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<li id="toc-Physics_of_Earth's_deformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics_of_Earth's_deformation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Physics of Earth's deformation</span> </div> </a> <ul id="toc-Physics_of_Earth's_deformation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radius_and_local_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radius_and_local_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Radius and local conditions</span> </div> </a> <ul id="toc-Radius_and_local_conditions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extrema:_equatorial_and_polar_radii" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extrema:_equatorial_and_polar_radii"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Extrema: equatorial and polar radii</span> </div> </a> <ul id="toc-Extrema:_equatorial_and_polar_radii-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Location-dependent_radii" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Location-dependent_radii"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Location-dependent radii</span> </div> </a> <button aria-controls="toc-Location-dependent_radii-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Location-dependent radii subsection</span> </button> <ul id="toc-Location-dependent_radii-sublist" class="vector-toc-list"> <li id="toc-Geocentric_radius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geocentric_radius"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Geocentric radius</span> </div> </a> <ul id="toc-Geocentric_radius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radii_of_curvature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radii_of_curvature"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Radii of curvature</span> </div> </a> <ul id="toc-Radii_of_curvature-sublist" class="vector-toc-list"> <li id="toc-Principal_radii_of_curvature" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Principal_radii_of_curvature"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Principal radii of curvature</span> </div> </a> <ul id="toc-Principal_radii_of_curvature-sublist" class="vector-toc-list"> <li id="toc-Meridional" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Meridional"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.1</span> <span>Meridional</span> </div> </a> <ul id="toc-Meridional-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_vertical" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Prime_vertical"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.2</span> <span>Prime vertical</span> </div> </a> <ul id="toc-Prime_vertical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polar_and_equatorial_radius_of_curvature" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Polar_and_equatorial_radius_of_curvature"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.3</span> <span>Polar and equatorial radius of curvature</span> </div> </a> <ul id="toc-Polar_and_equatorial_radius_of_curvature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.4</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Combined_radii_of_curvature" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Combined_radii_of_curvature"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Combined radii of curvature</span> </div> </a> <ul id="toc-Combined_radii_of_curvature-sublist" class="vector-toc-list"> <li id="toc-Azimuthal" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Azimuthal"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2.1</span> <span>Azimuthal</span> </div> </a> <ul id="toc-Azimuthal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-directional" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Non-directional"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2.2</span> <span>Non-directional</span> </div> </a> <ul id="toc-Non-directional-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Global_radii" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Global_radii"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Global radii</span> </div> </a> <button aria-controls="toc-Global_radii-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Global radii subsection</span> </button> <ul id="toc-Global_radii-sublist" class="vector-toc-list"> <li id="toc-Arithmetic_mean_radius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_mean_radius"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Arithmetic mean radius</span> </div> </a> <ul id="toc-Arithmetic_mean_radius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Authalic_radius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Authalic_radius"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Authalic radius</span> </div> </a> <ul id="toc-Authalic_radius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volumetric_radius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volumetric_radius"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Volumetric radius</span> </div> </a> <ul id="toc-Volumetric_radius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rectifying_radius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rectifying_radius"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Rectifying radius</span> </div> </a> <ul id="toc-Rectifying_radius-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Topographical_radii" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Topographical_radii"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Topographical radii</span> </div> </a> <button aria-controls="toc-Topographical_radii-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Topographical radii subsection</span> </button> <ul id="toc-Topographical_radii-sublist" class="vector-toc-list"> <li id="toc-Topographical_extremes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topographical_extremes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Topographical extremes</span> </div> </a> <ul id="toc-Topographical_extremes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topographical_global_mean" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topographical_global_mean"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Topographical global mean</span> </div> </a> <ul id="toc-Topographical_global_mean-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Derived_quantities:_diameter,_circumference,_arc-length,_area,_volume" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derived_quantities:_diameter,_circumference,_arc-length,_area,_volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Derived quantities: diameter, circumference, arc-length, area, volume</span> </div> </a> <ul id="toc-Derived_quantities:_diameter,_circumference,_arc-length,_area,_volume-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nominal_radii" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Nominal_radii"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Nominal radii</span> </div> </a> <ul id="toc-Nominal_radii-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Published_values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Published_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Published values</span> </div> </a> <ul id="toc-Published_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Earth radius</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 33 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-33" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">33 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Aardradius" title="Aardradius – Afrikaans" lang="af" hreflang="af" data-title="Aardradius" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B5%D9%81_%D9%82%D8%B7%D8%B1_%D8%A7%D9%84%D8%A3%D8%B1%D8%B6" title="نصف قطر الأرض – Arabic" lang="ar" hreflang="ar" data-title="نصف قطر الأرض" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%83%E0%A6%A5%E0%A6%BF%E0%A6%AC%E0%A7%80%E0%A6%B0_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B8%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A7" title="পৃথিবীর ব্যাসার্ধ – Bangla" lang="bn" hreflang="bn" data-title="পৃথিবীর ব্যাসার্ধ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%97%D0%B5%D0%BC%D0%B5%D0%BD_%D1%80%D0%B0%D0%B4%D0%B8%D1%83%D1%81" title="Земен радиус – Bulgarian" lang="bg" hreflang="bg" data-title="Земен радиус" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Radi_de_la_Terra" title="Radi de la Terra – Catalan" lang="ca" hreflang="ca" data-title="Radi de la Terra" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Erdradius" title="Erdradius – German" lang="de" hreflang="de" data-title="Erdradius" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/R%C3%A2%C7%B5_edla_T%C3%A8ra" title="Râǵ edla Tèra – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Râǵ edla Tèra" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Radio_terrestre" title="Radio terrestre – Spanish" lang="es" hreflang="es" data-title="Radio terrestre" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tera_radiuso" title="Tera radiuso – Esperanto" lang="eo" hreflang="eo" data-title="Tera radiuso" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%D8%B9%D8%A7%D8%B9_%D8%B2%D9%85%DB%8C%D9%86" title="شعاع زمین – Persian" lang="fa" hreflang="fa" data-title="شعاع زمین" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Rayon_de_la_Terre" title="Rayon de la Terre – French" lang="fr" hreflang="fr" data-title="Rayon de la Terre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%80%EA%B5%AC_%EB%B0%98%EA%B2%BD" title="지구 반경 – Korean" lang="ko" hreflang="ko" data-title="지구 반경" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%83%E0%A4%A5%E0%A5%8D%E0%A4%B5%E0%A5%80_%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="पृथ्वी त्रिज्या – Hindi" lang="hi" hreflang="hi" data-title="पृथ्वी त्रिज्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jari-jari_Bumi" title="Jari-jari Bumi – Indonesian" lang="id" hreflang="id" data-title="Jari-jari Bumi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Raggio_terrestre" title="Raggio terrestre – Italian" lang="it" hreflang="it" data-title="Raggio terrestre" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/%C3%84erdradius" title="Äerdradius – Luxembourgish" lang="lb" hreflang="lb" data-title="Äerdradius" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jejari_bumi" title="Jejari bumi – Malay" lang="ms" hreflang="ms" data-title="Jejari bumi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%99%E1%80%B9%E1%80%98%E1%80%AC_%E1%80%A1%E1%80%81%E1%80%BB%E1%80%84%E1%80%BA%E1%80%B8%E1%80%9D%E1%80%80%E1%80%BA" title="ကမ္ဘာ အချင်းဝက် – Burmese" lang="my" hreflang="my" data-title="ကမ္ဘာ အချင်းဝက်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9C%B0%E7%90%83%E5%8D%8A%E5%BE%84" title="地球半径 – Japanese" lang="ja" hreflang="ja" data-title="地球半径" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yer_radiusi" title="Yer radiusi – Uzbek" lang="uz" hreflang="uz" data-title="Yer radiusi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Promie%C5%84_Ziemi" title="Promień Ziemi – Polish" lang="pl" hreflang="pl" data-title="Promień Ziemi" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Raio_terrestre" title="Raio terrestre – Portuguese" lang="pt" hreflang="pt" data-title="Raio terrestre" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Raza_P%C4%83m%C3%A2ntului" title="Raza Pământului – Romanian" lang="ro" hreflang="ro" data-title="Raza Pământului" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D1%83%D1%81_%D0%97%D0%B5%D0%BC%D0%BB%D0%B8" title="Радиус Земли – Russian" lang="ru" hreflang="ru" data-title="Радиус Земли" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Earth_radius" title="Earth radius – Simple English" lang="en-simple" hreflang="en-simple" data-title="Earth radius" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zemljin_polmer" title="Zemljin polmer – Slovenian" lang="sl" hreflang="sl" data-title="Zemljin polmer" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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development, see <a href="/wiki/Spherical_Earth" title="Spherical Earth">Spherical Earth</a>. For its determination, see <a href="/wiki/Arc_measurement" title="Arc measurement">Arc measurement</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Distance from the Earth surface to a point near its center</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Earth radius</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:WGS84_mean_Earth_radius.