Earth radius

Template:For multi Template:Short description Template:Infobox quantity Template:Infobox unit</math>, <math>\mathcal{R}^\mathrm N_{p\mathrm{E}}</math> | units1 = SI base unit | inunits1 = Template:Val<ref name = "IAU XXIX">Template:Cite arXiv</ref> | units2 = Metric system | inunits2 = Template:Val | units3 = English units | inunits3 = Template:Val }} Template:Geodesy

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 Template:Cvt to a minimum (polar radius, denoted b) of nearly Template:Cvt.

A globally-average value is usually considered to be Template:Convert 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 (R₁) 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 (R₂); and the volumetric radius, which is the radius of a sphere having the same volume as the ellipsoid (R₃).<ref name="Moritz" /> All three values are about Template:Convert.

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 <math>\mathcal{R}^\mathrm N_\mathrm{E}</math>) 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).<ref name="IAU XXIX" />

Introduction

File:Earth oblateness to scale.svg

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 Template:Cvt above sea level, while the yellow area denotes the altitude range of the ISS in low Earth orbit.

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Earth's rotation, internal density variations, and external tidal forces cause its shape to deviate systematically from a perfect sphere.<ref group=lower-alpha>For details see figure of the Earth, geoid, and Earth tide.</ref> 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<ref group="lower-alpha">There is no single center to the geoid; it varies according to local geodetic conditions. Where there is land, the geoid is generally below ground; it represents where the sea level would be if water could reach it from the ocean via an imaginary canal.</ref>
• A spheroid, also called an ellipsoid of revolution, geocentric to model the entire Earth, or else geodetic for regional work<ref group=lower-alpha>In 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.</ref>
• 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".<ref group=lower-alpha>The value of the radius is completely dependent upon the latitude in the case of an ellipsoid model, and nearly so on the geoid.</ref> 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 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 Template:Mvar is larger than the polar radius Template:Mvar by approximately Template:Mvar. The oblateness constant Template:Mvar is given by

<math>q=\frac{a^3 \omega^2}{GM},</math>

where Template:Mvar is the angular frequency, Template:Mvar is the gravitational constant, and Template:Mvar is the mass of the planet.Template:Refn For the Earth Template:Math, which is close to the measured inverse flattening Template:Math. 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.<ref>Satellites Reveal A Mystery Of Large Change In Earth's Gravity Field , Aug. 1, 2002, Goddard Space Flight Center. </ref>

File:Lowresgeoidheight.jpg

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 Template:Convert 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).<ref>NASA's Grace Finds Greenland Melting Faster, 'Sees' Sumatra Quake, December 20, 2005, Goddard Space Flight Center.</ref>

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

File:Abu Reyhan Biruni-Earth Circumference.svg

Al-Biruni's (973 – Template:C.) 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 Template:Convert of reference ellipsoid height, and to within Template:Convert 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 radiiTemplate:Anchor

The following radii are derived from the World Geodetic System 1984 (WGS-84) reference ellipsoid.<ref name=tr8350_2>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref name="NGA">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.Template:Unclear-inline

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 Template:Mvar, or semi-major axis,<ref name="Snyder manual"/>Template:Rp is the distance from its center to the equator and equals Template:Convert.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> The equatorial radius is often used to compare Earth with other planets.

• The Earth's polar radius Template:Mvar, or semi-minor axis<ref name="Snyder manual"/>Template:Rp is the distance from its center to the North and South Poles, and equals Template:Convert.

Location-dependent radiiTemplate:Anchor

File:EarthEllipRadii.svg

Three different radii as a function of Earth's latitude. Template:Mvar is the geocentric radius; Template:Mvar is the meridional radius of curvature; and Template:Mvar is the prime vertical radius of curvature.

