Introduction
The Earth radius (R or R_E) denotes the linear distance from the planet’s center to a point on or near the surface. Under the common oblate‑spheroid approximation this distance varies with latitude and provides the primary link between a surface location, its vertical position, and many geodetic calculations. The spheroidal approximation is characterized by two principal extremes: the equatorial semi‑major axis a ≈ 6,378 km and the polar semi‑minor axis b ≈ 6,357 km; these parameters define the reference ellipsoid used in geodesy.
For many applications a single global value is convenient: a widely used mean is 6,371 km (±≈10 km, about 0.3% variability). The International Union of Geodesy and Geophysics (IUGG) also specifies three canonical spherical radii—R1 (mean of selected radii), R2 (authalic radius, equal surface area), and R3 (volumetric radius, equal volume)—all numerically close to 6,371 km. In planetary comparisons a nominal Earth radius (R_E^N) is used as a fixed conversion; the International Astronomical Union recommends using the equatorial radius when no other choice is stated.
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Numerous alternative radius definitions are used in geodetic practice. Radii of curvature of the reference spheroid depend on direction and latitude; radii that incorporate actual topography or localized curvature can yield values outside the simple polar–equatorial interval because they include elevation and local relief. Thus the “radius” is not unique but depends on the chosen geometric or physical definition.
The study and application of Earth radius concepts sit within the broader disciplines of geodesy, geodynamics, geomatics and the history of these fields. Core geodetic concepts that connect position and planetary form include geographical distance, the geoid (the equipotential surface approximating mean sea level), the figure of the Earth, geodetic coordinates and datums, geodesics (shortest paths on the ellipsoid), map projections, satellite geodesy and spatial reference systems. Modern realizations of coordinates and heights are underpinned by Global Navigation Satellite Systems (GNSS)—notably GPS, GLONASS, BeiDou (BDS), Galileo, NAVIC and QZSS—and by complementary methods such as Discrete Global Grid systems, geocoding schemes, and planar representations (e.g., UTM).
Geodetic practice has evolved through a sequence of regional and global standards and datums (selected examples with approximate years): Sea Level Datum of 1929 (NGVD 29); OSGB36 (1936); SK‑42 (1942); ED50 (1950); SAD69 (1969); GRS 80 (1980); ISO 6709 (1983); NAD83 (1983); WGS84 (1984); NAVD88 (1988); ETRS89 (1989); GCJ‑02 (2002); and Geo URI (2010). Complementary global frameworks include the International Terrestrial Reference System (ITRS), Spatial Reference System Identifiers (SRIDs) used in geospatial databases, and the Universal Transverse Mercator (UTM) projection for zonal planar coordinates.
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Because different definitions of “radius” serve different purposes, unambiguous positional and distance work requires explicit specification of the reference ellipsoid or nominal radius, the geodetic datum, and the vertical reference (for example, geoid versus ellipsoidal heights). Only with these references stated can spatial data and computations be interpreted and exchanged consistently.
A scale diagram of the 2003 IERS reference ellipsoid (north at top) illustrates Earth’s oblateness: a light-blue circle is shown alongside a dark-blue outer ellipse whose minor axis matches that circle while its eccentricity equals Earth’s; the graphic also marks the Kármán line at 100 km above sea level and shades the altitude band occupied by the International Space Station in low Earth orbit. This visualization underscores that Earth’s shape systematically departs from a perfect sphere under the influence of rotation, spatial variations in internal density, and external tidal forces; local topography adds further, high-frequency complexity that must be simplified for many practical purposes.
To manage this complexity, geographers and geodesists use a hierarchy of models. At the most exact level is the actual terrestrial surface; next is the geoid, defined as mean sea level mapped onto the real surface; farther simplified are spheroids (ellipsoids of revolution), employed either as a geocentric ellipsoid for global representation or as a geodetic ellipsoid for regional work; and, at the simplest level, a sphere. Associated with these models is a geometric notion of radius: for the geoid and ellipsoids, the distance from a model point to its chosen center is conventionally treated as the “radius of the Earth at that point,” while spherical models use a single mean radius. In contrast, the irregular true surface is more usefully described by elevations above or below sea level rather than by a radius—strictly speaking, only a sphere possesses a single unambiguous radius, though the broader geoscientific usage of “Earth’s radius” is standard.
Geocentric radii for common Earth models range from about 6,357 km at the poles to about 6,378 km at the equator, a difference of roughly one third of one percent. That small fractional departure often justifies the spherical approximation and the generic phrase “radius of the Earth” in many contexts. The modeling framework and terminology outlined here—actual surface, geoid, spheroid/ellipsoid (geocentric versus geodetic), sphere, and geocentric radii, together with the physical causes of non‑sphericity—apply equally to other major planets.