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/WGS84_mean_Earth_radius.svg/220px-WGS84_mean_Earth_radius.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/WGS84_mean_Earth_radius.svg/330px-WGS84_mean_Earth_radius.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/WGS84_mean_Earth_radius.svg/440px-WGS84_mean_Earth_radius.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span><div class="infobox-caption">Equatorial (<i>a</i>), polar (<i>b</i>) and arithmetic mean Earth radii as defined in the 1984 <a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a> revision (not to scale)</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other names</div></th><td class="infobox-data">terrestrial radius</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><i>R</i><sub>🜨</sub>, <i>R</i><sub>E</sub>, <i>a</i>, <i>b</i>, <i>a</i><sub>E</sub>, <i>b</i><sub>E</sub>, <i>R</i><sub><i>e</i>E</sub>, <i>R</i><sub><i>p</i>E</sub></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">meters</td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data"><a href="/wiki/Metre" title="Metre">m</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Behaviour under<br /><span class="nowrap"><a href="/wiki/Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">coord transformation</a></span></div></th><td class="infobox-data">scalar</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d25704a85c37e68d78dd9f549587912bf314b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.26ex; height:2.176ex;" alt="{\displaystyle {\mathsf {L}}}" /></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Quantity_value" class="mw-redirect" title="Quantity value">Value</a></th><td class="infobox-data"><i>Equatorial radius</i>: <span class="texhtml mvar" style="font-style:italic;">a</span> = (<span class="nowrap"><span data-sort-value="7006637813700000000♠"></span>6<span style="margin-left:.25em;">378</span><span style="margin-left:.25em;">137</span>.0 m</span>) <br /> <i>Polar radius</i>: <span class="texhtml mvar" style="font-style:italic;">b</span> = (<span class="nowrap"><span data-sort-value="7006635675230000000♠"></span>6<span style="margin-left:.25em;">356</span><span style="margin-left:.25em;">752</span>.3 m</span>)</td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><table class="infobox ib-unit"><tbody><tr><th colspan="2" class="infobox-above">Nominal Earth radius</th></tr><tr><td colspan="2" class="infobox-image notheme" style="background-color: #f8f9fa;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:EarthPieSlice.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/EarthPieSlice.png/220px-EarthPieSlice.png" decoding="async" width="220" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/EarthPieSlice.png/330px-EarthPieSlice.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/EarthPieSlice.png/440px-EarthPieSlice.png 2x" data-file-width="2220" data-file-height="2459" /></a></span><div class="infobox-caption">Cross section of Earth's Interior</div></td></tr><tr><th colspan="2" class="infobox-header">General information</th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/System_of_units_of_measurement" title="System of units of measurement">Unit system</a></th><td class="infobox-data"><a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Geophysics" title="Geophysics">geophysics</a></td></tr><tr><th scope="row" class="infobox-label">Unit of</th><td class="infobox-data"><a href="/wiki/Distance" title="Distance">distance</a></td></tr><tr><th scope="row" class="infobox-label">Symbol</th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094e54a9094cace8bcae1656a510becd421f1481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.436ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{e\mathrm {E} }^{\mathrm {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{e\mathrm {E} }^{\mathrm {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ae8c46119839647709ae9807a41bfa8af21249" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.088ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}_{e\mathrm {E} }^{\mathrm {N} }}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{p\mathrm {E} }^{\mathrm {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{p\mathrm {E} }^{\mathrm {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda48aa8d409d36e16eea551724845314eedb7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.149ex; height:3.509ex;" alt="{\displaystyle {\mathcal {R}}_{p\mathrm {E} }^{\mathrm {N} }}" /></span></td></tr><tr><th colspan="2" class="infobox-header">Conversions </th></tr><tr class="nowrap"><td>1 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094e54a9094cace8bcae1656a510becd421f1481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.436ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}" /></span> <i>in ...</i></td><td><i>... is equal to ...</i></td></tr><tr style="display:none"><th colspan="2"> </th></tr><tr><th scope="row" class="infobox-label"><span class="nowrap">   </span><a href="/wiki/SI_base_unit" title="SI base unit">SI base unit</a></th><td class="infobox-data"><span class="nowrap">   </span><span class="nowrap"><span data-sort-value="7006637810000000000♠"></span>6.3781<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>6</sup> m</span><sup id="cite_ref-IAU_XXIX_1-0" class="reference"><a href="#cite_note-IAU_XXIX-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr><tr><th scope="row" class="infobox-label"><span class="nowrap">   </span><a href="/wiki/Metric_system" title="Metric system">Metric system</a></th><td class="infobox-data"><span class="nowrap">   </span><span class="nowrap"><span data-sort-value="7006635700000000000♠"></span>6,357 to 6,378 km</span></td></tr><tr><th scope="row" class="infobox-label"><span class="nowrap">   </span><a href="/wiki/English_units" title="English units">English units</a></th><td class="infobox-data"><span class="nowrap">   </span><span class="nowrap"><span data-sort-value="7006635690880000000♠"></span>3,950 to 3,963 mi</span></td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r1236303919">@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-has-images-with-white-backgrounds img{background:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-has-images-with-white-backgrounds img{background:white}}</style> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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.sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Globe_Atlantic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/120px-Globe_Atlantic.svg.png" decoding="async" width="120" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/180px-Globe_Atlantic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/240px-Globe_Atlantic.svg.png 2x" data-file-width="717" data-file-height="721" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></li> <li><a href="/wiki/Geodynamics" title="Geodynamics">Geodynamics</a></li> <li><a href="/wiki/Geomatics" title="Geomatics">Geomatics</a></li> <li><a href="/wiki/History_of_geodesy" title="History of geodesy">History</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Concepts</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Geographical_distance" title="Geographical distance">Geographical distance</a></li> <li><a href="/wiki/Geoid" title="Geoid">Geoid</a></li> <li><a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">Figure of the Earth</a> <small>(<a class="mw-selflink selflink">radius</a> and <a href="/wiki/Earth%27s_circumference" title="Earth's circumference">circumference</a>)</small></li> <li><a href="/wiki/Geodetic_coordinates" title="Geodetic coordinates">Geodetic coordinates</a></li> <li><a href="/wiki/Geodetic_datum" title="Geodetic datum">Geodetic datum</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Horizontal_position_representation" title="Horizontal position representation">Horizontal position representation</a></li> <li><span class="nowrap"><a href="/wiki/Latitude" title="Latitude">Latitude</a> / <a href="/wiki/Longitude" title="Longitude">Longitude</a></span></li> <li><a href="/wiki/Map_projection" title="Map projection">Map projection</a></li> <li><a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">Reference ellipsoid</a></li> <li><a href="/wiki/Satellite_geodesy" title="Satellite geodesy">Satellite geodesy</a></li> <li><a href="/wiki/Spatial_reference_system" title="Spatial reference system">Spatial reference system</a></li> <li><a href="/wiki/Spatial_relation" title="Spatial relation">Spatial relations</a></li> <li><a href="/wiki/Vertical_position" title="Vertical position">Vertical positions</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Technologies</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Satellite_navigation" title="Satellite navigation">Global Nav. Sat. Systems (GNSSs)</a></li> <li><a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Pos. System (GPS)</a></li> <li><a href="/wiki/GLONASS" title="GLONASS">GLONASS <span style="font-size:85%;">(Russia)</span></a></li> <li><a href="/wiki/BeiDou" title="BeiDou">BeiDou (BDS) <span style="font-size:85%;">(China)</span></a></li> <li><a href="/wiki/Galileo_(satellite_navigation)" title="Galileo (satellite navigation)">Galileo <span style="font-size:85%;">(Europe)</span></a></li> <li><a href="/wiki/Indian_Regional_Navigation_Satellite_System" title="Indian Regional Navigation Satellite System">NAVIC <span style="font-size:85%;">(India)</span></a></li> <li><a href="/wiki/Quasi-Zenith_Satellite_System" title="Quasi-Zenith Satellite System">Quasi-Zenith Sat. Sys. (QZSS) <span style="font-size:85%;">(Japan)</span></a></li> <li><a href="/wiki/Discrete_Global_Grid" class="mw-redirect" title="Discrete Global Grid">Discrete Global Grid and Geocoding</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed plainlist"><div class="sidebar-list-title" style="color: var(--color-base)">Standards (history)</div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="nowrap" style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.2em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Sea_Level_Datum_of_1929" class="mw-redirect" title="Sea Level Datum of 1929">NGVD 29</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Sea Level Datum 1929</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/OSGB36" class="mw-redirect" title="OSGB36">OSGB36</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Ordnance Survey Great Britain 1936</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/SK-42_reference_system" title="SK-42 reference system">SK-42</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Systema Koordinat 1942 goda</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/ED50" title="ED50">ED50</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> European Datum 1950</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/South_American_Datum#SAD69" title="South American Datum">SAD69</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> South American Datum 1969</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/GRS_80" class="mw-redirect" title="GRS 80">GRS 80</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Geodetic Reference System 1980</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/ISO_6709" title="ISO 6709">ISO 6709</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Geographic point coord. 