Earth radius

The Earth's radius isn't a single, fixed number; it's a fascinating dance between rotation, gravity, and the very ground beneath our feet, forcing scientists to create clever models to define our planet's size. Unraveling these different "radii" reveals not only the Earth's true shape but also how we measure and understand our home in the cosmos. Join us as we explore the many ways to measure our magnificent planet. Earth's radius is not uniform; it varies from the equator to the poles due to rotation, making it an oblate spheroid. Scientists use several idealized models for Earth's radius, each suited for different applications like mapping, navigation, or planetary comparison. Local topography, gravitational anomalies, and even lunar tides introduce constant, subtle changes, making a single, universally "precise" radius impossible.

Source: Wikipedia

AI Summary

Earth's radius is not uniform; it varies from the equator to the poles due to rotation, making it an oblate spheroid.
Scientists use several idealized models for Earth's radius, each suited for different applications like mapping, navigation, or planetary comparison.
Local topography, gravitational anomalies, and even lunar tides introduce constant, subtle changes, making a single, universally "precise" radius impossible.

What is Earth's Radius?

When we talk about the Earth's radius—often denoted as R🜨 or RE—we're referring to the distance from our planet's center to a point on or near its surface. Sounds simple, right? But Earth is not a perfect sphere, which makes defining its radius a surprisingly complex endeavor.

Imagine a spinning ball of dough. As it whirls, it flattens at the poles and bulges at the equator. That's essentially what happens to Earth! Our planet is an "oblate ellipsoid" or "spheroid," meaning it's slightly squashed. This means the radius isn't the same everywhere.

At its widest point, the equatorial radius (denoted 'a') is about 6,378 kilometers (3,963 miles). At its narrowest, from the center to the poles, the polar radius (denoted 'b') is nearly 6,357 kilometers (3,950 miles). That's a difference of over 21 kilometers!

Finding a Global Average

Because of this variation, a single, universally accepted "average" radius is often needed for global calculations. Most commonly, this average is taken to be approximately 6,371 kilometers (3,959 miles), with only about 0.3% variability.

The International Union of Geodesy and Geophysics (IUGG) provides several important reference values for this global average. These include the mean radius, the authalic radius (for equal surface area), and the volumetric radius (for equal volume). All these fall very close to that 6,371 kilometer mark.

Beyond these averages, other specialized definitions exist. Some focus on the Earth's curvature at specific points, while others account for the actual topography—the mountains and valleys—of our planet. These can yield values slightly outside the equatorial and polar extremes.

In astronomy and geophysics, a "nominal Earth radius" is often used as a standard unit of measurement. It acts as a conversion factor, helping us express properties of other planets as multiples of Earth's size. Unless specified, this usually refers to the nominal equatorial radius.

The Earth's Dynamic Shape

Why isn't Earth a perfect sphere? It's a combination of factors: our planet's rotation, the varying density of its internal layers, and even external gravitational tugs from the Moon and Sun. These forces constantly reshape Earth, giving it a profoundly complex surface.

To make sense of this complexity, scientists use simplified models. Think of it like a set of nested approximations: the actual surface, the geoid (mean sea level), a spheroid (flattened sphere), and finally, a perfect sphere. Each model sacrifices some detail for practical use.

For the geoid and spheroid models, we can talk about a "radius at that point." But for the chaotic real surface, "elevation above sea level" is usually more practical than a "radius." Despite the variations, the Earth deviates from a perfect sphere by only about one-third of a percent, which is why a simple spherical model often works well enough.

Physics of Earth's Flattening

A planet's rotation causes it to bulge at the equator and flatten at the poles. The degree of this flattening is described by an "oblateness constant," 'q', which depends on the planet's equatorial radius, angular velocity, gravitational constant, and mass.

For Earth, this constant 'q' is approximately 1/289. This calculated value aligns closely with the measured inverse flattening of 1/298.257, confirming that rotation is the primary reason for our planet's oblate shape. Intriguingly, even this bulge can subtly change over time, possibly due to shifting ocean currents.

q={\frac {a^{3}\omega ^{2}}{GM}}

Beyond rotation, variations in Earth's internal density and crustal thickness cause gravity to fluctuate across the surface. This means that "mean sea level" isn't perfectly smooth; it forms an undulating surface called the "geoid." The geoid can differ from the idealized ellipsoid by as much as 110 meters globally.