Planetary rotation induces a centrifugal potential that deforms a self‑gravitating body into an oblate spheroid, producing an equatorial bulge so that the equatorial radius a exceeds the polar radius b by approximately a − b ≈ a q. The dimensionless oblateness parameter q quantifies the rotational contribution and is given exactly by q = a^3 ω^2/(G M), where ω is the angular frequency of rotation, G the gravitational constant and M the planetary mass. For Earth the theoretical value implied by q (1/q ≈ 289) is close to, but does not fully account for, the observed inverse flattening (1/f ≈ 298.257 with f = (a − b)/a), indicating that rotation dominates global flattening while other factors produce residual differences. Temporal changes in the equatorial bulge occur on decadal and shorter timescales; for example, an observed decrease followed by an increase since 1998 has been linked in part to ocean mass redistribution by currents, demonstrating that mass movements in the fluid envelope modify the planet’s figure. Heterogeneities in internal density and crustal thickness generate spatial and temporal variations in gravity so that mean sea level departs from the reference ellipsoid; this departure, the geoid height, can be positive or negative and on Earth ranges in magnitude up to roughly 110 m, though it can change abruptly in response to large mass redistributions such as coseismic deformation from major earthquakes (e.g., the Sumatra–Andaman event) or ice‑mass loss (e.g., Greenland). In addition to internal mass changes, time‑varying external gravitational forces from the Moon and Sun produce elastic Earth tides, yielding surface displacements on the order of tenths of a meter with a semi‑diurnal (~12 h) periodicity.
Local values of the Earth’s radius are not uniquely defined because the planet’s surface departs from a simple sphere: topography and the geoid (the local deviation of mean sea level from any mathematical surface) produce spatially varying heights. To make practical, repeatable measurements and maps, geodesy therefore adopts idealized reference surfaces—principally reference ellipsoids and the geoid—that provide consistent, usable radii for positioning and mapping even though they are simplifications of the true surface.
Historically, estimates of Earth’s size progressed from broad, multi‑site methods (e.g., Eratosthenes) to techniques suited to limited locales. Al‑Biruni (973–c.1050) pioneered a single‑site approach that removed the need for long baseline surveys by inferring Earth’s curvature from measurements made at one location, thereby reducing logistical difficulties in his era. Over succeeding centuries surveyors typically chose ellipsoids fitted to particular regions so as to minimize local mapping errors; these regional reference ellipsoids yielded superior accuracy over their coverage at the cost of global consistency.
Curvature at a point on the Earth is anisotropic: there are two principal curvatures, one tighter and one flatter, analogous to the cross directions on a torus. On most of the globe the meridional (north–south) direction exhibits the greatest curvature (i.e., the smallest radius of curvature), while the zonal or parallel (east–west) direction is flatter and has a larger radius. Because curvature depends both on location and on direction, derived quantities that rely on local curvature—such as the distance to the true horizon—vary with azimuth; for example, horizon distance at the equator is marginally shorter when measured toward the poles than along an east–west line.
Modern, general‑purpose Earth models are referenced to a globally consistent reference ellipsoid and refined so that surface heights match that ellipsoid to on the order of 5 m, and match mean sea level to roughly 100 m—figures that do not account for local geoid undulation. The advent of satellite remote sensing and global navigation systems (notably GPS) has enabled truly global models that best approximate the whole Earth, trading some regional best‑fit accuracy for global uniformity. For precise local geodetic work, however, locally tuned ellipsoids and high‑resolution geoid models remain important to capture regional deviations from the global idealization.
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The values commonly cited for Earth’s equatorial and polar radii are parameters of the WGS‑84 reference ellipsoid, an idealized mathematical surface used to approximate the planet’s shape. The equatorial (semi‑major) axis a is defined as the distance from Earth’s centre to the equator and is 6,378,137.0 m (6,378.1370 km). The polar (semi‑minor) axis b is the distance from Earth’s centre to the poles and is 6,356,752.3 m (6,356.7523 km). The equatorial radius is often employed when comparing Earth’s size with that of other planets.
These ellipsoidal dimensions are derived from geodetic measurements that carry an uncertainty on the order of ±2 m in both equatorial and polar directions. Local departures from the ellipsoid—topography (mountains, trenches) and geoid undulations—can produce considerably larger local discrepancies, so reporting additional digits of precision for the WGS‑84 radii seldom yields a corresponding improvement in real positional accuracy. For applications demanding the utmost numerical precision, users should consult the complete WGS‑84 specification for exact constants and ancillary parameters, while bearing in mind that measurement uncertainty and local topographic/geoid effects typically limit achievable accuracy more than the final decimal places of the ellipsoidal radii.
Geocentric radius R(φ) is the straight-line distance from the Earth’s center to a point on the surface of the reference ellipsoid (an oblate spheroid) at geodetic latitude φ. For an ellipsoid of revolution with equatorial semiaxis a and polar semiaxis b this distance is given exactly by
R(φ) = sqrt(((a^2 cos φ)^2 + (b^2 sin φ)^2) / ((a cos φ)^2 + (b sin φ)^2)),
with a and b in linear units and φ entered consistently (radians or degrees) into the trigonometric functions. The value R(φ) therefore inherits the units of a and b and depends only on the chosen ellipsoid and latitude.