1983</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/North_American_Datum#North_American_Datum_of_1983" title="North American Datum">NAD 83</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> North American Datum 1983</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/World_Geodetic_System" title="World Geodetic System">WGS 84</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> World Geodetic System 1984</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/North_American_Vertical_Datum_of_1988" title="North American Vertical Datum of 1988">NAVD 88</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> N. American Vertical Datum 1988</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/European_Terrestrial_Reference_System_1989" title="European Terrestrial Reference System 1989">ETRS89</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> European Terrestrial Ref. Sys. 1989</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Restrictions_on_geographic_data_in_China" title="Restrictions on geographic data in China">GCJ-02</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Chinese obfuscated datum 2002</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Geo_URI_scheme" title="Geo URI scheme">Geo URI</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Internet link to a point 2010</td></tr></tbody></table> <ul><li><a href="/wiki/International_Terrestrial_Reference_System" class="mw-redirect" title="International Terrestrial Reference System">International Terrestrial Reference System</a></li> <li><a href="/wiki/SRID" class="mw-redirect" title="SRID">Spatial Reference System Identifier (SRID)</a></li> <li><a href="/wiki/Universal_Transverse_Mercator_coordinate_system" title="Universal Transverse Mercator coordinate system">Universal Transverse Mercator (UTM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Geodesy" title="Template:Geodesy"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Geodesy" title="Template talk:Geodesy"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geodesy" title="Special:EditPage/Template:Geodesy"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Earth radius</b> (denoted as <i>R</i><sub>🜨</sub> or <i>R</i><sub>E</sub>) is the distance from the center of <a href="/wiki/Earth" title="Earth">Earth</a> to a point on or near its surface. Approximating the <a href="/wiki/Figure_of_Earth" class="mw-redirect" title="Figure of Earth">figure of Earth</a> by an <a href="/wiki/Earth_spheroid" class="mw-redirect" title="Earth spheroid">Earth spheroid</a> (an <a href="/wiki/Oblate_ellipsoid" class="mw-redirect" title="Oblate ellipsoid">oblate ellipsoid</a>), the radius ranges from a maximum (<b>equatorial radius</b>, denoted <i>a</i>) of nearly 6,378 km (3,963 mi) to a minimum (<b>polar radius</b>, denoted <i>b</i>) of nearly 6,357 km (3,950 mi). </p><p>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 <a href="/wiki/International_Union_of_Geodesy_and_Geophysics" title="International Union of Geodesy and Geophysics">International Union of Geodesy and Geophysics</a> (IUGG) provides three reference values: the <i>mean radius</i> (<i>R</i><sub>1</sub>) of three radii measured at two equator points and a pole; the <i>authalic radius</i>, which is the radius of a sphere with the same surface area (<i>R</i><sub>2</sub>); and the <i>volumetric radius</i>, which is the radius of a sphere having the same volume as the ellipsoid (<i>R</i><sub>3</sub>).<sup id="cite_ref-Moritz_2-0" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> All three values are about 6,371 kilometres (3,959 mi). </p><p>Other ways to define and measure the Earth's radius involve either the spheroid's <a href="/wiki/Radius_of_curvature" title="Radius of curvature">radius of curvature</a> or the actual <a href="/wiki/Topography" title="Topography">topography</a>. A few definitions yield values outside the range between the <a href="/wiki/Geographical_pole" title="Geographical pole">polar</a> radius and <a href="/wiki/Equator" title="Equator">equatorial</a> radius because they account for localized effects. </p><p>A <i>nominal Earth radius</i> (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094e54a9094cace8bcae1656a510becd421f1481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.436ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {E} }^{\mathrm {N} }}" /></span>) is sometimes used as a <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit of measurement</a> in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> and <a href="/wiki/Geophysics" title="Geophysics">geophysics</a>, a <a href="/wiki/Conversion_factor" class="mw-redirect" title="Conversion factor">conversion factor</a> 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 <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">International Astronomical Union</a> (IAU).<sup id="cite_ref-IAU_XXIX_1-1" class="reference"><a href="#cite_note-IAU_XXIX-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Earth_oblateness_to_scale.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Earth_oblateness_to_scale.svg/300px-Earth_oblateness_to_scale.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Earth_oblateness_to_scale.svg/450px-Earth_oblateness_to_scale.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Earth_oblateness_to_scale.svg/600px-Earth_oblateness_to_scale.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>A scale diagram of the <a href="/wiki/Flattening" title="Flattening">oblateness</a> of the 2003 <a href="/wiki/IERS" class="mw-redirect" title="IERS">IERS</a> <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>, with north at the top. The light blue region is a circle. The outer edge of the dark blue line is an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> with the same <a href="/wiki/Minor_axis" class="mw-redirect" title="Minor axis">minor axis</a> as the circle and the same <a href="/wiki/Eccentricity_(mathematics)#Ellipses" title="Eccentricity (mathematics)">eccentricity</a> as the Earth. The red line represents the <a href="/wiki/Karman_line" class="mw-redirect" title="Karman line">Karman line</a> 100 km (62 mi) above <a href="/wiki/Sea_level" title="Sea level">sea level</a>, while the yellow area denotes the <a href="/wiki/Apsis" title="Apsis">altitude</a> range of the <a href="/wiki/International_Space_Station" title="International Space Station">ISS</a> in <a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">low Earth orbit</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">Figure of the Earth</a>, <a href="/wiki/Earth_ellipsoid" title="Earth ellipsoid">Earth ellipsoid</a>, and <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">Reference ellipsoid</a></div> <p><a href="/wiki/Earth%27s_rotation" title="Earth's rotation">Earth's rotation</a>, internal density variations, and external <a href="/wiki/Tidal_force" title="Tidal force">tidal forces</a> cause its shape to deviate systematically from a perfect sphere.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Local <a href="/wiki/Topography" title="Topography">topography</a> 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. </p><p>Each of the models in common use involve some notion of the geometric <a href="/wiki/Radius" title="Radius">radius</a>. Strictly speaking, spheres are the only solids to have radii, but broader uses of the term <i>radius</i> 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: </p> <ul><li>The actual surface of Earth</li> <li>The <a href="/wiki/Geoid" title="Geoid">geoid</a>, defined by <a href="/wiki/Mean_sea_level" class="mw-redirect" title="Mean sea level">mean sea level</a> at each point on the real surface<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup></li> <li>A <a href="/wiki/Spheroid" title="Spheroid">spheroid</a>, also called an <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> of revolution, <a href="/wiki/Geodetic_system#Geodetic_versus_geocentric_latitude" class="mw-redirect" title="Geodetic system">geocentric</a> to model the entire Earth, or else <a href="/wiki/Geodetic_system#Geodetic_versus_geocentric_latitude" class="mw-redirect" title="Geodetic system">geodetic</a> for regional work<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup></li> <li>A <a href="/wiki/Sphere" title="Sphere">sphere</a></li></ul> <p>In the case of the geoid and ellipsoids, the fixed distance from any point on the model to the specified center is called <i>"a radius of the Earth"</i> or <i>"the radius of the Earth at that point"</i>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> It is also common to refer to any <i><a href="#Mean_radii">mean radius</a></i> of a spherical model as <i>"the radius of the earth"</i>. 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. </p><p>Regardless of the model, any of these <i>geocentric</i> 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 <a href="/wiki/Planet" title="Planet">planet</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Physics_of_Earth's_deformation"><span id="Physics_of_Earth.27s_deformation"></span>Physics of Earth's deformation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=2" title="Edit section: Physics of Earth's deformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Equatorial_bulge" title="Equatorial bulge">Equatorial bulge</a></div> <p>Rotation of a planet causes it to approximate an <i><a href="/wiki/Spheroid" title="Spheroid">oblate ellipsoid</a>/spheroid</i> with a bulge at the <a href="/wiki/Equator" title="Equator">equator</a> and flattening at the <a href="/wiki/North_Pole" title="North Pole">North</a> and <a href="/wiki/South_Pole" title="South Pole">South Poles</a>, so that the <i>equatorial radius</i> <span class="texhtml mvar" style="font-style:italic;">a</span> is larger than the <i>polar radius</i> <span class="texhtml mvar" style="font-style:italic;">b</span> by approximately <span class="texhtml mvar" style="font-style:italic;">aq</span>. The <i>oblateness constant</i> <span class="texhtml mvar" style="font-style:italic;">q</span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318f3ef055d946153757cae6dd8d2a1a1d88f0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.435ex; height:5.843ex;" alt="{\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}},}" /></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">ω</span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>, <span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>, and <span class="texhtml mvar" style="font-style:italic;">M</span> is the mass of the planet.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> For the Earth <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>q</i></span></span>⁠</span> ≈ 289</span>, which is close to the measured inverse <a href="/wiki/Flattening" title="Flattening">flattening</a> <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>f</i></span></span>⁠</span> ≈ 298.257</span>. 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.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Lowresgeoidheight.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Lowresgeoidheight.jpg/400px-Lowresgeoidheight.jpg" decoding="async" width="400" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/Lowresgeoidheight.jpg 1.5x" data-file-width="580" data-file-height="284" /></a><figcaption></figcaption></figure> <p>The variation in <a href="/wiki/Density" title="Density">density</a> and <a href="/wiki/Crust_(geology)" title="Crust (geology)">crustal</a> 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 <i><a href="/wiki/Geoid" title="Geoid">geoid</a> height</i>, 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 <a href="/wiki/2004_Indian_Ocean_earthquake" class="mw-redirect" title="2004 Indian Ocean earthquake">Sumatra-Andaman earthquake</a>) or reduction in ice masses (such as <a href="/wiki/Greenland" title="Greenland">Greenland</a>).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>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 <a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Radius_and_local_conditions">Radius and local conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=3" title="Edit section: Radius and local conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Abu_Reyhan_Biruni-Earth_Circumference.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Abu_Reyhan_Biruni-Earth_Circumference.svg/220px-Abu_Reyhan_Biruni-Earth_Circumference.svg.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Abu_Reyhan_Biruni-Earth_Circumference.svg/330px-Abu_Reyhan_Biruni-Earth_Circumference.