Geocentric radiusTemplate:Anchor

The geocentric radius is the distance from the Earth's center to a point on the spheroid surface at geodetic latitude Template:Mvar, given by the formula <ref>Template:Cite book</ref>

<math>R(\varphi)=\sqrt{\frac{(a^2\cos\varphi)^2+(b^2\sin\varphi)^2}{(a\cos\varphi)^2+(b\sin\varphi)^2}},</math>

where Template:Mvar and Template:Mvar 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 curvatureTemplate:Anchor

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 Template:Mvar is<ref name="Jekeli">Template:Cite book</ref>

<math>M(\varphi)=\frac{(ab)^2}{\big((a\cos\varphi)^2+(b\sin\varphi)^2\big)^\frac32}

=\frac{a(1-e^2)}{(1-e^2\sin^2\varphi)^\frac32} =\frac{1-e^2}{a^2} N(\varphi)^3,</math> where <math>e</math> is the eccentricity of the earth. This is the radius that Eratosthenes measured in his arc measurement.

Prime vertical

File:Geodetic latitude and the length of Normal.svg

The length PQ, called the prime vertical radius, is <math>N(\phi).</math> The length IQ is equal to <math>e^2 N(\phi).</math> <math>R = (X, Y, Z).</math>

If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.<ref group=lower-alpha name=curvprim>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.</ref>

This Earth's prime-vertical radius of curvature, also called the Earth's transverse radius of curvature, is defined perpendicular (orthogonal) to Template:Mvar at geodetic latitude Template:Mvar<ref group=lower-alpha>Template:Mvar 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.</ref> and is<ref name="Jekeli"/>

<math>N(\varphi)=\frac{a^2}{\sqrt{(a\cos\varphi)^2+(b\sin\varphi)^2}}

=\frac{a}{\sqrt{1-e^2\sin^2\varphi}}.</math>

N can also be interpreted geometrically as the normal distance from the ellipsoid surface to the polar axis.<ref name="Bowring">Template:Cite journal </ref> The radius of a parallel of latitude is given by <math>p=N\cos(\varphi)</math>.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>

Polar and equatorial radius of curvature

The Earth's meridional radius of curvature at the equator equals the meridian's semi-latus rectum:

<math>M(0\text{°})=\frac{b^2}{a}=6,335.439 \text{ km.}</math>

The Earth's prime-vertical radius of curvature at the equator equals the equatorial radius, <math>N(0\text{°})=a</math>

The Earth's polar radius of curvature (either meridional or prime-vertical) is <math>M(90\text{°})=N(90\text{°})=\frac{a^2}{b}=6,399.594 \text{ km.}</math>

Derivation

Template:Collapse top The principal curvatures are the roots of Equation (125) in:<ref name="Lass">Template:Cite book</ref>

<math>(E G - F^2) \kappa^2 - (e G + g E - 2 f F) \kappa + (e g - f^2) = 0

= \det(A - \kappa B),</math>

where in the first fundamental form for a surface (Equation (112) in<ref name="Lass"></ref>):

<math>ds^2 = \sum_{ij} a_{ij} dw^i dw^j = E \,d\varphi^2 + 2 F \,d\varphi \,d\lambda + G \,d\lambda^2,</math>

E, F, and G are elements of the metric tensor:

<math> 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},</math>

<math>r = [r^1, r^2, r^3]^T = [x, y, z]^T</math>, <math>w^1 = \varphi</math>, <math>w^2 = \lambda,</math>

in the second fundamental form for a surface (Equation (123) in<ref name="Lass"></ref>):

<math> 2 D = \sum_{ij} b_{ij} dw^i dw^j = e \,d\varphi^2 + 2 f \,d\varphi \,d\lambda + g \,d\lambda^2,</math>

e, f, and g are elements of the shape tensor:

<math>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},</math>

<math>n = \frac{N}{|N|}</math> is the unit normal to the surface at <math>r</math>, and because <math>\frac{\partial r}{\partial \varphi}</math> and <math>\frac{\partial r}{\partial \lambda}</math> are tangents to the surface,

<math>N = \frac{\partial r}{\partial \varphi} \times \frac{\partial r}{\partial \lambda}</math>

is normal to the surface at <math>r</math>.