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Evaluating the expression at principal latitudes yields the extrema: at the Equator (φ = 0°) R = a, the maximum geocentric radius, and at the poles (φ = ±90°) R = b, the minimum. These extrema occur at the vertices of the meridional ellipse of the ellipsoid and reflect the geometric fact that for an oblate spheroid a > b.
The latitude variation of the geocentric radius contrasts with that of the ellipsoid’s radii of curvature: the latter attain their maximum at the poles and minimum at the Equator, i.e., their extrema occur in the opposite regions. This complementary behavior arises from the ellipsoid’s oblateness—an increased equatorial semiaxis increases the center-to-surface distance at low latitudes while producing larger curvature radii toward the poles.
Principal radii of curvature
The local bending of a smooth surface is captured by two principal radii of curvature, each associated with one of the orthogonal normal sections through the point: the meridional (north–south) and the prime‑vertical (east–west). These radii are the reciprocals of the principal curvatures κ (R = 1/κ), which quantify the extremal normal curvatures of the surface.
Algebraically the principal curvatures are the roots of a generalized eigenvalue problem for the pair formed by the first and second fundamental forms. Writing the metric (first fundamental form) in surface coordinates w^1 = φ, w^2 = λ as ds^2 = E dφ^2 + 2F dφ dλ + G dλ^2 gives the matrix A = [[E, F],[F, G]]; the shape (second fundamental) form 2D = e dφ^2 + 2f dφ dλ + g dλ^2 corresponds to B = [[e, f],[f, g]]. The principal curvatures satisfy det(A − κ B) = 0, equivalently the quadratic (EG − F^2) κ^2 − (eG + gE − 2fF) κ + (eg − f^2) = 0. Here the entries of A arise from the inner products of tangent vectors ∂r/∂φ and ∂r/∂λ, while the entries of B are obtained by projecting the second derivatives of the position vector r onto the unit normal n = (r_φ × r_λ)/|r_φ × r_λ|.
Viewed geometrically, the second fundamental form is the quadratic measure of how the surface departs from its tangent plane: for an infinitesimal displacement dr in the tangent coordinates, 2D gives the signed normal distance from the displaced point r + dr to the tangent plane at r, and its associated quadratic form encodes the local normal deviation and hence curvature.
For the common geodetic model of an oblate spheroid the mixed coefficients vanish (F = f = 0), which diagonalizes the problem and yields the two principal curvatures in simple form κ1 = g/G and κ2 = e/E; their reciprocals are the meridional and prime‑vertical radii of curvature, respectively. In geodetic terms these radii therefore quantify curvature in the north–south and east–west normal sections and are fundamental for describing local geometric and geophysical behavior of the Earth’s surface.
Meridional radius of curvature M(φ) characterizes the curvature of a meridian (a line of constant longitude) in the north–south plane at latitude φ: it is the radius of the osculating circle that best fits the meridian in the meridional section. For an oblate ellipsoid with semi‑major axis a (equatorial radius), semi‑minor axis b (polar radius) and eccentricity e defined by e^2 = (a^2 − b^2)/a^2, M(φ) has the exact closed form
M(φ) = a(1 − e^2) / (1 − e^2 sin^2 φ)^{3/2}.
This expression can be written compactly in terms of the prime‑vertical radius N(φ) = a / sqrt(1 − e^2 sin^2 φ) as
M(φ) = ((1 − e^2)/a^2) · N(φ)^3,
which follows directly from the algebraic relation between N and the ellipsoid parameters.
Because M(φ) depends on sin^2 φ, it varies with latitude: for an oblate Earth (a > b) the meridional radius attains its minimum at the equator, M(0) = a(1 − e^2) = b^2/a, and increases monotonically to its maximum at the poles, M(±90°) = a^2/b. Historically, this meridional radius is the effective radius used in classical measurements of a meridian arc; early determinations of Earth’s size (notably Eratosthenes’ measurement) thus implicitly measured a quantity equivalent to M(φ).
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The prime-vertical radius of curvature, denoted N(φ) and sometimes called the Earth’s prime‑vertical or transverse radius of curvature, is the characteristic radius governing east–west curvature at geodetic latitude φ when nearby points are aligned due east–west. A secondary linear quantity IQ is proportional to the squared first eccentricity, with IQ = e^2 N(φ), where e^2 = (a^2 − b^2)/a^2 and a, b are the ellipsoid semi‑major and semi‑minor axes. N(φ) admits two algebraically equivalent expressions:
N(φ) = a^2 / sqrt((a cosφ)^2 + (b sinφ)^2) = a / sqrt(1 − e^2 sin^2φ),
which emphasize respectively its relation to the principal axes and its common form in geodetic formulae. Geometrically, N(φ) equals the distance measured along the surface normal from the ellipsoid to the polar axis, a property that explains its role in describing both east–west curvature and normal offsets. The radius p of the parallel (the circle obtained by intersecting the ellipsoid with a plane parallel to the equatorial plane at latitude φ) is p = N(φ) cosφ. These quantities enter routinely into transformations between geodetic and geocentric systems, in which a position is expressed by the Cartesian vector R = (X, Y, Z).