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Abu_Reyhan_Biruni-Earth_Circumference.svg/440px-Abu_Reyhan_Biruni-Earth_Circumference.svg.png 2x" data-file-width="1000" data-file-height="900" /></a><figcaption><a href="/wiki/Al-Biruni#Geodesy_and_geography" title="Al-Biruni">Al-Biruni</a>'s (973 – <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1050</span>) method for calculation of the Earth's radius simplified measuring the circumference compared to taking measurements from two locations distant from each other.</figcaption></figure> <p>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). </p><p>Additionally, the radius can be estimated from the curvature of the Earth at a point. Like a <a href="/wiki/Torus" title="Torus">torus</a>, the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The corresponding <a href="/wiki/Radius_of_curvature_(applications)" class="mw-redirect" title="Radius of curvature (applications)">radius of curvature</a> depends on the location and direction of measurement from that point. A consequence is that a distance to the <a href="/wiki/Horizon" title="Horizon">true horizon</a> at the equator is slightly shorter in the north–south direction than in the east–west direction. </p><p>In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a>, many models have been created. Historically, these models were based on regional topography, giving the best <a href="/wiki/Figure_of_the_Earth#Historical_Earth_ellipsoids" title="Figure of the Earth">reference ellipsoid</a> for the area under survey. As satellite <a href="/wiki/Remote_sensing" title="Remote sensing">remote sensing</a> and especially the <a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Positioning System</a> gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole. </p> <div class="mw-heading mw-heading2"><h2 id="Extrema:_equatorial_and_polar_radii">Extrema: equatorial and polar radii<span class="anchor" id="Fixed_radius"></span><span class="anchor" id="Equatorial_radius"></span><span class="anchor" id="Polar_radius"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=4" title="Edit section: Extrema: equatorial and polar radii"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following radii are derived from the <a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a> 1984 (<a href="/wiki/WGS-84" class="mw-redirect" title="WGS-84">WGS-84</a>) <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>.<sup id="cite_ref-tr8350_2_11-0" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> 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 <a href="/wiki/Accuracy" class="mw-redirect" title="Accuracy">accuracy</a>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (April 2021)">clarification needed</span></a></i>]</sup> </p><p>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. </p> <ul><li>The Earth's <i>equatorial radius</i> <span class="texhtml mvar" style="font-style:italic;">a</span>, or <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a>,<sup id="cite_ref-Snyder_manual_13-0" class="reference"><a href="#cite_note-Snyder_manual-13"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 11">: 11 </span></sup> is the distance from its center to the <a href="/wiki/Equator" title="Equator">equator</a> and equals 6,378.1370 km (3,963.1906 mi).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The equatorial radius is often used to compare Earth with other <a href="/wiki/Planet#Attributes" title="Planet">planets</a>.</li> <li>The Earth's <i>polar radius</i> <span class="texhtml mvar" style="font-style:italic;">b</span>, or <a href="/wiki/Semi-minor_axis" class="mw-redirect" title="Semi-minor axis">semi-minor axis</a><sup id="cite_ref-Snyder_manual_13-1" class="reference"><a href="#cite_note-Snyder_manual-13"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 11">: 11 </span></sup> is the distance from its center to the North and South Poles, and equals 6,356.7523 km (3,949.9028 mi).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Location-dependent_radii">Location-dependent radii<span class="anchor" id="Radii_with_location_dependence"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=5" title="Edit section: Location-dependent radii"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EarthEllipRadii.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/EarthEllipRadii.svg/220px-EarthEllipRadii.svg.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/EarthEllipRadii.svg/330px-EarthEllipRadii.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/EarthEllipRadii.svg/440px-EarthEllipRadii.svg.png 2x" data-file-width="780" data-file-height="611" /></a><figcaption>Three different radii as a function of Earth's latitude. <span class="texhtml mvar" style="font-style:italic;">R</span> is the geocentric radius; <span class="texhtml mvar" style="font-style:italic;">M</span> is the meridional radius of curvature; and <span class="texhtml mvar" style="font-style:italic;">N</span> is the prime vertical radius of curvature.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Geocentric_radius">Geocentric radius<span class="anchor" id="Geocentric"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=6" title="Edit section: Geocentric radius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Geocentric_distance" class="mw-redirect" title="Geocentric distance">Geocentric distance</a>.</div> <p>The <i>geocentric radius</i> is the distance from the Earth's center to a point on the spheroid surface at <a href="/wiki/Geodetic_latitude" class="mw-redirect" title="Geodetic latitude">geodetic latitude</a> <span class="texhtml mvar" style="font-style:italic;">φ</span>, given by the formula <sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\varphi )={\sqrt {\frac {(a^{2}\cos \varphi )^{2}+(b^{2}\sin \varphi )^{2}}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(\varphi )={\sqrt {\frac {(a^{2}\cos \varphi )^{2}+(b^{2}\sin \varphi )^{2}}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64f616bd6c63c20fed6252ba7934262d2f16a09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.457ex; height:7.509ex;" alt="{\displaystyle R(\varphi )={\sqrt {\frac {(a^{2}\cos \varphi )^{2}+(b^{2}\sin \varphi )^{2}}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}},}" /></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are, respectively, the equatorial radius and the polar radius. </p><p>The extrema geocentric radii on the ellipsoid coincide with the equatorial and polar radii. They are <a href="/wiki/Vertex_(curve)" title="Vertex (curve)">vertices</a> of the ellipse and also coincide with minimum and maximum radius of curvature. </p> <div class="mw-heading mw-heading3"><h3 id="Radii_of_curvature">Radii of curvature<span class="anchor" id="Radius_of_curvature"></span><span class="anchor" id="Curvature"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=7" title="Edit section: Radii of curvature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Spheroid#Curvature" title="Spheroid">Spheroid § Curvature</a></div> <div class="mw-heading mw-heading4"><h4 id="Principal_radii_of_curvature">Principal radii of curvature</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=8" title="Edit section: Principal radii of curvature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two <a href="/wiki/Principal_curvature" title="Principal curvature">principal radii of curvature</a>: along the meridional and prime-vertical <a href="/wiki/Normal_section" class="mw-redirect" title="Normal section">normal sections</a>. </p> <div class="mw-heading mw-heading5"><h5 id="Meridional">Meridional</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=9" title="Edit section: Meridional"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In particular, the <i>Earth's <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridional</a> radius of curvature</i> (in the north–south direction) at <span class="texhtml mvar" style="font-style:italic;">φ</span> is<sup id="cite_ref-Jekeli_16-0" class="reference"><a href="#cite_note-Jekeli-16"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24dd959e2ceb84ce9d1309242083ffd66d07c8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:71.916ex; height:8.176ex;" alt="{\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},}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /></span> is the <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a> of the earth. This is the radius that <a href="/wiki/Eratosthenes#Measurement_of_the_Earth" title="Eratosthenes">Eratosthenes</a> measured in his <a href="/wiki/Arc_measurement" title="Arc measurement">arc measurement</a>. </p> <div class="mw-heading mw-heading5"><h5 id="Prime_vertical">Prime vertical</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=10" title="Edit section: Prime vertical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geodetic_latitude_and_the_length_of_Normal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Geodetic_latitude_and_the_length_of_Normal.svg/220px-Geodetic_latitude_and_the_length_of_Normal.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Geodetic_latitude_and_the_length_of_Normal.svg/330px-Geodetic_latitude_and_the_length_of_Normal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Geodetic_latitude_and_the_length_of_Normal.svg/440px-Geodetic_latitude_and_the_length_of_Normal.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>The length PQ, called the <i>prime vertical radius</i>, is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7143e417ccafb66605e0e7dec2ed44f44d24c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.905ex; height:2.843ex;" alt="{\displaystyle N(\phi ).}" /></span> The length IQ is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2}N(\phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2}N(\phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe10a9c607882ff39c8247d40ff7ea0e3c070d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.043ex; height:3.176ex;" alt="{\displaystyle e^{2}N(\phi ).}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=(X,Y,Z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=(X,Y,Z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad48e7a4ac7b28a418d7ddc99d0d970020b2866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.82ex; height:2.843ex;" alt="{\displaystyle R=(X,Y,Z).}" /></span></figcaption></figure> <p>If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.<sup id="cite_ref-curvprim_17-0" class="reference"><a href="#cite_note-curvprim-17"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup> </p><p>This <i>Earth's <a href="/wiki/Prime_vertical" title="Prime vertical">prime-vertical</a> radius of curvature</i>, also called the <i>Earth's transverse radius of curvature</i>, is defined perpendicular (<a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a>) to <span class="texhtml mvar" style="font-style:italic;">M</span> at geodetic latitude <span class="texhtml mvar" style="font-style:italic;">φ</span><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>g<span class="cite-bracket">]</span></a></sup> and is<sup id="cite_ref-Jekeli_16-1" class="reference"><a href="#cite_note-Jekeli-16"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\varphi )={\frac {a^{2}}{\sqrt {(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\varphi }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <mo stretchy="false">(</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\varphi )={\frac {a^{2}}{\sqrt {(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\varphi }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d0f1832f82ce05d8d24ba1f3db1b8b195d39f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:52.252ex; height:8.509ex;" alt="{\displaystyle N(\varphi )={\frac {a^{2}}{\sqrt {(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\varphi }}}.}" /></span></dd></dl> <p><i>N</i> can also be interpreted geometrically as the <a href="/wiki/Normal_distance" class="mw-redirect" title="Normal distance">normal distance</a> from the ellipsoid surface to the polar axis.