Even external forces play a role! The gravitational pull from the Moon and Sun causes "Earth tides," which can deform the surface by tenths of a meter over a roughly 12-hour period. These constant, subtle changes highlight why a single, unchanging "radius" is an elusive concept.

Measuring Local Curvature

Given these many local and transient influences on surface height, a "general purpose" model for Earth's radius must balance global precision with practical application. The values we use are refined to within meters of a reference ellipsoid and hundreds of meters of mean sea level.

Another way to think about the Earth's radius is through its curvature at any given point. Like the surface of a donut, the Earth's curvature is not uniform. It's greatest (tightest) in the north-south direction along meridians and smallest (flattest) in the east-west direction along parallels of latitude.

This difference in curvature means, for example, that the true horizon at the equator is slightly shorter when looking north or south compared to looking east or west. This intricate geometry makes defining a single "radius of curvature" at a point quite specific to direction.

In summary, local terrain variations prevent us from defining a single, perfectly "precise" radius. From Eratosthenes' ancient measurements to modern satellite remote sensing and GPS, scientists have continually developed new models—first regional, then global—to best approximate our planet's multifaceted shape.

The Extremes: Equatorial and Polar Radii

The World Geodetic System 1984 (WGS-84) provides the most widely used reference ellipsoid for Earth. It's an idealized surface, and the measurements used to define it have an uncertainty of about ±2 meters for both equatorial and polar dimensions.

Under WGS-84, the Earth's equatorial radius, 'a', also known as the semi-major axis, is precisely 6,378.1370 kilometers (3,963.1906 miles). This value is often used when comparing Earth's size to other planets.

The Earth's polar radius, 'b', or semi-minor axis, measures the distance from the center to the North and South Poles, totaling 6,356.7523 kilometers (3,949.9028 miles). These two values define the basic dimensions of our planet's shape.

Location-Dependent Radii

Geocentric Radius

The "geocentric radius" measures the distance from Earth's center to a specific point on the spheroid's surface at a given geodetic latitude, denoted as φ. This radius naturally changes as you move from the equator to the poles.

The formula for the geocentric radius R(φ) uses both the equatorial radius 'a' and the polar radius 'b'. The maximum geocentric radius occurs at the equator, matching 'a', and the minimum occurs at the poles, matching 'b'.

R(\varphi )={\sqrt {\frac {(a^{2}\cos \varphi )^{2}+(b^{2}\sin \varphi )^{2}}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}

Principal Radii of Curvature

Instead of direct distance from the center, we can define radii based on how much the Earth's surface curves at a specific point. There are two primary directions for this: along a meridian (north-south) and perpendicular to it (east-west), known as the meridional and prime-vertical radii of curvature.

The Earth's meridional radius of curvature, M(φ), describes the curvature in the north-south direction at latitude φ. This is the kind of radius Eratosthenes effectively measured when he calculated Earth's circumference thousands of years ago.

Here, 'e' represents the Earth's eccentricity, a measure of how much it deviates from a perfect circle. This value for M(φ) is crucial for understanding how distances are measured along lines of longitude.

M(\varphi )={\frac {a(1-e^{2})}{(1-e^{2}\sin ^{2}\varphi )^{\frac {3}{2}}}}

Perpendicular to the meridional direction is the prime-vertical radius of curvature, N(φ), sometimes called the transverse radius. This describes the Earth's curvature in the east-west direction at latitude φ.

N(φ) can also be seen as the normal distance from the ellipsoid surface to the polar axis. Its value is essential for calculations involving distances along parallels of latitude.

N(\varphi )={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\varphi }}}

Azimuthal Radius

What if you're looking in a direction that's not exactly north-south or east-west? The azimuthal radius of curvature, Rc, combines the principal radii to give you the curvature in any arbitrary direction (azimuth α) at a given latitude.

This formula, derived from Euler's curvature theorem, shows how the curvature in any direction is a blend of the meridional (M) and prime-vertical (N) curvatures, weighted by the azimuth angle.

R_{\mathrm {c} }^{-1}=M^{-1}\cos ^{2}\alpha +N^{-1}\sin ^{2}\alpha

Article

Earth radius

Earth radius (denoted as R🜨 or RE) 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 6,378 km (3,963 mi) to a minimum (polar radius, denoted b) of nearly 6,357 km (3,950 mi).