The azimuthal radius of curvature R_c at a point on the reference ellipsoid quantifies the radius of curvature of a normal section taken in a horizontal direction specified by azimuth α (measured clockwise from geographic north) at geodetic latitude φ. It is given by
R_c = 1 / ( (cos^2 α) / M(φ) + (sin^2 α) / N(φ) ),
or, equivalently, by its reciprocal (the curvature)
R_c^{-1} = cos^2 α / M(φ) + sin^2 α / N(φ).
This form shows that the curvature in any tangent‑plane direction is a weighted sum of the two principal curvatures 1/M and 1/N, with weights cos^2α and sin^2α; it is therefore a direct application of Euler’s theorem on normal curvature for a smooth surface, here with the principal directions aligned with the meridian and the prime vertical.
The principal radii themselves vary with latitude: for an ellipsoid with semi‑major axis a and eccentricity e,
M(φ) = a(1 − e^2) / (1 − e^2 sin^2 φ)^{3/2},
N(φ) = a / (1 − e^2 sin^2 φ)^{1/2}.
Consequently R_c depends on both azimuth and latitude. In the principal directions the general formula reduces to the expected values: α = 0° (meridional) yields R_c = M, and α = 90° (prime vertical) yields R_c = N. For intermediate directions, for example α = 45°, the curvature is the arithmetic mean of the principal curvatures, R_c^{-1} = (M^{-1} + N^{-1})/2, so R_c is the reciprocal of that mean.
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Latitudinal variation is systematic because M and N depend on sin φ: at the equator (sin φ = 0) one has N = a and M = a(1 − e^2), while toward the poles the relative sizes of M and N change according to the ellipsoid’s eccentricity, altering the directional curvature for a given α. The azimuthal curvature formula is fundamental in geodesy and surveying for evaluating local curvature in any heading, with direct use in horizon and line‑of‑sight calculations, curvature corrections in engineering, and precise assessment of directional scale and distortion in map projections.
Non‑directional scalar radii summarize the local curvature of an ellipsoidal surface at latitude φ by combining the two principal radii of curvature (commonly denoted M and N) without reference to a particular azimuth. The azimuthal radius of curvature Rc(α) varies with direction α around the surface normal; two canonical non‑directional combinations are the Gaussian radius of curvature Ra(φ) and the radius of mean curvature Rm(φ), obtained by integrating Rc or its reciprocal around a full circle.
The Gaussian radius Ra is defined by Ra = 1/√K, where K is the Gaussian curvature K = κ1κ2 (with κi = 1/M,1/N). Equivalently Ra is the circular average of the azimuthal radius,
Ra(φ) = (1/2π) ∫0^{2π} Rc(α) dα,
and algebraically equals the geometric mean of the principal radii, Ra = √(MN). For an ellipsoid one obtains the closed form
Ra(φ) = (a^2 b)/((a cosφ)^2 + (b sinφ)^2) = a√(1−e^2)/(1−e^2 sin^2φ),
so Ra corresponds to the radius of the osculating sphere that best fits the surface at latitude φ.
The radius of mean curvature Rm is the harmonic mean of the principal radii,
Rm = 2/(1/M + 1/N),
so its reciprocal equals the arithmetic mean of the principal curvatures,
Rm^{-1} = (M^{-1} + N^{-1})/2 = (1/2π) ∫0^{2π} Rc^{-1}(α) dα.
Thus Rm characterizes the averaged curvature scale obtained by integrating the curvature (the reciprocal of Rc) around the normal.
Both Ra and Rm depend on latitude φ through cosφ and sinφ and on the ellipsoid parameters a, b and eccentricity e (entering as 1−e^2 sin^2φ). Conceptually, Ra captures the osculating‑sphere length scale via the geometric mean √(MN), while Rm captures the mean curvature scale via the harmonic mean 2/(1/M+1/N); each is produced by averaging Rc(α) or Rc^{-1}(α) over the full 0–2π azimuthal interval.
Equatorial radii of curvature
At the equator (latitude 0°) the ellipsoid is described by the equatorial radius a, the polar radius b, and the meridian semi‑latus rectum ℓ, defined by ℓ = b²/a. This fixed locus determines the curvature values evaluated below.
The meridional radius of curvature at the equator is M(0°) = b²/a = ℓ. It equals 6,335.439 km and represents the radius of curvature of a north–south (meridional) section of the ellipsoid at latitude 0°, coinciding with the meridian’s semi‑latus rectum.