<sup id="cite_ref-Bowring_19-0" class="reference"><a href="#cite_note-Bowring-19"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The radius of a <a href="/wiki/Parallel_of_latitude" class="mw-redirect" title="Parallel of latitude">parallel of latitude</a> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=N\cos(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=N\cos(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6dbc469e5b92e30b240a91a24dfef9461889569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.249ex; height:2.843ex;" alt="{\displaystyle p=N\cos(\varphi )}" /></span>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading5"><h5 id="Polar_and_equatorial_radius_of_curvature">Polar and equatorial radius of curvature</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=11" title="Edit section: Polar and equatorial radius of curvature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>Earth's meridional radius of curvature at the equator</i> equals the meridian's <a href="/wiki/Semi-latus_rectum" class="mw-redirect" title="Semi-latus rectum">semi-latus rectum</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(0{\text{°}})={\frac {b^{2}}{a}}=6,335.439{\text{ km.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>°</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mn>6</mn> <mo>,</mo> <mn>335.439</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> km.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(0{\text{°}})={\frac {b^{2}}{a}}=6,335.439{\text{ km.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e96b946c5ccb961584d999e51778b8bc37e096c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.87ex; height:5.676ex;" alt="{\displaystyle M(0{\text{°}})={\frac {b^{2}}{a}}=6,335.439{\text{ km.}}}" /></span></dd></dl> <p>The <i>Earth's prime-vertical radius of curvature at the equator</i> equals the equatorial radius, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(0{\text{°}})=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>°</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(0{\text{°}})=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc5d90bdedb0c61b7423d98d82769c451d0a35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.526ex; height:2.843ex;" alt="{\displaystyle N(0{\text{°}})=a}" /></span> </p><p>The <i>Earth's polar radius of curvature</i> (either meridional or prime-vertical) is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(90{\text{°}})=N(90{\text{°}})={\frac {a^{2}}{b}}=6,399.594{\text{ km.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>°</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>°</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mn>6</mn> <mo>,</mo> <mn>399.594</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> km.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(90{\text{°}})=N(90{\text{°}})={\frac {a^{2}}{b}}=6,399.594{\text{ km.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af7cf04bad537f7b9f40a432b9cb4fd0a5c8eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.724ex; height:5.843ex;" alt="{\displaystyle M(90{\text{°}})=N(90{\text{°}})={\frac {a^{2}}{b}}=6,399.594{\text{ km.}}}" /></span> </p> <div class="mw-heading mw-heading5"><h5 id="Derivation">Derivation</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=12" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1256386598">.mw-parser-output .cot-header-mainspace{background:#F0F2F5;color:inherit}.mw-parser-output .cot-header-other{background:#CCFFCC;color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-night .mw-parser-output .cot-header-other{background:#003500;color:inherit}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-os .mw-parser-output .cot-header-other{background:#003500;color:inherit}}</style> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Extended content</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> <p>The principal curvatures are the roots of Equation (125) in:<sup id="cite_ref-Lass_22-0" class="reference"><a href="#cite_note-Lass-22"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (EG-F^{2})\kappa ^{2}-(eG+gE-2fF)\kappa +(eg-f^{2})=0=\det(A-\kappa B),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mi>G</mi> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>G</mi> <mo>+</mo> <mi>g</mi> <mi>E</mi> <mo>−</mo> <mn>2</mn> <mi>f</mi> <mi>F</mi> <mo stretchy="false">)</mo> <mi>κ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>g</mi> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>κ</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (EG-F^{2})\kappa ^{2}-(eG+gE-2fF)\kappa +(eg-f^{2})=0=\det(A-\kappa B),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627579f6795a32f0f8535d0eee94313f14d8ea6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.963ex; height:3.176ex;" alt="{\displaystyle (EG-F^{2})\kappa ^{2}-(eG+gE-2fF)\kappa +(eg-f^{2})=0=\det(A-\kappa B),}" /></span></dd></dl> <p>where in the <a href="/wiki/First_fundamental_form" title="First fundamental form">first fundamental form</a> for a surface (Equation (112) in<sup id="cite_ref-Lass_22-1" class="reference"><a href="#cite_note-Lass-22"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=\sum _{ij}a_{ij}dw^{i}dw^{j}=E\,d\varphi ^{2}+2F\,d\varphi \,d\lambda +G\,d\lambda ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mi>E</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>F</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>φ</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>λ</mi> <mo>+</mo> <mi>G</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=\sum _{ij}a_{ij}dw^{i}dw^{j}=E\,d\varphi ^{2}+2F\,d\varphi \,d\lambda +G\,d\lambda ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a08e70824dfbaa419c97bd5d8eb8b605ea98b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.58ex; height:5.843ex;" alt="{\displaystyle ds^{2}=\sum _{ij}a_{ij}dw^{i}dw^{j}=E\,d\varphi ^{2}+2F\,d\varphi \,d\lambda +G\,d\lambda ^{2},}" /></span></dd></dl> <p><i>E</i>, <i>F</i>, and <i>G</i> are elements of the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>E</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>G</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0553ba4c0ff46ea2509038b676abc047e20cd0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.612ex; height:6.676ex;" alt="{\displaystyle 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}},}" /></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=[r^{1},r^{2},r^{3}]^{T}=[x,y,z]^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=[r^{1},r^{2},r^{3}]^{T}=[x,y,z]^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bbe1c37804b95a79f311e99dcbe998b2d2e9a08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.629ex; height:3.176ex;" alt="{\displaystyle r=[r^{1},r^{2},r^{3}]^{T}=[x,y,z]^{T}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{1}=\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{1}=\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f97216b999ea199120095943350552726dfe5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.337ex; height:3.176ex;" alt="{\displaystyle w^{1}=\varphi }" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{2}=\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{2}=\lambda ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20f307c6138d7035e5885191034dff6a0d9872d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.819ex; height:3.009ex;" alt="{\displaystyle w^{2}=\lambda ,}" /></span> </p><p>in the <a href="/wiki/Second_fundamental_form" title="Second fundamental form">second fundamental form</a> for a surface (Equation (123) in<sup id="cite_ref-Lass_22-2" class="reference"><a href="#cite_note-Lass-22"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2D=\sum _{ij}b_{ij}dw^{i}dw^{j}=e\,d\varphi ^{2}+2f\,d\varphi \,d\lambda +g\,d\lambda ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>D</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mi>e</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>φ</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>λ</mi> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2D=\sum _{ij}b_{ij}dw^{i}dw^{j}=e\,d\varphi ^{2}+2f\,d\varphi \,d\lambda +g\,d\lambda ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60a072d08ce73305d626cc67cf504040bb997b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.209ex; height:5.843ex;" alt="{\displaystyle 2D=\sum _{ij}b_{ij}dw^{i}dw^{j}=e\,d\varphi ^{2}+2f\,d\varphi \,d\lambda +g\,d\lambda ^{2},}" /></span></dd></dl> <p><i>e</i>, <i>f</i>, and <i>g</i> are elements of the shape tensor: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munder> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi mathvariant="normal">∂</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>e</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> </mtd> <mtd> <mi>g</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3622f1fcd4cabbb818eb85c8d0cecae7fb96a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.017ex; height:6.843ex;" alt="{\displaystyle 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}},}" /></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\frac {N}{|N|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {N}{|N|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c29aba8e334dd970fd1b93252b839db6848a61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.687ex; height:6.009ex;" alt="{\displaystyle n={\frac {N}{|N|}}}" /></span> is the unit normal to the surface at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>, and because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial r}{\partial \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial r}{\partial \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826d0f50ed82dc5e3f3aade814d4a8bcea498843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:3.674ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial r}{\partial \varphi }}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial r}{\partial \lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial r}{\partial \lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/357dc5f471375eabc506214cee6f620d69c316da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.509ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial r}{\partial \lambda }}}" /></span> are tangents to the surface, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\frac {\partial r}{\partial \varphi }}\times {\frac {\partial r}{\partial \lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\frac {\partial r}{\partial \varphi }}\times {\frac {\partial r}{\partial \lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d119ac59bdc49b699582e4994c063b800009916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.186ex; height:6.009ex;" alt="{\displaystyle N={\frac {\partial r}{\partial \varphi }}\times {\frac {\partial r}{\partial \lambda }}}" /></span></dd></dl> <p>is normal to the surface at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>. </p><p>With <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=f=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0173adee5635370ef07e1a084b5c723338d58f51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.379ex; height:2.509ex;" alt="{\displaystyle F=f=0}" /></span> for an oblate spheroid, the curvatures are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{1}={\frac {g}{G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mi>G</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{1}={\frac {g}{G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27e33bded75d3983e4f4fe2abf6a5fe37ca8865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.155ex; height:5.009ex;" alt="{\displaystyle \kappa _{1}={\frac {g}{G}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{2}={\frac {e}{E}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi>E</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{2}={\frac {e}{E}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af59466df1b1dc503c05bc4550ea755915e63c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.75ex; height:4.676ex;" alt="{\displaystyle \kappa _{2}={\frac {e}{E}},}" /></span></dd></dl> <p>and the principal radii of curvature are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}={\frac {1}{\kappa _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}={\frac {1}{\kappa _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87b6a35c7860fd7015af46fc2a09d37c11c0b0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.146ex; height:5.509ex;" alt="{\displaystyle R_{1}={\frac {1}{\kappa _{1}}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2}={\frac {1}{\kappa _{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{2}={\frac {1}{\kappa _{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4d6f4b7f88623288e6736a1df298c701726fe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.793ex; height:5.509ex;" alt="{\displaystyle R_{2}={\frac {1}{\kappa _{2}}}.