A globally-average value is usually considered to be 6,371 kilometres (3,959 mi) with a 0.3% variability (±10 km) for the following reasons. The International Union of Geodesy and Geophysics (IUGG) provides three reference values: the mean radius (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). All three values are about 6,371 kilometres (3,959 mi).

Other ways to define and measure the Earth's radius involve either the spheroid's radius of curvature or the actual topography. A few definitions yield values outside the range between the polar radius and equatorial radius because they account for localized effects.

A nominal Earth radius (denoted ${\displaystyle {\mathcal {R}}{\mathrm {E} }^{\mathrm {N} }}$) is sometimes used as a unit of measurement in astronomy and geophysics, a conversion factor used when expressing planetary properties as multiples or fractions of a constant terrestrial radius; if the choice between equatorial or polar radii is not explicit, the equatorial radius is to be assumed, as recommended by the International Astronomical Union (IAU).

Introduction

Earth radius

A scale diagram of the oblateness of the 2003 IERS reference ellipsoid, with north at the top. The light blue region is a circle. The outer edge of the dark blue line is an ellipse with the same minor axis as the circle and the same eccentricity as the Earth. The red line represents the Karman line 100 km (62 mi) above sea level, while the yellow area denotes the altitude range of the ISS in low Earth orbit.

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:

• The actual surface of Earth • The geoid, defined by mean sea level at each point on the real surface • A spheroid, also called an ellipsoid of revolution, geocentric to model the entire Earth, or else geodetic for regional work • 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". 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 a is larger than the polar radius b by approximately aq. The oblateness constant q is given by

${\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}},}$

where ω is the angular frequency, G is the gravitational constant, and M is the mass of the planet. For the Earth ⁠1/q⁠ ≈ 289, which is close to the measured inverse flattening ⁠1/f⁠ ≈ 298.257. Additionally, the bulge at the equator shows slow variations. The bulge had been decreasing, but since 1998 the bulge has increased, possibly due to redistribution of ocean mass via currents.

The variation in density and crustal thickness causes gravity to vary across the surface and in time, so that the mean sea level differs from the ellipsoid. This difference is the geoid height, positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under 110 m (360 ft) on Earth. The geoid height can change abruptly due to earthquakes (such as the Sumatra-Andaman earthquake) or reduction in ice masses (such as Greenland).

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

Al-Biruni's (973 – c. 1050) method for calculation of the Earth's radius simplified measuring the circumference compared to taking measurements from two locations distant from each other.

Given local and transient influences on surface height, the values defined below are based on a "general purpose" model, refined as globally precisely as possible within 5 m (16 ft) of reference ellipsoid height, and to within 100 m (330 ft) of mean sea level (neglecting geoid height).

Additionally, the radius can be estimated from the curvature of the Earth at a point. Like a torus, the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The corresponding radius of curvature depends on the location and direction of measurement from that point. A consequence is that a distance to the true horizon at the equator is slightly shorter in the north–south direction than in the east–west direction.

In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by Eratosthenes, many models have been created. Historically, these models were based on regional topography, giving the best reference ellipsoid for the area under survey. As satellite remote sensing and especially the Global Positioning System gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole.

Extrema: equatorial and polar radii

Earth radius

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.

• The Earth's equatorial radius a, or semi-major axis, is the distance from its center to the equator and equals 6,378.1370 km (3,963.1906 mi). The equatorial radius is often used to compare Earth with other planets. • The Earth's polar radius b, or semi-minor axis is the distance from its center to the North and South Poles, and equals 6,356.7523 km (3,949.9028 mi).

Location-dependent radii

Earth radius

Various radii as a function of latitude.

Geocentric radius

The geocentric radius is the distance from the Earth's center to a point on the spheroid surface at geodetic latitude φ, given by the formula

${\displaystyle R(\varphi )={\sqrt {\frac {(a^{2}\cos \varphi )^{2}+(b^{2}\sin \varphi )^{2}}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}},}$

where a and b are, respectively, the equatorial radius and the polar radius.

