Because the Earth, like all planets, is not a perfect sphere, the radius of Earth can vary at different places on the surface. The radius of the Earth at a point on the surface is the distance from the center of the Earth to the mean sea level at that point. This value varies from about 6,357.750 km — 6,379.135 km (≈3,949.501 — 3,962.189 mi), values between the polar radius and the equatorial radius (with few exceptions). The radius of the Earth can also refer to other fixed radii as well as to various mean radii, outlined below. For all planets the sources of the distortion from spherical are rotation, variation of mass density within the planet, and tidal forces. ^[1] This article is about Earth as a planet. ... The eight planets and three dwarf planets of the Solar System. ... A sphere is a perfectly symmetrical geometrical object. ... Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... This article is about Earth as a planet. ... For considerations of sea level change, in particular rise associated with possible global warming, see sea level rise. ...

Introduction

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 where the oblateness constant q is In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe. ... For other uses, see North Pole (disambiguation). ... For other uses, see South Pole (disambiguation). ...

where ω is the angular frequency, G is the gravitational constant, and M is the mass of the planet. ^[2] For the Earth , which is close the measured inverse flattening . Additionally, the bulge at the equator shows slow variations. The bulge had been declining, but since 1998 the bulge has increased, possibly due to redistribution of ocean mass via currents. ^[3] It has been suggested that this article or section be merged into Angular velocity. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ...

The variation in density and crustal thickness causes gravity to vary on the surface, so that the mean sea level will differ 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 on Earth. The geoid height can have abrupt changes due to earthquakes (such as the Sumatra-Andaman earthquake) or reduction in ice masses (such as Greenland). ^[4] Image File history File links No higher resolution available. ... In physics, density is mass m per unit volume V. For the common case of a homogeneous substance, it is expressed as: where, in SI units: ρ (rho) is the density of the substance, measured in kg·m-3 m is the mass of the substance, measured in kg V is... Earth cutaway from core to exosphere. ... The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... The 2004 Indian Ocean earthquake, known by the scientific community as the Sumatra-Andaman earthquake,[1] was a great undersea earthquake that occurred at 00:58:53 UTC (07:58:53 local time) December 26, 2004 with an epicentre off the west coast of Sumatra, Indonesia. ...

The tides from the gravity of the Moon and Sun cause the surface of the Earth to rise and fall by tenths of meters at a point over a nearly 12 hr period. It has been suggested that this article or section be merged into Tide. ...

Therefore, the values defined below are based on a "general purpose" model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m 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 largest (tightest) in one direction (North-South on Earth) and smallest (flattest) perpendicularly (East-West). The corresponding radius of curvature depends on 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 geometry, a torus (pl. ... The distance from the center of a sphere or ellipsoid to its surface is its radius. ... Horizon. ...

In summary, Local variations in terrain prevent the definition of a single absolutely "precise" radius. One can only find mathematically precise values based on a given model. Since the estimate by Eratosthenes, a plethora of models have been created, some accommodating or based on regional topography. The advancements in measuring technology, now including satellites, mean that different reference ellipsoid models have made their way into general usage over the years, providing slightly different values. Eratosthenes (Greek ; 276 BC - 194 BC) was a Greek mathematician, geographer and astronomer. ... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ...

(Note: Earth radius is sometimes used as a unit of distance, especially in astronomy and geology. It is usually denoted by R_E.)

Fixed radii

The following radii are fixed, and do not include a variable location dependence.

Equatorial radius: a

The Earth's equatorial radius, or semi-major axis, is the distance from its center to the equator and equals 6,378.135 km (≈3,963.189 mi; ≈3,443.917 nmi). At 0° S 121.83° E, the geoid height rises to 63.42 m above the reference ellipsoid (WGS-84), giving a total radius of 6,378.200 km. The equatorial radius is often used to compare Earth with other planets. The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe. ... km redirects here. ... A mile is a unit of length, usually used to measure distance, in a number of different systems, including Imperial units, United States customary units and Norwegian/Swedish mil. ... A nautical mile or sea mile is a unit of length. ... The World Geodetic System defines a reference frame for the earth, for use in geodesy and navigation. ... The eight planets and three dwarf planets of the Solar System. ...

Polar radius: b

The Earth's polar radius, or semi-minor axis, is the distance from its center to the North and South Poles, and equals 6,356.750 km (≈3,949.901 mi; ≈3,432.370 nmi). The geoid height (WGS-84) at the North Pole is 13.6 m above the reference ellipsoid, and at the South Pole 29.5 m below the reference, giving the more exact 6,356.766 km and 6,356.723 km, respectively. In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ...