}" /></span></dd></dl> <p>The first and second radii of curvature correspond, respectively, to the Earth's meridional and prime-vertical radii of curvature. </p><p>Geometrically, the second fundamental form gives the distance from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+dr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>d</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+dr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dab06b0f538372d626953daa8f3a1859fd87837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.154ex; height:2.343ex;" alt="{\displaystyle r+dr}" /></span> to the plane tangent at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>. </p> </td></tr></tbody></table></div> <div class="mw-heading mw-heading4"><h4 id="Combined_radii_of_curvature">Combined radii of curvature</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=13" title="Edit section: Combined radii of curvature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading5"><h5 id="Azimuthal">Azimuthal<span class="anchor" id="Directional"></span></h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=14" title="Edit section: Azimuthal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Earth's <i>azimuthal radius of curvature</i>, along an <a href="/wiki/Earth_normal_section" class="mw-redirect" title="Earth normal section">Earth normal section</a> at an <a href="/wiki/Azimuth" title="Azimuth">azimuth</a> (measured clockwise from north) <span class="texhtml mvar" style="font-style:italic;">α</span> and at latitude <span class="texhtml mvar" style="font-style:italic;">φ</span>, is derived from <a href="/wiki/Euler%27s_theorem_(differential_geometry)" title="Euler's theorem (differential geometry)">Euler's curvature formula</a> as follows:<sup id="cite_ref-Torge_23-0" class="reference"><a href="#cite_note-Torge-23"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 97">: 97 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {c} }={\frac {1}{{\dfrac {\cos ^{2}\alpha }{M}}+{\dfrac {\sin ^{2}\alpha }{N}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>M</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>N</mi> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {c} }={\frac {1}{{\dfrac {\cos ^{2}\alpha }{M}}+{\dfrac {\sin ^{2}\alpha }{N}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7188044aacc795e04abc894fa74ca61b15fad529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:23.645ex; height:8.843ex;" alt="{\displaystyle R_{\mathrm {c} }={\frac {1}{{\dfrac {\cos ^{2}\alpha }{M}}+{\dfrac {\sin ^{2}\alpha }{N}}}}.}" /></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Non-directional">Non-directional</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=15" title="Edit section: Non-directional"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to combine the principal radii of curvature above in a non-directional manner. </p><p><span class="anchor" id="Gaussian"></span><span class="anchor" id="Gaussian_radius_of_curvature"></span>The <i>Earth's <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian radius of curvature</a></i> at latitude <span class="texhtml mvar" style="font-style:italic;">φ</span> is<sup id="cite_ref-Torge_23-1" class="reference"><a href="#cite_note-Torge-23"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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^{2}b}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}={\frac {a{\sqrt {1-e^{2}}}}{1-e^{2}\sin ^{2}\varphi }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>K</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>M</mi> <mi>N</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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^{2}b}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}={\frac {a{\sqrt {1-e^{2}}}}{1-e^{2}\sin ^{2}\varphi }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58160063579185a4b514bd64ad7263d3f3aa0d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:86.613ex; height:7.176ex;" alt="{\displaystyle 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^{2}b}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}={\frac {a{\sqrt {1-e^{2}}}}{1-e^{2}\sin ^{2}\varphi }},}" /></span></dd></dl> <p>where <i>K</i> is the <i>Gaussian curvature</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\kappa _{1}\,\kappa _{2}={\frac {\det B}{\det A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mi>B</mi> </mrow> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\kappa _{1}\,\kappa _{2}={\frac {\det B}{\det A}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d198cc617a772f249236063f2b482327d9675552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.3ex; height:5.509ex;" alt="{\displaystyle K=\kappa _{1}\,\kappa _{2}={\frac {\det B}{\det A}}.}" /></span> </p><p><span class="anchor" id="Mean_radius_of_curvature"></span>The <i>Earth's <a href="/wiki/Mean_curvature" title="Mean curvature">mean radius of curvature</a></i> at latitude <span class="texhtml mvar" style="font-style:italic;">φ</span> is<sup id="cite_ref-Torge_23-2" class="reference"><a href="#cite_note-Torge-23"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 97">: 97 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\text{m}}={\frac {2}{{\dfrac {1}{M}}+{\dfrac {1}{N}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\text{m}}={\frac {2}{{\dfrac {1}{M}}+{\dfrac {1}{N}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c35b2582680feae1e6b51087e0e3b54839ac21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:16.965ex; height:8.343ex;" alt="{\displaystyle R_{\text{m}}={\frac {2}{{\dfrac {1}{M}}+{\dfrac {1}{N}}}}.}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Global_radii">Global radii<span class="anchor" id="Mean_radii"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=16" title="Edit section: Global radii"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 <a href="/wiki/WGS-84" class="mw-redirect" title="WGS-84">WGS-84</a> ellipsoid;<sup id="cite_ref-tr8350_2_11-1" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> namely, </p> <dl><dd><i>Equatorial radius</i>: <span class="texhtml mvar" style="font-style:italic;">a</span> = (<span class="nowrap"><span data-sort-value="7006637813700000000♠"></span>6<span style="margin-left:.25em;">378</span>.1370 km</span>)</dd> <dd><i>Polar radius</i>: <span class="texhtml mvar" style="font-style:italic;">b</span> = (<span class="nowrap"><span data-sort-value="7006635675230000000♠"></span>6<span style="margin-left:.25em;">356</span>.7523 km</span>)</dd></dl> <p>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. </p> <div class="mw-heading mw-heading3"><h3 id="Arithmetic_mean_radius">Arithmetic mean radius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=17" title="Edit section: Arithmetic mean radius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:WGS84_mean_Earth_radius.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/WGS84_mean_Earth_radius.svg/220px-WGS84_mean_Earth_radius.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/WGS84_mean_Earth_radius.svg/330px-WGS84_mean_Earth_radius.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/WGS84_mean_Earth_radius.svg/440px-WGS84_mean_Earth_radius.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Equatorial (<i>a</i>), polar (<i>b</i>) and arithmetic mean Earth radii as defined in the 1984 <a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a> revision (not to scale)</figcaption></figure> <p>In geophysics, the <a href="/wiki/International_Union_of_Geodesy_and_Geophysics" title="International Union of Geodesy and Geophysics">International Union of Geodesy and Geophysics</a> (IUGG) defines the <i>Earth's <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> radius</i> (denoted <span class="texhtml"><i>R</i><sub>1</sub></span>) to be<sup id="cite_ref-Moritz_2-1" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}={\frac {2a+b}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}={\frac {2a+b}{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6a99d0ae6eebbb7bf5ab80042eb12141994bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.63ex; height:5.343ex;" alt="{\displaystyle R_{1}={\frac {2a+b}{3}}.}" /></span></dd></dl> <p>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 6,371.0088 km (3,958.7613 mi).<sup id="cite_ref-Moritz2000_24-0" class="reference"><a href="#cite_note-Moritz2000-24"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Authalic_radius">Authalic radius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=18" title="Edit section: Authalic radius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Authalic_latitude" class="mw-redirect" title="Authalic latitude">Authalic latitude</a></div> <p><i>Earth's authalic radius</i> (meaning <a href="/wiki/Equal-area_projection" title="Equal-area projection">"equal area"</a>) is the radius of a hypothetical perfect sphere that has the same surface area as the <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>. The <a href="/wiki/IUGG" class="mw-redirect" title="IUGG">IUGG</a> denotes the authalic radius as <span class="texhtml"><i>R</i><sub>2</sub></span>.<sup id="cite_ref-Moritz_2-2" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A closed-form solution exists for a spheroid:<sup id="cite_ref-Snyder_manual_13-2" class="reference"><a href="#cite_note-Snyder_manual-13"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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 }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>e</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> </mrow> <mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>e</mi> </mrow> <mi>e</mi> </mfrac> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>A</mi> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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 }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7358c01700ed23568d45ac2cd1febd05dd0d91c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:63.658ex; height:8.176ex;" alt="{\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 }}},}" /></span></dd></dl> <p>where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle e={\sqrt {a^{2}-b^{2}}}{\big /}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mi>a</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle e={\sqrt {a^{2}-b^{2}}}{\big /}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba264a415e730e2122a2e0d366f8d967a44fc9b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.255ex; height:3.676ex;" alt="{\displaystyle \textstyle e={\sqrt {a^{2}-b^{2}}}{\big /}a}" /></span>⁠</span> is the eccentricity, and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>⁠</span> is the surface area of the spheroid. </p><p>For the Earth, the authalic radius is 6,371.0072 km (3,958.7603 mi).<sup id="cite_ref-Moritz2000_24-1" class="reference"><a href="#cite_note-Moritz2000-24"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The authalic radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f571121c264178676d1df8ab899f238a39bc2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{2}}" /></span> also corresponds to the <i>radius of (global) mean curvature</i>, obtained by averaging the Gaussian curvature, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /></span>, over the surface of the ellipsoid. Using the <a href="/wiki/Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a>, this gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\int K\,dA}{A}}={\frac {4\pi }{A}}={\frac {1}{R_{2}^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mi>K</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>A</mi> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\int K\,dA}{A}}={\frac {4\pi }{A}}={\frac {1}{R_{2}^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a3a58a72cf1f1c688a12ffe3cc3de290a47804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.882ex; height:6.843ex;" alt="{\displaystyle {\frac {\int K\,dA}{A}}={\frac {4\pi }{A}}={\frac {1}{R_{2}^{2}}}.