The maximum and minimum geocentric radii on the ellipsoid coincide, respectively, with the equatorial and polar radii, at the vertices of the ellipse. However, the maximum and minimum radii of curvature coincide with the poles and Equator, in opposite regions than the geocentric radii, due to Earth's oblateness.

Principal radii of curvature

There are two principal radii of curvature: along the meridional and prime-vertical normal sections.

$${\displaystyle (EG-F^{2})\kappa ^{2}-(eG+gE-2fF)\kappa +(eg-f^{2})=0=\det(A-\kappa B),}$$

Meridional

In particular, the Earth's meridional radius of curvature (in the north–south direction) at φ is

${\displaystyle M(\varphi )={\frac {(ab)^{2}}{{\big (}(a\cos \varphi )^{2}+(b\sin \varphi )^{2}{\big )}^{\frac {3}{2}}}}={\frac {a(1-e^{2})}{(1-e^{2}\sin ^{2}\varphi )^{\frac {3}{2}}}}={\frac {1-e^{2}}{a^{2}}}N(\varphi )^{3},}$

where ${\displaystyle e}$ is the eccentricity of the earth. This is the radius that Eratosthenes measured in his arc measurement.

Prime vertical

The length PQ, called the prime vertical radius, is ${\displaystyle N(\phi ).}$ The length IQ is equal to ${\displaystyle e^{2}N(\phi ).}$ ${\displaystyle R=(X,Y,Z).}$

If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction. This Earth's prime-vertical radius of curvature, also called the Earth's transverse radius of curvature, is defined perpendicular (orthogonal) to M at geodetic latitude φ and is

${\displaystyle N(\varphi )={\frac {a^{2}}{\sqrt {(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\varphi }}}.}$

N can also be interpreted geometrically as the normal distance from the ellipsoid surface to the polar axis. The radius of a parallel of latitude is given by ${\displaystyle p=N\cos(\varphi )}$.

Azimuthal

Euler radius vs. azimuth at various latitudes

Earth's azimuthal radius of curvature of an Earth normal section at an azimuth (measured clockwise from north) α and at latitude φ, is derived from Euler's curvature formula as follows:

${\displaystyle R{\mathrm {c} }={\frac {1}{{\dfrac {\cos ^{2}\alpha }{M}}+{\dfrac {\sin ^{2}\alpha }{N}}}}}$

${\displaystyle R{\mathrm {c} }^{-1}=M^{-1}\cos ^{2}\alpha +N^{-1}\sin ^{2}\alpha ,}$

where ${\displaystyle R{\mathrm {c} }^{-1}}$ is the azimuthal curvature of the ellipsoid.

Non-directional (Gaussian and mean)

Difference between mean and Gaussian radii of curvature

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

Earth's Gaussian radius of curvature at latitude φ is

${\displaystyle R{\text{a}}(\varphi )={\frac {1}{\sqrt {K}}}={\frac {1}{2\pi }}\int {0}^{2\pi }R{\text{c}}(\alpha )\,d\alpha ,}$

${\displaystyle R{\text{a}}(\varphi )={\sqrt {MN}}={\frac {a^{2}b}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}={\frac {a{\sqrt {1-e^{2}}}}{1-e^{2}\sin ^{2}\varphi }},}$

where K is the Gaussian curvature, ${\displaystyle K=\kappa {1}\,\kappa {2}.}$ The Gaussian radius of curvature is defined in terms of the product of the principal radii of curvature and it integrates the azimuthal radius of curvature ${\displaystyle R{\mathrm {c} }}$ over the full circle. It corresponds to the radius of the osculating sphere that best fits the ellipsoid locally.

${\displaystyle R{\text{m}}={\frac {2}{{\dfrac {1}{M}}+{\dfrac {1}{N}}}}}$

${\displaystyle R{\text{m}}^{-1}={\frac {M^{-1}+N^{-1}}{2}}={\frac {1}{2\pi }}\int {0}^{2\pi }R{c}^{-1}(\theta )\;d\theta .}$

The mean curvature ${\displaystyle R{\text{m}}^{-1}}$ equals the arithmetic mean of the two principal curvatures and integrates the azimuthal curvature ${\displaystyle R{\mathrm {c} }^{-1}}$ over the full circle.