The prime‑vertical radius of curvature at the equator is N(0°) = a, i.e., the prime‑vertical curvature at latitude 0° is exactly the equatorial radius a, corresponding to the ellipsoid’s radius in the equatorial plane.
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The Gaussian radius of curvature at the equator reduces to R_a(0°) = b, so the Gaussian curvature evaluated at latitude 0° corresponds to the curvature associated with the polar axis rather than the equatorial radius.
The mean radius of curvature at the equator is the harmonic‑type mean of a and ℓ, given by R_m(0°) = 2aℓ/(a+ℓ) and numerically equal to 6,356.716 km.
Polar radii of curvature
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For a reference ellipsoid with semi‑major axis a (equatorial) and semi‑minor axis b (polar), the radii of curvature at a given latitude are intrinsic properties of the ellipsoidal surface computed from a and b rather than straight‑line distances from the ellipsoid centre. At the geographic poles (latitude ±90°) the two principal surface curvatures coincide, so the meridional radius of curvature M and the prime‑vertical radius of curvature N take the same value
M(90°) = N(90°) = a²/b,
which for the reference ellipsoid considered evaluates to 6,399.594 km. The Gaussian radius of curvature Rₐ and the mean radius of curvature Rₘ likewise equal this common principal value at the poles,
Rₐ(90°) = Rₘ(90°) = a²/b,
so both Gaussian and mean measures of curvature coincide with the principal radii there. Geometrically this quantity a²/b is a surface radius of curvature at the poles and should not be conflated with the polar semi‑axis b; they differ for an oblate ellipsoid and only become identical in the special case a = b (a sphere).
The Earth is represented by a hierarchy of geometric models chosen according to application: a sphere provides a simple idealization, an oblate spheroid (ellipsoid) supplies a more accurate geometric reference, and the geoid denotes the true equipotential sea‑level surface. The sphere is therefore a coarse approximation of the spheroid, which in turn approximates the geoid’s shape and gravity‑defined surface.
For global and mapping purposes the WGS‑84 reference ellipsoid is used as the standard numerical basis. Its principal axes are the equatorial (semi‑major) radius a = 6378.1370 km and the polar (semi‑minor) radius b = 6356.7523 km; these two values define the ellipsoidal geometry from which representative radii are computed. The disparity between a and b expresses the ellipsoid’s oblateness (flattening) relative to a perfect sphere.
When reporting sphere‑ or ellipsoid‑based radii in general geographic contexts, values are given in kilometres rather than millimetres to avoid implying the millimetre‑scale precision required by geodetic treatments of the geoid. Any subsequent mean radii or other derived radii referenced here are explicitly computed from the WGS‑84 parameters a = 6378.1370 km and b = 6356.7523 km, which thus provide the consistent notation and numerical foundation for global‑scale geometric calculations.
Arithmetic mean radius
The World Geodetic System 1984 (WGS 84) represents Earth’s size in schematic diagrams by three characteristic radii—equatorial (a), polar (b), and the arithmetic mean radius—diagrams that are conventionally drawn not to scale for clarity in geodetic and geophysical exposition. The International Union of Geodesy and Geophysics (IUGG) defines the arithmetic mean radius, R1, by the relation R1 = (2a + b) / 3, where a is the equatorial radius and b the polar radius. The factor of two on a arises from the spheroid’s biaxial symmetry: the body has two equal equatorial axes and a distinct polar axis, so a spheroid can be regarded as a special case of a triaxial ellipsoid with two identical axes. Using the conventional a and b values adopted for global reference, the IUGG and the U.S. National Geospatial-Intelligence Agency (NGA) report R1 = 6,371.0087714 km (3,958.7613160 mi); this single-number average is widely used in geodetic and geophysical calculations that require a representative global radius.
Authalic radius
The authalic radius (denoted R2 by the IUGG) is the radius of a hypothetical sphere whose surface area equals that of a given reference ellipsoid. Numerically for the Earth R2 = 6,371.0072 km (3,958.7603 mi).
For a spheroid with semi-axes a (semimajor, equatorial) and b (semiminor, polar) the eccentricity is e = sqrt(a^2 − b^2)/a. Closed-form, exact analytic expressions exist for the authalic radius in terms of a and b, and equivalently it can be obtained directly from the spheroid’s exact surface area A; these formulae involve no approximation to the area input.
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By definition R2 = sqrt(A/(4π)), so the authalic radius is the unique spherical radius that reproduces the ellipsoid’s total surface area. This property makes R2 the appropriate single scalar for area-preserving comparisons between an ellipsoid and a sphere and for replacing an ellipsoid by an equal-area sphere in theoretical or applied work.