With <math>F = f = 0</math> for an oblate spheroid, the curvatures are

<math>\kappa_1 = \frac{g}{G}</math> and <math>\kappa_2 = \frac{e}{E},</math>

and the principal radii of curvature are

<math>R_1 = \frac{1}{\kappa_1}</math> and <math>R_2 = \frac{1}{\kappa_2}.</math>

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 <math>r + dr</math> to the plane tangent at <math>r</math>. Template:Collapse bottom

Combined radii of curvature

AzimuthalTemplate:Anchor

The Earth's azimuthal radius of curvature, along an Earth normal section at an azimuth (measured clockwise from north) Template:Mvar and at latitude Template:Mvar, is derived from Euler's curvature formula as follows:<ref name=Torge/>Template:Rp

<math>R_\mathrm{c}=\frac{1}{\dfrac{\cos^2\alpha}{M}+\dfrac{\sin^2\alpha}{N}}.</math>

Non-directional

It is possible to combine the principal radii of curvature above in a non-directional manner.

Template:AnchorTemplate:AnchorThe Earth's Gaussian radius of curvature at latitude Template:Mvar is<ref name=Torge>Template:Cite book</ref>

<math>R_\text{a}(\varphi) = \frac{1}{\sqrt{K}} = \frac{1}{2\pi}\int_{0}^{2\pi}R_\text{c}(\alpha)\,d\alpha = \sqrt{MN} = \frac{a^2b}{(a\cos\varphi)^2+(b\sin\varphi)^2}

= \frac{a\sqrt{1-e^2}}{1-e^2\sin^2\varphi},</math> where K is the Gaussian curvature, <math>K = \kappa_1\,\kappa_2 = \frac{\det B}{\det A}.</math>

Template:AnchorThe Earth's mean radius of curvature at latitude Template:Mvar is<ref name=Torge />Template:Rp

<math>R_\text{m} = \frac{2}{\dfrac{1}{M} + \dfrac{1}{N}}.</math>

Global radiiTemplate:Anchor

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;<ref name=tr8350_2 /> namely,

Equatorial radius: Template:Mvar = (Template:Val)

Polar radius: Template:Mvar = (Template:Val)

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.

Arithmetic mean radius

File:WGS84 mean Earth radius.svg

Equatorial (a), polar (b) and arithmetic mean Earth radii as defined in the 1984 World Geodetic System revision (not to scale)

In geophysics, the International Union of Geodesy and Geophysics (IUGG) defines the Earth's arithmetic mean radius (denoted Template:Math) to be<ref name="Moritz">Moritz, H. (1980). Geodetic Reference System 1980 Template:Webarchive, by resolution of the XVII General Assembly of the IUGG in Canberra.</ref>

<math>R_1 = \frac{2a+b}{3}.</math>

The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid.<ref name="Moritz2000">Template:Cite journal</ref> For Earth, the arithmetic mean radius is published by IUGG and NGA as Template:Convert.<ref name="IAU XXIX"/><ref name="NGA"/>

Authalic radius

Template:See also 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 Template:Math.<ref name="Moritz"/> A closed-form solution exists for a spheroid:<ref name="Snyder manual">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.</ref>

<math>R_2

=\sqrt{\frac12\left(a^2+\frac{b^2}{e}\ln{\frac{1+e}{b/a}} \right) } =\sqrt{\frac{a^2}2+\frac{b^2}2\frac{\tanh^{-1}e}e} =\sqrt{\frac{A}{4\pi}},</math> where Template:Tmath is the eccentricity, and Template:Tmath is the surface area of the spheroid.