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Volumetric_radius">Volumetric radius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=19" title="Edit section: Volumetric radius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another spherical model is defined by the <i>Earth's volumetric radius</i>, which is the radius of a sphere of volume equal to the ellipsoid. The <a href="/wiki/IUGG" class="mw-redirect" title="IUGG">IUGG</a> denotes the volumetric radius as <span class="texhtml"><i>R</i><sub>3</sub></span>.<sup id="cite_ref-Moritz_2-3" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{3}={\sqrt[{3}]{a^{2}b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{3}={\sqrt[{3}]{a^{2}b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c304c12d43772df15ea6ea3be8e835d270b47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.169ex; height:3.343ex;" alt="{\displaystyle R_{3}={\sqrt[{3}]{a^{2}b}}.}" /></span></dd></dl> <p>For Earth, the volumetric radius equals 6,371.0008 km (3,958.7564 mi).<sup id="cite_ref-Moritz2000_24-2" class="reference"><a href="#cite_note-Moritz2000-24"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Rectifying_radius">Rectifying radius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=20" title="Edit section: Rectifying radius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quarter_meridian" class="mw-redirect" title="Quarter meridian">Quarter meridian</a> and <a href="/wiki/Rectifying_latitude" class="mw-redirect" title="Rectifying latitude">Rectifying latitude</a></div> <p>Another global radius is the <i>Earth's rectifying radius</i>, giving a sphere with circumference equal to the <a href="/wiki/Circumference" title="Circumference">perimeter</a> of the ellipse described by any polar cross section of the ellipsoid. This requires an <a href="/wiki/Circumference#Ellipse" title="Circumference">elliptic integral</a> to find, given the polar and equatorial radii: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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 .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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 .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da3dfc49d996dc1c07bcee019ba1bdaf6e3f2e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.403ex; height:6.509ex;" alt="{\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 .}" /></span></dd></dl> <p>The rectifying radius is equivalent to the meridional mean, which is defined as the average value of <span class="texhtml mvar" style="font-style:italic;">M</span>:<sup id="cite_ref-Snyder_manual_13-3" class="reference"><a href="#cite_note-Snyder_manual-13"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{r}}={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}M(\varphi )\,d\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mi>M</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{r}}={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}M(\varphi )\,d\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0dcd6b9d2274a2951c9751e5452fba972a4955a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.011ex; height:6.509ex;" alt="{\displaystyle M_{\text{r}}={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}M(\varphi )\,d\varphi .}" /></span></dd></dl> <p>For integration limits of [0,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>], 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). </p><p>The meridional mean is well approximated by the semicubic mean of the two axes,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2020)">citation needed</span></a></i>]</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{r}}\approx \left({\frac {a^{\frac {3}{2}}+b^{\frac {3}{2}}}{2}}\right)^{\frac {2}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> <mo>≈</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{r}}\approx \left({\frac {a^{\frac {3}{2}}+b^{\frac {3}{2}}}{2}}\right)^{\frac {2}{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5a7a312adf176d07c8d45343bc3ec270110322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.054ex; height:9.509ex;" alt="{\displaystyle M_{\text{r}}\approx \left({\frac {a^{\frac {3}{2}}+b^{\frac {3}{2}}}{2}}\right)^{\frac {2}{3}},}" /></span></dd></dl> <p>which differs from the exact result by less than 1 μm (4<span style="margin:0 .15em 0 .25em">×</span>10<sup>−5</sup> in); the mean of the two axes, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{r}}\approx {\frac {a+b}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{r}}\approx {\frac {a+b}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d76b66cd56d5a496970b910d5ad7fe409a4a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.78ex; height:5.343ex;" alt="{\displaystyle M_{\text{r}}\approx {\frac {a+b}{2}},}" /></span></dd></dl> <p>about 6,367.445 km (3,956.547 mi), can also be used. </p> <div class="mw-heading mw-heading2"><h2 id="Topographical_radii">Topographical radii</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=21" title="Edit section: Topographical radii"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Earth#Size_and_shape" title="Earth">Earth § Size and shape</a></div> <p>The mathematical expressions above apply over the surface of the ellipsoid. The cases below considers Earth's <a href="/wiki/Topography" title="Topography">topography</a>, above or below a <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>. As such, they are <i>topographical <a href="/wiki/Geocentric_distance" class="mw-redirect" title="Geocentric distance">geocentric distances</a></i>, <i>R</i><sub>t</sub>, which depends not only on latitude. </p> <div class="mw-heading mw-heading3"><h3 id="Topographical_extremes">Topographical extremes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=22" title="Edit section: Topographical extremes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Maximum <i>R</i><sub>t</sub>: the summit of <a href="/wiki/Chimborazo" title="Chimborazo">Chimborazo</a> is 6,384.4 km (3,967.1 mi) from the Earth's center.</li> <li>Minimum <i>R</i><sub>t</sub>: the floor of the <a href="/wiki/Arctic_Ocean" title="Arctic Ocean">Arctic Ocean</a> is 6,352.8 km (3,947.4 mi) from the Earth's center.<sup id="cite_ref-extrema_25-0" class="reference"><a href="#cite_note-extrema-25"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Topographical_global_mean">Topographical global mean</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=23" title="Edit section: Topographical global mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>topographical mean geocentric distance</i> averages elevations everywhere, resulting in a value <span class="nowrap"><span data-sort-value="7002230000000000000♠"></span>230 m</span> larger than the <a href="#Arithmetic_mean_radius">IUGG mean radius</a>, the <a href="/wiki/Authalic_radius" class="mw-redirect" title="Authalic radius">authalic radius</a>, or the <a href="#Volumetric_radius">volumetric radius</a>. This topographical average is 6,371.230 km (3,958.899 mi) with uncertainty of 10 m (33 ft).<sup id="cite_ref-chambat_26-0" class="reference"><a href="#cite_note-chambat-26"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Derived_quantities:_diameter,_circumference,_arc-length,_area,_volume"><span id="Derived_quantities:_diameter.2C_circumference.2C_arc-length.2C_area.2C_volume"></span>Derived quantities: diameter, circumference, arc-length, area, volume <span class="anchor" id="Derived_quantities"></span><span class="anchor" id="Derived_geometric_quantities"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=24" title="Edit section: Derived quantities: diameter, circumference, arc-length, area, volume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Diameter"></span><b>Earth's <a href="/wiki/Diameter" title="Diameter">diameter</a></b> is simply twice Earth's radius; for example, <i>equatorial diameter</i> (2<i>a</i>) and <i>polar diameter</i> (2<i>b</i>). For the WGS84 ellipsoid, that's respectively: </p> <dl><dd><span class="texhtml">2<i>a</i> = 12,756.2740 km (7,926.3812 mi)</span>,</dd> <dd><span class="texhtml">2<i>b</i> = 12,713.5046 km (7,899.8055 mi)</span>.</dd></dl> <p><i><a href="/wiki/Earth%27s_circumference" title="Earth's circumference">Earth's circumference</a></i> equals the <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> length. The <i>equatorial circumference</i> is simply the <a href="/wiki/Circle_perimeter" class="mw-redirect" title="Circle perimeter">circle perimeter</a>: <i>C</i><sub>e</sub> = 2<i>πa</i>, in terms of the equatorial radius <i>a</i>. The <i>polar circumference</i> equals <i>C</i><sub>p</sub> = 4<i>m</i><sub>p</sub>, four times the <a href="/wiki/Quarter_meridian" class="mw-redirect" title="Quarter meridian">quarter meridian</a> <i>m</i><sub>p</sub> = <i>aE</i>(<i>e</i>), where the polar radius <i>b</i> enters via the eccentricity <i>e</i> = (1 − <i>b</i><sup>2</sup>/<i>a</i><sup>2</sup>)<sup>0.5</sup>; see <a href="/wiki/Ellipse#Circumference" title="Ellipse">Ellipse#Circumference</a> for details. </p><p><a href="/wiki/Arc_length" title="Arc length">Arc length</a> of more general <a href="/wiki/Surface_curve" class="mw-redirect" title="Surface curve">surface curves</a>, such as <a href="/wiki/Meridian_arc" title="Meridian arc">meridian arcs</a> and <a href="/wiki/Earth_geodesics" class="mw-redirect" title="Earth geodesics">geodesics</a>, can also be derived from Earth's equatorial and polar radii. </p><p>Likewise for <a href="/wiki/Surface_area" title="Surface area">surface area</a>, either based on a <a href="/wiki/Map_projection" title="Map projection">map projection</a> or a <a href="/wiki/Geodesic_polygon" class="mw-redirect" title="Geodesic polygon">geodesic polygon</a>. </p><p><span class="anchor" id="Volume"></span><b>Earth's volume</b>, or that of the reference ellipsoid, is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\tfrac {4}{3}}\pi a^{2}b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\tfrac {4}{3}}\pi a^{2}b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6d9aca4d371f255c1f6cb74122bccf21c214d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.804ex; height:3.676ex;" alt="{\displaystyle V={\tfrac {4}{3}}\pi a^{2}b.}" /></span> Using the parameters from <a href="/wiki/WGS84" class="mw-redirect" title="WGS84">WGS84</a> ellipsoid of revolution, <span class="nowrap"><i>a</i> = 6,378.137 km</span> and <span class="nowrap"><i>b</i> = <span class="nowrap"><span data-sort-value="7006635675231420000♠"></span>6<span style="margin-left:.25em;">356</span>.752<span style="margin-left:.25em;">3142</span> km</span></span>, <span class="nowrap"><i>V</i> = 1.08321<span style="margin:0 .15em 0 .25em">×</span>10<sup>12</sup> km<sup>3</sup> (2.5988<span style="margin:0 .15em 0 .25em">×</span>10<sup>11</sup> cu mi)</span>.<sup id="cite_ref-earth_fact_sheet_27-0" class="reference"><a href="#cite_note-earth_fact_sheet-27"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Nominal_radii">Nominal radii<span class="anchor" id="Nominal"></span><span class="anchor" id="Nominal_radius"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=25" title="Edit section: Nominal radii"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In astronomy, the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">International Astronomical Union</a> denotes the <i>nominal equatorial Earth radius</i> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\text{eE}}^{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>eE</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\text{eE}}^{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9ce43d9005404d3654534daa978785e414155f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.052ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}_{\text{eE}}^{\text{N}}}" /></span>, which is defined to be exactly 6,378.1 km (3,963.2 mi).