There is an intrinsic curvature interpretation: R2 is the radius corresponding to the surface-averaged Gaussian curvature of the ellipsoid. From the Gauss–Bonnet relation one obtains (∫ K dA)/A = 4π/A = 1/R2^2, so the mean of the Gaussian curvature over the surface equals the reciprocal square of the authalic radius.
Practically and theoretically, the authalic radius provides an exact, convenient measure for area-preserving map projections and for geodetic or geophysical analyses that require an equivalent-area spherical model; the spheroidal closed-form expressions permit computation directly from the ellipsoid semi-axes (a, b) or from the known surface area A.
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Volumetric radius (R₃)
The volumetric radius is the radius of a sphere whose volume equals that of a chosen reference ellipsoid; it furnishes a single spherical parameter that conserves the ellipsoid’s total volume for applications that require an equivalent sphere rather than an ellipsoid. The International Union of Geodesy and Geophysics (IUGG) denotes this quantity by R₃.
Mathematically, R₃ is defined in terms of the ellipsoid semi‑major axis a (equatorial radius) and semi‑minor axis b (polar radius) by
R₃ = (a² b)^(1/3),
i.e., the cube root of the product a squared times b. To compute R₃ for any terrestrial reference ellipsoid, substitute that ellipsoid’s values of a and b into the formula and evaluate the cube root. Using standard Earth ellipsoid parameters, R₃ = 6,371.0008 km (≈ 3,958.7564 mi).
Rectifying radius
The rectifying radius Mr of the Earth is the radius of a sphere whose great‑circle circumference equals the perimeter of any polar cross‑section ellipse of the Earth’s oblate ellipsoid. Because an ellipse’s perimeter has no elementary closed form, Mr is defined by an elliptic integral of the ellipsoid semi‑axes a (equatorial) and b (polar). An exact representation is
Mr = (2/π) ∫_0^{π/2} sqrt(a^2 cos^2 φ + b^2 sin^2 φ) dφ,
which can be interpreted as the quarter‑period average of the radial distance of the ellipse.
The same numerical value arises if Mr is expressed as the mean of the meridional radius of curvature M(φ) over latitudes 0 ≤ φ ≤ π/2:
Mr = (2/π) ∫_0^{π/2} M(φ) dφ.
Thus the rectifying radius coincides with the meridional mean for the usual quarter‑period integration limits.
For the standard Earth ellipsoid the integral evaluates to Mr ≈ 6,367.4491 km (3,956.5494 mi). Practical closed‑form approximations exist: the semicubic mean
Mr ≈ ((a^(3/2) + b^(3/2)) / 2)^(2/3)
matches the exact integral to within about 1 μm (≈4×10^−5 in), effectively indistinguishable for geodetic purposes. The simple arithmetic mean (a + b)/2 gives a coarser value (≈6,367.445 km for Earth) that is nevertheless very close to the true rectifying radius.
Because Mr is fundamentally determined by an elliptic integral, exact computation requires numerical integration or special functions; however, the exceptional fidelity of the semicubic mean and the small discrepancy of the arithmetic mean explain why simple axis averages are often sufficient in many geodetic estimates, while the semicubic formula provides near‑machine precision.
Topographical radii
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Geodetic practice treats a rotationally symmetric reference ellipsoid as the computational baseline; this idealized surface, defined by a semi‑major axis a and a measure of flattening (f or eccentricity e), supplies closed‑form relations for coordinates and radii of points that lie on that mathematical surface. Real topography, however, departs from the ellipsoid, and quantifying a real point’s position relative to the Earth’s center requires the topographical geocentric distance Rt — the straight‑line distance from the planet’s geometric centre to a point on the actual surface (whether above or below the chosen ellipsoid).
Rt is computed from the geodetic (ellipsoidal) height h measured normal to the reference ellipsoid: h>0 for terrain above the ellipsoid (e.g. mountains) and h<0 for features below it (e.g. ocean trenches). Given geodetic latitude φ, longitude λ and ellipsoidal height h, one first evaluates the prime‑vertical radius of curvature
N(φ) = a / sqrt(1 − e^2 sin^2φ).
The Earth‑centred Cartesian coordinates then follow as
X = (N(φ) + h) cosφ cosλ,
Y = (N(φ) + h) cosφ sinλ,
Z = [(1 − e^2) N(φ) + h] sinφ,
and the topographical geocentric distance is Rt = sqrt(X^2 + Y^2 + Z^2).
Because N(φ) varies with latitude on a flattened ellipsoid, Rt depends explicitly on φ as well as on h; longitude λ alters the Cartesian components but, for an axisymmetric reference ellipsoid, the radial magnitude is invariant to λ only when h is constant. In addition, geodetic latitude φ (the angle of the surface normal) differs from geocentric latitude θ (the central angle) according to the ellipsoid’s eccentricity, with tanθ = (1 − e^2) tanφ. That distinction matters when converting between surface‑based angular coordinates and Earth‑centred vectors or when interpreting angular positions together with radial distances.