For the Earth, the authalic radius is Template:Convert.<ref name=Moritz2000/>

The authalic radius <math>R_2</math> also corresponds to the radius of (global) mean curvature, obtained by averaging the Gaussian curvature, <math>K</math>, over the surface of the ellipsoid. Using the Gauss–Bonnet theorem, this gives

<math> \frac{\int K \,dA}{A} = \frac{4\pi}{A} = \frac{1}{R_2^2}.</math>

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 Template:Math.<ref name="Moritz"/>

<math>R_3 = \sqrt[3]{a^2b}.</math>

For Earth, the volumetric radius equals Template:Convert.<ref name=Moritz2000/>

Rectifying radius

Template:See also 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:

<math>M_\text{r} = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \sqrt{a^2 \cos^2\varphi + b^2 \sin^2\varphi} \,d\varphi.</math>

The rectifying radius is equivalent to the meridional mean, which is defined as the average value of Template:Mvar:<ref name="Snyder manual"/>

<math>M_\text{r} = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} M(\varphi) \,d\varphi.</math>

For integration limits of [0,Template:Sfrac], the integrals for rectifying radius and mean radius evaluate to the same result, which, for Earth, amounts to Template:Convert.

The meridional mean is well approximated by the semicubic mean of the two axes,Template:Cn

<math>M_\text{r} \approx \left(\frac{a^\frac32 + b^\frac32}{2}\right)^\frac23,</math>

which differs from the exact result by less than Template:Convert; the mean of the two axes,

<math>M_\text{r} \approx \frac{a + b}{2},</math>

about Template:Convert, can also be used.

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, R_t, which depends not only on latitude.

Topographical extremes

• Maximum R_t: the summit of Chimborazo is Template:Convert from the Earth's center.
• Minimum R_t: the floor of the Arctic Ocean is Template:Convert from the Earth's center.<ref name=extrema>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Topographical global mean

The topographical mean geocentric distance averages elevations everywhere, resulting in a value Template:Val larger than the IUGG mean radius, the authalic radius, or the volumetric radius. This topographical average is Template:Convert with uncertainty of Template:Convert.<ref name="chambat">Template:Cite journal</ref>

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

Template:AnchorEarth's diameter is simply twice Earth's radius; for example, equatorial diameter (2a) and polar diameter (2b). For the WGS84 ellipsoid, that's respectively:

Template:Math,

Template:Math.

Earth's circumference equals the perimeter length. The equatorial circumference is simply the circle perimeter: C_e = 2πa, in terms of the equatorial radius a. The polar circumference equals C_p = 4m_p, four times the quarter meridian m_p = 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.

Template:AnchorEarth's volume, or that of the reference ellipsoid, is <math>V = \tfrac{4}{3}\pi a^2 b.</math> Using the parameters from WGS84 ellipsoid of revolution, Template:Nowrap and Template:Nowrap, Template:Nowrap.<ref name="earth_fact_sheet">Template:Citation</ref>

Nominal radiiTemplate:Anchor

In astronomy, the International Astronomical Union denotes the nominal equatorial Earth radius as <math>\mathcal{R}^\text{N}_\text{eE}</math>, which is defined to be exactly Template:Convert.<ref name = "IAU XXIX"/>Template:Rp The nominal polar Earth radius is defined exactly as <math>\mathcal{R}^\text{N}_\text{pE}</math> = Template:Convert. 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.<ref name = "IAU XXIX"/>Template:Rp The nominal radius serves as a unit of length for astronomy. (The notation is defined such that it can be easily generalized for other planets; e.g., <math>\mathcal{R}^\text{N}_\text{pJ}</math> for the nominal polar Jupiter radius.)

