Earth radius

Earth radius (denoted by the symbol or by formula_1) 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, denoted "a") to a minimum of nearly (polar radius, denoted "b"). 
A "nominal Earth radius" is sometimes used as a unit of measurement in astronomy and geophysics, which is recommended by the International Astronomical Union 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 equatorial radius because they include local or geoidal topography 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. 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:
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". 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.
Physics of Earth's deformation.
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" is larger than the "polar radius" by approximately . The "oblateness constant" is given by
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 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) 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 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 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.
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 :
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) meridian direction) at is:
where formula_5 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.
This "Earth's prime-vertical radius of curvature", also called the "Earth's transverse radius of curvature", is defined perpendicular (normal or orthogonal) to at geodetic latitude is:
B. R. Bowring gives a geometric proof that this is the perpendicular distance from the surface to the polar axis.
Particular values.
The "Earth's meridional radius of curvature at the equator" equals the meridian's semi-latus rectum:
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:
Derivation.
The principal curvatures are the roots of Equation (125) in:
where in the first fundamental form for a surface (Equation (112) in ):
E, F, and G are elements of the metric tensor:
formula_10, formula_11, formula_12
in the second fundamental form for a surface (Equation (123) in ):
e, f, and g are elements of the shape tensor:
formula_15 is the unit normal to the surface at formula_16, and because
formula_17 and formula_18 are tangents to the surface,
is normal to the surface at formula_16.
With formula_21 for an oblate spheroid, the curvatures are
and the principal radii of curvature are
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 formula_26 to the plane tangent at formula_16.
Combined radii of curvature.
Azimuthal.
The Earth's "azimuthal radius of curvature", along a course at an azimuth (measured clockwise from north) at , is derived from Euler's curvature formula as follows:
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:
Where "K" is the "Gaussian curvature", formula_30.
The "Earth's mean radius of curvature" at latitude is:
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,
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 denotes the "nominal equatorial Earth radius" as formula_32, which is defined to be . The "nominal polar Earth radius" is defined as formula_33 = . 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.
Mean radius.
In geophysics, the International Union of Geodesy and Geophysics (IUGG) defines the "Earth's mean radius" (denoted ) to be
The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid.
For Earth, the 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:
where and is the surface area of the spheroid.
For the Earth, the authalic radius is .
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 .
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:
For integration limits of [0,], 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,
which differs from the exact result by less than ; the mean of the two axes,
about , can also be used.
Global mean radius of curvature.
The "global mean radius of curvature", averaged in all azimuths and at all points on the surface, is given by the area-weighted global mean Gaussian curvature:
For the WGS 84 ellipsoid, the mean curvature equals .
Topographical radii.
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 distances", "Rt", which depends not only on latitude.
Topographical global mean.
The "topographical mean geocentric distance" averages elevations everywhere, resulting in a value larger than the IUGG mean radius, the authalic radius, or the volumetric radius. This topographical average is with uncertainty of .
Derived quantities: diameter, circumference, arc-length, area, volume.
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:
"Earth's circumference" equals the perimeter length. The "equatorial circumference" is simply the circle perimeter: "Ce=2πa", in terms of the equatorial radius, "a". The "polar circumference" equals "Cp=4mp", four times the quarter meridian "mp=aE(e)", where the polar radius "b" enters via the eccentricity, "e=(1-b2/a2)0.5"; see Ellipse#Circumference for details.
Arc length of more general surface curves, such as meridian arcs 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.
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 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.