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Accurate evaluation of Rt is essential in any application referenced to the Earth’s centre — satellite altimetry, gravity field modelling, line‑of‑sight and geocentric distance calculations among them — and requires a consistent choice of ellipsoid parameters (a and e or f), precise geodetic latitudes and longitudes, and reliable ellipsoidal heights (including negative values where topography lies below the reference ellipsoid).
The greatest center-to-surface distance on Earth is 6,384.4 km (3,967.1 mi), attained at the summit of Chimborazo, while the smallest is 6,352.8 km (3,947.4 mi), occurring at the floor of the Arctic Ocean. The difference between these extrema is 31.6 km (19.7 mi), which quantifies the maximum variation in radial distance from Earth’s center to its surface. Expressed relative to the overall radius, this range is about 31.6/6,384.4 ≈ 0.495%, a small but geophysically significant departure from sphericity.
These radial measures combine both topographic elevation above mean sea level and oceanic depth below it, so the point farthest from the center is not necessarily the point of greatest elevation above mean sea level. The spatial pattern of the radial variation reflects Earth’s oblate spheroidal form: rotationally driven equatorial bulging places equatorial highs farther from the center than polar or high-latitude lows. Thus the ~31.6 km span provides a direct, quantitative illustration of how rotational flattening modifies absolute center-to-surface distances across different geographic regions.
Topographical global mean
The topographical mean geocentric distance is the global average distance from Earth’s center to the real surface, incorporating both continental elevations and ocean depths. Its measured value is 6,371.230 km (3,958.899 mi) with an uncertainty of ±10 m (±33 ft), which expresses the precision of this estimate.
Compared with commonly used smooth-Earth reference radii, the topographical mean exceeds each by 230 m. Those reference measures—the IUGG mean radius (the conventional international mean Earth radius), the authalic radius (radius of a sphere with the same total surface area), and the volumetric radius (radius of a sphere with the same volume)—are geometric or area/volume constructs that do not account for actual surface roughness. The 230 m excess therefore quantifies the net contribution of mountains and ocean basins to the mean distance from Earth’s center and is consequential for applications requiring high-fidelity representations of the true Earth surface, including precise geodesy, satellite altimetry, and global-scale geographic modeling.
The Earth is conventionally represented by an oblate spheroid (an ellipsoid of revolution) whose principal measures are the equatorial radius a and the polar radius b; diameters are twice these radii (2a and 2b), here 2a = 12,756.2740 km (7,926.3812 mi) and 2b = 12,713.5046 km (7,899.8055 mi). Using the equatorial radius the equatorial circumference follows from the circular relation Ce = 2πa, giving Ce = 40,075.0167 km (24,901.461 mi) for the adopted reference ellipsoid. The meridional (polar) circumference is obtained from the quarter‑meridian integral mp = a E(e) (E(e) is the complete elliptic integral of the second kind) so Cp = 4 mp; numerically Cp ≈ 40,007.8629 km (24,859.733 mi).
The ellipsoid eccentricity e = sqrt(1 − b^2/a^2) quantifies the flattening and enters the elliptic integrals and geodetic formulae that govern meridian integrals, true geodesics and other arc lengths on the surface (for the WGS84 parameters e ≈ 0.0818191908). More general surface distances, including meridian arcs and shortest paths between arbitrary surface points, are derived from a, b and e by using elliptic integrals or by applying standard geodetic solution algorithms referenced to the chosen ellipsoid.
Surface area of the reference ellipsoid can be computed consistently in multiple ways: closed‑form analytic expressions in terms of a, b and e; area estimations via map projections that preserve or approximate area; or by summing the areas of geodesic polygons on the ellipsoid. Volume is given analytically for an oblate spheroid by V = (4/3) π a^2 b; with the WGS84 values a = 6,378.137 km and b = 6,356.7523142 km this yields V ≈ 1.08321×10^12 km^3 (2.5988×10^11 cu mi). In practical geodetic and geographic work these WGS84 radii serve as the reference basis from which diameters, circumferences, eccentricity, meridian integrals, arc lengths, surface area and volume are derived.
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Nominal radii
The International Astronomical Union (IAU) has adopted fixed, exact values for Earth’s nominal radii to provide a standardized length scale in astronomical work. These are expressed using a subscript/superscript notation (for Earth, R_{e}^{N} and R_{p}^{N}) and are set exactly to 6,378.1 km (3,963.2 mi) for the equatorial radius and 6,356.8 km (3,949.9 mi) for the polar radius. Both values are referenced to the zero Earth‑tide convention, i.e., they correspond to the tidal reference state adopted for reporting these constants. By practice, the equatorial value is taken as the default “nominal radius” whenever a single Earth radius is required; the polar value is used only when a polar‑specific dimension is needed. Treated as exact constants, the nominal radii function as convenient units of length that promote consistency in size comparisons and calculations across astronomical literature and datasets. The IAU’s notation is designed to be extendable to other planetary bodies (for example, a nominal polar radius for Jupiter can be written in the same subscript/superscript form), enabling a uniform convention for nominal planetary radii.
