sqrt frac a 2 cos varphi 2 b 2 sin varphi 2 a cos varphi 2 b sin varphi 2 where a and b are respectively the equatorial radius and the polar radius the maximum and minimum geocentric radii on the ellipsoid coincide respectively with the equatorial and polar radii at the vertices of the ellipse however the maximum and minimum radii of curvature coincide with the poles and equator in opposite regions than the geocentric radii due to earth s oblateness radii of curvature edit see also spheroid curvature principal radii of curvature edit there are two principal radii of curvature along the meridional and prime vertical normal sections derivation the principal curvatures are the roots of equation 125 in 11 e g f 2 κ 2 e g g e 2 f f κ e g f 2 0 det a κ b displaystyle eg f 2 kappa 2 eg ge 2ff kappa eg f 2 0 det a kappa b where in the first fundamental form for a surface equation 112 in 11 d s 2 i j a i j d w i d w j e d φ 2 2 f d φ d λ g d λ 2 displaystyle ds 2 sum _ ij a_ ij dw i dw j e d varphi 2 2f d varphi d lambda g d lambda 2 e f and g are elements of the metric tensor a a i j ν r ν w i r ν w j e f f g displaystyle 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 r r 1 r 2 r 3 t x y z t displaystyle r r 1 r 2 r 3 t x y z t w 1 φ displaystyle w 1 varphi w 2 λ displaystyle w 2 lambda in the second fundamental form for a surface equation 123 in 11 2 d i j b i j d w i d w j e d φ 2 2 f d φ d λ g d λ 2 displaystyle 2d sum _ ij b_ ij dw i dw j e d varphi 2 2f d varphi d lambda g d lambda 2 e f and g are elements of the shape tensor b b i j ν n ν 2 r ν w i w j e f f g displaystyle 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 n n n displaystyle n frac n n is the unit normal to the surface at r displaystyle r and because r φ displaystyle frac partial r partial varphi and r λ displaystyle frac partial r partial lambda are tangents to the surface n r φ r λ displaystyle n frac partial r partial varphi times frac partial r partial lambda is normal to the surface at r displaystyle r with f f 0 displaystyle f f 0 for an oblate spheroid the curvatures are κ 1 g g displaystyle kappa _ 1 frac g g and κ 2 e e displaystyle kappa _ 2 frac e e and the principal radii of curvature are r 1 1 κ 1 displaystyle r_ 1 frac 1 kappa _ 1 and r 2 1 κ 2 displaystyle r_ 2 frac 1 kappa _ 2 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 d r displaystyle r dr to the plane tangent at r displaystyle r meridional edit in particular the earth s meridional radius of curvature in the north south direction at φ is 12 m φ a b 2 a cos φ 2 b sin φ 2 3 2 a 1 e 2 1 e 2 sin 2 φ 3 2 1 e 2 a 2 n φ 3 displaystyle m varphi frac ab 2 big a cos varphi 2 b sin varphi 2 big frac 3 2 frac a 1 e 2 1 e 2 sin 2 varphi frac 3 2 frac 1 e 2 a 2 n varphi 3 where e displaystyle e is the eccentricity of the earth this is the radius that eratosthenes measured in his arc measurement prime vertical edit the length pq called the prime vertical radius is n ϕ displaystyle n phi the length iq is equal to e 2 n ϕ displaystyle e 2 n phi r x y z displaystyle r x y z if one point had appeared due east of the other one finds the approximate curvature in the east west direction f this earth s prime vertical radius of curvature also called the earth s transverse radius of curvature is defined perpendicular orthogonal to m at geodetic latitude φ g and is 12 n φ a 2 a cos φ 2 b sin φ 2 a 1 e 2 sin 2 φ displaystyle n varphi frac a 2 sqrt a cos varphi 2 b sin varphi 2 frac a sqrt 1 e 2 sin 2 varphi n can also be interpreted geometrically as the normal distance from the ellipsoid surface to the polar axis 13 the radius of a parallel of latitude is given by p n cos φ displaystyle p n cos varphi 14 15 combined radii of curvature edit azimuthal edit euler radius vs azimuth at various latitudes earth s azimuthal radius of curvature of an earth normal section at an azimuth measured clockwise from north α and at latitude φ is derived from euler s curvature formula ...