Published values

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

Agency	Description	Value (in meters)	Ref
IAU	nominal "zero tide" equatorial	Template:Val	Template:R
IAU	nominal "zero tide" polar	Template:Val	Template:R
IUGG	equatorial radius	Template:Val	Template:R
IUGG	semiminor axis (b)	Template:Val	Template:R
IUGG	polar radius of curvature (c)	Template:Val	Template:R
IUGG	mean radius (R1)	Template:Val	Template:R
IUGG	radius of sphere of same surface (R2)	Template:Val	Template:R
IUGG	radius of sphere of same volume (R3)	Template:Val	Template:R
NGA	WGS-84 ellipsoid, semi-major axis (a)	Template:Val	Template:R
NGA	WGS-84 ellipsoid, semi-minor axis (b)	Template:Val	Template:R
NGA	WGS-84 ellipsoid, polar radius of curvature (c)	Template:Val	Template:R
NGA	WGS-84 ellipsoid, Mean radius of semi-axes (R1)	Template:Val	Template:R
NGA	WGS-84 ellipsoid, Radius of Sphere of Equal Area (R2)	Template:Val	Template:R
NGA	WGS-84 ellipsoid, Radius of Sphere of Equal Volume (R3)	Template:Val	Template:R
	GRS 80 semi-major axis (a)	Template:Val
	GRS 80 semi-minor axis (b)	Template:Val
	Spherical Earth Approx. of Radius (RE)	Template:Val	<ref name="Phillips">Template:Cite book</ref>
	meridional radius of curvature at the equator	Template:Val
	Maximum (the summit of Chimborazo)	Template:Val	Template:R
	Minimum (the floor of the Arctic Ocean)	Template:Val	Template:R
	Average distance from center to surface	Template:Val	Template:R

History

The first published reference to the Earth's size appeared around 350 BC, when Aristotle reported in his book On the Heavens<ref>Template:Cite book</ref> 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<ref> Template:Cite journal</ref> to almost double the true value.<ref> Template:Cite EB1911</ref> The first known scientific measurement and calculation of the circumference of the Earth was performed by Eratosthenes in about 240 BC. Estimates of the error of Eratosthenes's measurement range from 0.5% to 17%.<ref> Template:Cite encyclopedia</ref> For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant.

Around 100 BC, Posidonius of Apamea recomputed Earth's radius, and found it to be close to that by Eratosthenes,<ref>Posidonius, fragment 202</ref> but later Strabo incorrectly attributed him a value about 3/4 of the actual size.<ref>Cleomedes (in Fragment 202) stated that if the distance is measured by some other number the result will be different, and using 3,750 instead of 5,000 produces this estimation: 3,750 x 48 = 180,000; see Fischer I., (1975), Another Look at Eratosthenes' and Posidonius' Determinations of the Earth's Circumference, Ql. J. of the Royal Astron. Soc., Vol. 16, p. 152.</ref> Claudius Ptolemy around 150 AD gave empirical evidence supporting a spherical Earth,<ref>Template:Cite book</ref> but he accepted the lesser value attributed to Posidonius. His highly influential work, the Almagest,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> left no doubt among medieval scholars that Earth is spherical, but they were wrong about its size.

By 1490, Christopher Columbus believed that traveling 3,000 miles west from the west coast of the Iberian Peninsula would let him reach the eastern coasts of Asia.<ref>John Freely, Before Galileo: The Birth of Modern Science in Medieval Europe (2013), Template:ISBN</ref> However, the 1492 enactment of that voyage brought his fleet to the Americas. The Magellan expedition (1519–1522), which was the first circumnavigation of the World, soundly demonstrated the sphericity of the Earth,<ref>Template:Cite book</ref> and affirmed the original measurement of Template:Convert by Eratosthenes.

Around 1690, Isaac Newton and Christiaan Huygens argued that Earth was closer to an oblate spheroid than to a sphere. However, around 1730, Jacques Cassini argued for a prolate spheroid instead, due to different interpretations of the Newtonian mechanics involved.<ref>Template:Cite book</ref> To settle the matter, the French Geodesic Mission (1735–1739) measured one degree of latitude at two locations, one near the Arctic Circle and the other near the equator. The expedition found that Newton's conjecture was correct:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> the Earth is flattened at the poles due to rotation's centrifugal force.

See also

• Earth's circumference
• Earth mass
• Effective Earth radius
• Geodesy
• Geographical distance
• Osculating sphere
• History of geodesy
• Planetary radius

External links

• {{#invoke:citation/CS1|citation

|CitationClass=web }}