Published values for Earth’s radius depend strongly on the chosen reference surface and the averaging or curvature metric applied. Simple nominal radii used for astronomical and geodetic conventions are provided by the IAU for a “zero‑tide” equipotential (i.e., excluding tidal deformation): equatorial ≈ 6 378 100 m and polar ≈ 6 356 800 m. These give a convenient equatorial–polar pair but do not capture more subtle distinctions among axis lengths, curvature, and equivalent spherical radii.
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The IUGG reference ellipsoid specifies the principal axis lengths (semi‑major a and semi‑minor b) and yields several different representative radii: a = 6 378 137.0 m; b = 6 356 752.3141 m; polar radius of curvature c ≈ 6 399 593.6259 m. From this ellipsoid one can derive alternative single‑value radii by different averaging criteria: mean of the semi‑axes R1 ≈ 6 371 008.7714 m; radius of a sphere with equal surface area R2 ≈ 6 371 007.1810 m; and radius of a sphere of equal volume R3 ≈ 6 371 000.79 m. These derived values distinguish axis lengths from radii that represent equivalent spheres by area or volume.
The WGS‑84 ellipsoid used in global positioning is effectively identical to the IUGG/IERS conventions: a = 6 378 137.0 m; b = 6 356 752.3142 m; c ≈ 6 399 593.6258 m, with R1, R2 and R3 matching the IUGG values to sub‑metre precision. GRS 80 likewise provides the same standard ellipsoidal axes (a = 6 378 137.0 m; b ≈ 6 356 752.31414 m), so these three systems define the common set of axis parameters used in modern geodesy.
For simplified or pedagogical treatments a single spherical radius is sometimes adopted; a representative value is R_E ≈ 6 366 707.0195 m, which averages the Earth’s figure into one scalar. Local curvature quantities also differ from axis lengths: for example, the meridional radius of curvature at the equator (the radius of curvature along a meridian at latitude 0°) is ≈ 6 335 439 m, reflecting local geometric behaviour rather than global axis dimensions.
Real topography and the ellipsoidal bulge produce extreme center‑to‑surface distances on Earth’s true figure. The farthest point from the centre is near the summit of Chimborazo in Ecuador, ≈ 6 384 400 m, while the nearest point on the seafloor occurs in parts of the Arctic Ocean at ≈ 6 352 800 m. A global mean distance from Earth’s centre to the surface, averaging actual topography and bathymetry, is approximately 6 371 230 ± 10 m, which provides a summary central value with stated uncertainty.
History
Ancient attempts to quantify Earth’s size produced estimates that varied widely because units and methods were uncertain. By about 350 BC Aristotle recorded a circulating value of 400,000 stadia for the planet’s circumference; modern assessments consider this range plausibly close to, or almost twice, the true value depending on which stadion length is assumed. The first explicit scientific measurement is attributed to Eratosthenes (c. 240 BC), who used geometrical reasoning to infer a circumference that later commentators equated with roughly 40,000 km (25,000 mi). Evaluations of Eratosthenes’ accuracy range from very good (~0.5% error) to moderately poor (~17%), the principal source of uncertainty being the ambiguous stadion unit used in transmission.
Subsequent ancient work produced similar but contested figures. Posidonius (c. 100 BC) obtained a radius comparable to Eratosthenes’ result, but a later, mistaken attribution by Strabo reduced Posidonius’ value to about three‑quarters of the true size; this diminished estimate became influential. In the second century AD Ptolemy, while firmly endorsing a spherical Earth, adopted the smaller circumference associated with Posidonius in his influential synthesis, so that medieval scholarship accepted Earth’s sphericity but often retained an underestimated magnitude.
Age‑of‑exploration voyages exposed these errors empirically. Christopher Columbus relied on the reduced contemporary estimates when projecting a short westward passage to Asia in 1492; his arrival in the Americas instead revealed that the accepted dimensions of the globe were too small. The Magellan circumnavigation (1519–1522) provided decisive practical confirmation of global continuity in longitude and distance and is commonly seen as vindicating Eratosthenes’ larger circumference estimate.
The conceptual model of Earth’s shape evolved with advances in physics. By about 1690 Isaac Newton and Christiaan Huygens argued from rotational dynamics that a spinning planet should be an oblate spheroid—flattened at the poles and bulging at the equator. Opposing readings of the new theory, most notably by Jacques Cassini around 1730, led some geodesists to favor a prolate figure (elongated at the poles), precipitating an empirical contest. The French Geodesic Mission (1735–1739) measured meridian arcs near the Arctic and the equator and resolved the dispute: the measured latitude dependence of the meridian confirmed Newton’s prediction that centrifugal force from rotation produces polar flattening.