Equatorial radii of curvature

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

${\displaystyle M(0{\text{°}})={\frac {b^{2}}{a}}=\ell =6,335.439{\text{ km.}}}$

Earth's prime-vertical radius of curvature at the equator equals the equatorial radius,

Earth's Gaussian radius of curvature at the equator simplifies to the polar radius (not the equatorial radius):

Earth's mean radius of curvature at the equator involves the semi-latus rectum:

${\displaystyle R{\text{m}}(0{\text{°}})=2(a\cdot \ell )/(a+\ell )=6,356.716{\text{ km.}}}$

Polar radii of curvature

Earth's radius of curvature at the poles (either meridional or prime-vertical) is

${\displaystyle M(90{\text{°}})=N(90{\text{°}})={\frac {a^{2}}{b}}=6,399.594{\text{ km.}}}$

Earth's Gaussian or mean radii of curvature at the poles also equal the principal radii of curvature at the poles (not the polar radius):

${\displaystyle R{\text{a}}(90{\text{°}})=R{\text{m}}(90{\text{°}})={\frac {a^{2}}{b}}}$

Global radii

Earth radius

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,

Equatorial radius: a = (6378.1370 km) Polar radius: b = (6356.7523 km)

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

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 R1) to be

The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid. For Earth, the arithmetic mean radius is published by IUGG and NGA as 6,371.0087714 km (3,958.7613160 mi).

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 R2. A closed-form solution exists for a spheroid:

${\displaystyle R{2}={\sqrt {{\frac {1}{2}}\left(a^{2}+{\frac {b^{2}}{e}}\ln {\frac {1+e}{b/a}}\right)}}={\sqrt {{\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}{\frac {\tanh ^{-1}e}{e}}}}={\sqrt {\frac {A}{4\pi }}},}$

where ⁠${\displaystyle \textstyle e={\sqrt {a^{2}-b^{2}}}{\big /}a}$⁠ is the eccentricity, and ⁠${\displaystyle A}$⁠ is the surface area of the spheroid.

For the Earth, the authalic radius is 6,371.0072 km (3,958.7603 mi).

The authalic radius ${\displaystyle R{2}}$ also corresponds to the radius of (global) mean curvature, obtained by averaging the Gaussian curvature, ${\displaystyle K}$, over the surface of the ellipsoid. Using the Gauss–Bonnet theorem, this gives

${\displaystyle {\frac {\int K\,dA}{A}}={\frac {4\pi }{A}}={\frac {1}{R{2}^{2}}}.}$

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 R3.

For Earth, the volumetric radius equals 6,371.0008 km (3,958.7564 mi).

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:

${\displaystyle M{\text{r}}={\frac {2}{\pi }}\int {0}^{\frac {\pi }{2}}{\sqrt {a^{2}\cos ^{2}\varphi +b^{2}\sin ^{2}\varphi }}\,d\varphi .}$

The rectifying radius is equivalent to the meridional mean, which is defined as the average value of M:

${\displaystyle M{\text{r}}={\frac {2}{\pi }}\int {0}^{\frac {\pi }{2}}M(\varphi )\,d\varphi .}$

For integration limits of [0,⁠π/2⁠], the integrals for rectifying radius and mean radius evaluate to the same result, which, for Earth, amounts to 6,367.4491 km (3,956.5494 mi).

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

${\displaystyle M{\text{r}}\approx \left({\frac {a^{\frac {3}{2}}+b^{\frac {3}{2}}}{2}}\right)^{\frac {2}{3}},}$

which differs from the exact result by less than 1 μm (4×10−5 in); the mean of the two axes,

${\displaystyle M{\text{r}}\approx {\frac {a+b}{2}},}$

about 6,367.445 km (3,956.547 mi), can also be used.

Topographical radii

Earth radius

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 extremes

• Maximum Rt: the summit of Chimborazo is 6,384.4 km (3,967.1 mi) from the Earth's center. • Minimum Rt: the floor of the Arctic Ocean is 6,352.8 km (3,947.4 mi) from the Earth's center.

Topographical global mean

The topographical mean geocentric distance averages elevations everywhere, resulting in a value 230 m larger than the IUGG mean radius, the authalic radius, or the volumetric radius. This topographical average is 6,371.230 km (3,958.899 mi) with uncertainty of 10 m (33 ft).

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

Earth radius

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

2a = 12,756.2740 km (7,926.3812 mi), 2b = 12,713.5046 km (7,899.8055 mi).