Radii with location dependence

Radius at a given geodetic latitude

The Earth's radius at geodetic latitude, , is:

Radius of curvature

These are based on a oblate ellipsoid. In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ...

Eratosthenes used two points, one exactly north of the other. The points are separated by distance D, and the vertical directions at the two points are known to differ by angle of θ, in radians. A formula based on Eratosthenes method is Eratosthenes (Greek ; 276 BC - 194 BC) was a Greek mathematician, geographer and astronomer. ... In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i. ...

which gives an estimate of radius based on the north-south curvature of the Earth.

Meridional

In particular the Earth's radius of curvature in the (north-south) meridian at is:

Normal

If one point had appeared due east of the other, one finds the approximate curvature in east-west direction. ^[5]

This radius of curvature in the prime vertical, which is perpendicular, or normal, to M at geodetic latitude is: ^[6]

Note that N=R at the equator: In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

The Earth's mean radius of curvature (averaging over all directions) at latitude is: Image File history File links No higher resolution available. ...

The Earth's radius of curvature along a course at geodetic bearing (measured clockwise from north) , at is: ^[7]

The Earth's equatorial radius of curvature in the meridian is:

= 6335.437 km

The Earth's polar radius of curvature is:

= 6399.592 km

Mean radii

Quadratic mean radius: Q_r

The ellipsoidal quadratic mean radius provides the best approximation of Earth's average transverse meridional radius and radius of curvature:

It is this radius that would be used to approximate the ellipsoid's average great ellipse (i.e., this is the equivalent spherical "great-circle" radius of the ellipsoid).
For Earth, Q_r equals 6,372.795477598 km (≈3,959.871 mi; ≈3,441.034 nmi).

Authalic mean radius: A_r

Earth's authalic ("equal area") mean radius is 6,371.005076123 km (≈3,958.759 mi; ≈3,440.067 nmi). This number is derived by square rooting the average (latitudinally cosine corrected) geometric mean of the meridional and transverse equatorial, or "normal" (i.e., perpendicular), arcradii of all surface points on the spheroid, which can be reduced to a closed-form solution: In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

where A is the authalic surface area of Earth. This would be the radius of a hypothetical perfect sphere which has the same, geometric mean oriented surface area as the spheroid.

Volumetric radius: V_r

Another, less utilized, sphericalization is that of the volumetric radius, which is the radius of a sphere of equal volume:

For Earth, the volumetric radius equals 6,370.998685023 km (≈3,958.755 mi; ≈3,440.064 nmi).

Meridional Earth radius

Another radius mean is the meridional mean, which equals the radius used in finding the perimeter of an ellipse. It can also be found by just finding the average value of M: The circumference is the distance around a closed curve. ...

For Earth, this works out to 6367.446988834 km (≈3,956.548 mi; ≈3,438.146 nmi).

See also

• Effective Earth radius
• Radius of curvature
• Figure of the Earth
• Earth tide

In telecommunication, effective Earth radius is the radius of a hypothetical Earth for which the distance to the radio horizon, assuming rectilinear propagation, is the same as that for the actual Earth with an assumed uniform vertical gradient of atmospheric refractive index. ... The distance from the center of a sphere or ellipsoid to its surface is its radius. ... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ... It has been suggested that this article or section be merged into Tide. ...

Notes and references

1. ^ The center of the Earth is somewhat model dependent. Exceptions to the cited range will occur near the South Pole and along the equator. Also, differences due to variation of mass density within the planet and tidal forces require data for the entire surface of the Earth and are not included here. For detail see Figure of the Earth, Geoid, and Earth tide.
2. ^ This follows from the International Astronomical Union definition rule (2): a planet assumes a shape due to hydrostatic equilibrium where gravity and centrifugal forces are nearly balanced. IAU 2006 General Assembly: Result of the IAU Resolution votes
3. ^ Satellites Reveal A Mystery Of Large Change In Earth's Gravity Field, Aug. 1, 2002, Goddard Space Flight Center.
4. ^ NASA's Grace Finds Greenland Melting Faster, 'Sees' Sumatra Quake, December 20, 2005, Goddard Space Flight Center.
5. ^ 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 exchanged for east in this discussion.
6. ^ N is defined as the radius of curvature in the plane which is normal to both the surface of the ellipsoid at, and the meridian passing through, the specific point of interest.
7. ^ A related application of M and N: if two nearby points have the difference in latitude of and longitude of (in radians) with mean latitude , then the distance D between them is

   

   The quantities inside the parentheses are approximately Dcosα and Dsinα, respectively. Thus dφ and dλ be estimated from D, M, and N.