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):

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 ${\displaystyle V={\tfrac {4}{3}}\pi a^{2}b.}$ Using the parameters from WGS84 ellipsoid of revolution, a = 6,378.137 km and b = 6356.7523142 km, it results in:

Nominal radii

Earth radius

In astronomy, the International Astronomical Union denotes the nominal equatorial Earth radius as ${\displaystyle {\mathcal {R}}{\text{eE}}^{\text{N}}}$, which is defined to be exactly 6,378.1 km (3,963.2 mi). The nominal polar Earth radius is defined exactly as ${\displaystyle {\mathcal {R}}{\text{pE}}^{\text{N}}}$ = 6,356.8 km (3,949.9 mi). 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. (The notation is defined such that it can be easily generalized for other planets; e.g., ${\displaystyle {\mathcal {R}}{\text{pJ}}^{\text{N}}}$ for the nominal polar Jupiter radius.)

Published values

Earth radius

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

<table><thead><tr><th>Agency</th><th>Description</th><th>Value (in meters)</th><th>Ref.</th></tr></thead><tbody><tr><td>IAU</td><td>nominal "zero tide" equatorial</td><td>6378100</td><td></td></tr><tr><td>IAU</td><td>nominal "zero tide" polar</td><td>6356800</td><td></td></tr><tr><td>IUGG</td><td>equatorial radius</td><td>6378137</td><td></td></tr><tr><td>IUGG</td><td>semiminor axis (b)</td><td>6356752.3141</td><td></td></tr><tr><td>IUGG</td><td>polar radius of curvature (c)</td><td>6399593.6259</td><td></td></tr><tr><td>IUGG</td><td>mean radius (R1)</td><td>6371008.7714</td><td></td></tr><tr><td>IUGG</td><td>radius of sphere of same surface (R2)</td><td>6371007.1810</td><td></td></tr><tr><td>IUGG</td><td>radius of sphere of same volume (R3)</td><td>6371000.7900</td><td></td></tr><tr><td>NGA</td><td>WGS-84 ellipsoid, semi-major axis (a)</td><td>6378137.0</td><td></td></tr><tr><td>NGA</td><td>WGS-84 ellipsoid, semi-minor axis (b)</td><td>6356752.3142</td><td></td></tr><tr><td>NGA</td><td>WGS-84 ellipsoid, polar radius of curvature (c)</td><td>6399593.6258</td><td></td></tr><tr><td>NGA</td><td>WGS-84 ellipsoid, Mean radius of semi-axes (R1)</td><td>6371008.7714</td><td></td></tr><tr><td>NGA</td><td>WGS-84 ellipsoid, Radius of Sphere of Equal Area (R2)</td><td>6371007.1809</td><td></td></tr><tr><td>NGA</td><td>WGS-84 ellipsoid, Radius of Sphere of Equal Volume (R3)</td><td>6371000.7900</td><td></td></tr><tr><td></td><td>GRS 80 semi-major axis (a)</td><td>6378137.0</td><td></td></tr><tr><td></td><td>GRS 80 semi-minor axis (b)</td><td>≈6356752.314140</td><td></td></tr><tr><td></td><td>Spherical Earth Approx. of Radius (RE)</td><td>6366707.0195</td><td></td></tr><tr><td></td><td>meridional radius of curvature at the equator</td><td>6335439</td><td></td></tr><tr><td></td><td>Maximum (the summit of Chimborazo)</td><td>6384400</td><td></td></tr><tr><td></td><td>Minimum (the floor of the Arctic Ocean)</td><td>6352800</td><td></td></tr><tr><td></td><td>Average distance from center to surface</td><td>6371230±10</td><td></td></tr></tbody></table>

History

Earth radius

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 error 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.

Around 100 BC, Posidonius of Apamea recomputed Earth's radius, and found it to be close to that by Eratosthenes, but later Strabo incorrectly attributed him a value about 3/4 of the actual size. Claudius Ptolemy around 150 AD gave empirical evidence supporting a spherical Earth, but he accepted the lesser value attributed to Posidonius. His highly influential work, the Almagest, 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. 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, and affirmed the original measurement of 40,000 km (25,000 mi) 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. 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: the Earth is flattened at the poles due to rotation's centrifugal force.