<sup id="cite_ref-IAU_XXIX_1-2" class="reference"><a href="#cite_note-IAU_XXIX-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 3">: 3 </span></sup> The <i>nominal polar Earth radius</i> is defined exactly as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\text{pE}}^{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>pE</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\text{pE}}^{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22d9e5db5a1cd92b415f722945585c768390675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.236ex; height:3.509ex;" alt="{\displaystyle {\mathcal {R}}_{\text{pE}}^{\text{N}}}" /></span> = 6,356.8 km (3,949.9 mi). These values correspond to the zero <a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a> convention. Equatorial radius is conventionally used as the nominal value unless the polar radius is explicitly required.<sup id="cite_ref-IAU_XXIX_1-3" class="reference"><a href="#cite_note-IAU_XXIX-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 4">: 4 </span></sup> The nominal radius serves as a <a href="/wiki/Unit_of_length" title="Unit of length">unit of length</a> for <a href="/wiki/Astronomical_system_of_units" title="Astronomical system of units">astronomy</a>. (The notation is defined such that it can be easily generalized for other <a href="/wiki/Planet#Size_and_shape" title="Planet">planets</a>; e.g., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\text{pJ}}^{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>pJ</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\text{pJ}}^{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41295f294f64ffbaea092d83a9eb88009d8f1dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.962ex; height:3.509ex;" alt="{\displaystyle {\mathcal {R}}_{\text{pJ}}^{\text{N}}}" /></span> for the nominal polar <a href="/wiki/Jupiter_radius" title="Jupiter radius">Jupiter radius</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Published_values">Published values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=26" title="Edit section: Published values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This table summarizes the accepted values of the Earth's radius. </p> <table class="wikitable sortable"> <tbody><tr> <th>Agency </th> <th>Description </th> <th>Value (in meters) </th> <th>Ref </th></tr> <tr> <td><a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">IAU</a> </td> <td>nominal "zero tide" equatorial </td> <td><span class="nowrap"><span data-sort-value="7006637810000000000♠"></span>6<span style="margin-left:.25em;">378</span><span style="margin-left:.25em;">100</span></span> </td> <td><sup id="cite_ref-IAU_XXIX_1-4" class="reference"><a href="#cite_note-IAU_XXIX-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>IAU </td> <td>nominal "zero tide" polar </td> <td><span class="nowrap"><span data-sort-value="7006635680000000000♠"></span>6<span style="margin-left:.25em;">356</span><span style="margin-left:.25em;">800</span></span> </td> <td><sup id="cite_ref-IAU_XXIX_1-5" class="reference"><a href="#cite_note-IAU_XXIX-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/International_Union_of_Geodesy_and_Geophysics" title="International Union of Geodesy and Geophysics">IUGG</a> </td> <td>equatorial radius </td> <td><span class="nowrap"><span data-sort-value="7006637813700000000♠"></span>6<span style="margin-left:.25em;">378</span><span style="margin-left:.25em;">137</span></span> </td> <td><sup id="cite_ref-Moritz_2-4" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>IUGG </td> <td>semiminor axis (<i>b</i>) </td> <td><span class="nowrap"><span data-sort-value="7006635675231410000♠"></span>6<span style="margin-left:.25em;">356</span><span style="margin-left:.25em;">752</span>.3141</span> </td> <td><sup id="cite_ref-Moritz_2-5" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>IUGG </td> <td>polar radius of curvature (<i>c</i>) </td> <td><span class="nowrap"><span data-sort-value="7006639959362590000♠"></span>6<span style="margin-left:.25em;">399</span><span style="margin-left:.25em;">593</span>.6259</span> </td> <td><sup id="cite_ref-Moritz_2-6" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>IUGG </td> <td>mean radius (<i>R<sub>1</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006637100877140000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">008</span>.7714</span> </td> <td><sup id="cite_ref-Moritz_2-7" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>IUGG </td> <td>radius of sphere of same surface (<i>R<sub>2</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006637100718100000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">007</span>.1810</span> </td> <td><sup id="cite_ref-Moritz_2-8" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>IUGG </td> <td>radius of sphere of same volume (<i>R<sub>3</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006637100079000000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">000</span>.7900</span> </td> <td><sup id="cite_ref-Moritz_2-9" class="reference"><a href="#cite_note-Moritz-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">NGA</a> </td> <td><a href="/wiki/WGS-84" class="mw-redirect" title="WGS-84">WGS-84</a> ellipsoid, semi-major axis (<i>a</i>) </td> <td><span class="nowrap"><span data-sort-value="7006637813700000000♠"></span>6<span style="margin-left:.25em;">378</span><span style="margin-left:.25em;">137</span>.0</span> </td> <td><sup id="cite_ref-tr8350_2_11-2" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>NGA </td> <td>WGS-84 ellipsoid, semi-minor axis (<i>b</i>) </td> <td><span class="nowrap"><span data-sort-value="7006635675231420000♠"></span>6<span style="margin-left:.25em;">356</span><span style="margin-left:.25em;">752</span>.3142</span> </td> <td><sup id="cite_ref-tr8350_2_11-3" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>NGA </td> <td>WGS-84 ellipsoid, polar radius of curvature (<i>c</i>) </td> <td><span class="nowrap"><span data-sort-value="7006639959362580000♠"></span>6<span style="margin-left:.25em;">399</span><span style="margin-left:.25em;">593</span>.6258</span> </td> <td><sup id="cite_ref-tr8350_2_11-4" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>NGA </td> <td>WGS-84 ellipsoid, Mean radius of semi-axes (<i>R<sub>1</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006637100877140000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">008</span>.7714</span> </td> <td><sup id="cite_ref-tr8350_2_11-5" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>NGA </td> <td>WGS-84 ellipsoid, Radius of Sphere of Equal Area (<i>R<sub>2</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006637100718090000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">007</span>.1809</span> </td> <td><sup id="cite_ref-tr8350_2_11-6" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>NGA </td> <td>WGS-84 ellipsoid, Radius of Sphere of Equal Volume (<i>R<sub>3</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006637100079000000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">000</span>.7900</span> </td> <td><sup id="cite_ref-tr8350_2_11-7" class="reference"><a href="#cite_note-tr8350_2-11"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td> </td> <td><a href="/wiki/GRS_80" class="mw-redirect" title="GRS 80">GRS 80</a> semi-major axis (<i>a</i>) </td> <td><span class="nowrap"><span data-sort-value="7006637813700000000♠"></span>6<span style="margin-left:.25em;">378</span><span style="margin-left:.25em;">137</span>.0</span> </td> <td> </td></tr> <tr> <td> </td> <td><a href="/wiki/GRS_80" class="mw-redirect" title="GRS 80">GRS 80</a> semi-minor axis (<i>b</i>) </td> <td><span class="nowrap"><span data-sort-value="7006635675231414000♠"></span>≈6<span style="margin-left:.25em;">356</span><span style="margin-left:.25em;">752</span>.314<span style="margin-left:.25em;">140</span></span> </td> <td> </td></tr> <tr> <td> </td> <td>Spherical Earth Approx. of Radius (<i>R<sub>E</sub></i>) </td> <td><span class="nowrap"><span data-sort-value="7006636670701950000♠"></span>6<span style="margin-left:.25em;">366</span><span style="margin-left:.25em;">707</span>.0195</span> </td> <td><sup id="cite_ref-Phillips_28-0" class="reference"><a href="#cite_note-Phillips-28"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td> </td> <td>meridional radius of curvature at the equator </td> <td><span class="nowrap"><span data-sort-value="7006633543900000000♠"></span>6<span style="margin-left:.25em;">335</span><span style="margin-left:.25em;">439</span></span> </td> <td> </td></tr> <tr> <td> </td> <td>Maximum (the summit of Chimborazo) </td> <td><span class="nowrap"><span data-sort-value="7006638440000000000♠"></span>6<span style="margin-left:.25em;">384</span><span style="margin-left:.25em;">400</span></span> </td> <td><sup id="cite_ref-extrema_25-1" class="reference"><a href="#cite_note-extrema-25"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td> </td> <td>Minimum (the floor of the Arctic Ocean) </td> <td><span class="nowrap"><span data-sort-value="7006635280000000000♠"></span>6<span style="margin-left:.25em;">352</span><span style="margin-left:.25em;">800</span></span> </td> <td><sup id="cite_ref-extrema_25-2" class="reference"><a href="#cite_note-extrema-25"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td> </td> <td>Average distance from center to surface </td> <td><span class="nowrap"><span data-sort-value="7006637123000000000♠"></span>6<span style="margin-left:.25em;">371</span><span style="margin-left:.25em;">230</span><span style="margin-left:0.3em;margin-right:0.15em;">±</span>10</span> </td> <td><sup id="cite_ref-chambat_26-1" class="reference"><a href="#cite_note-chambat-26"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=27" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_geodesy" title="History of geodesy">History of geodesy</a>, <a href="/wiki/Spherical_Earth#History" title="Spherical Earth">Spherical Earth § History</a>, <a href="/wiki/Earth%27s_circumference#History" title="Earth's circumference">Earth's circumference § History</a>, and <a href="/wiki/Meridian_arc#History" title="Meridian arc">Meridian arc § History</a></div> <p>The first published reference to the Earth's size appeared around 350 <a href="/wiki/Anno_Domini" title="Anno Domini">BC</a>, when <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> reported in his book <i><a href="/wiki/On_the_Heavens" title="On the Heavens">On the Heavens</a></i><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> that mathematicians had guessed the circumference of the Earth to be 400,000 <a href="/wiki/Stadia_(length)" class="mw-redirect" title="Stadia (length)">stadia</a>. Scholars have interpreted Aristotle's figure to be anywhere from highly accurate<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> to almost double the true value.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The first known scientific measurement and calculation of the circumference of the Earth was performed by <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a> in about 240 BC. Estimates of the error of Eratosthenes's measurement range from 0.5% to 17%.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant. </p><p>Around 100 BC, <a href="/wiki/Posidonius_of_Apamea" class="mw-redirect" title="Posidonius of Apamea">Posidonius of Apamea</a> recomputed Earth's radius, and found it to be close to that by Eratosthenes,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> but later <a href="/wiki/Strabo" title="Strabo">Strabo</a> incorrectly attributed him a value about 3/4 of the actual size.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Claudius_Ptolemy" class="mw-redirect" title="Claudius Ptolemy">Claudius Ptolemy</a> around 150 <a href="/wiki/Anno_Domini" title="Anno Domini">AD</a> gave empirical evidence supporting a <a href="/wiki/Spherical_Earth" title="Spherical Earth">spherical Earth</a>,<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> but he accepted the lesser value attributed to Posidonius. His highly influential work, the <i><a href="/wiki/Almagest" title="Almagest">Almagest</a></i>,<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> left no doubt among medieval scholars that Earth is spherical, but they were wrong about its size. </p><p>By 1490, <a href="/wiki/Christopher_Columbus" title="Christopher Columbus">Christopher Columbus</a> believed that traveling 3,000 miles west from the west coast of the <a href="/wiki/Iberian_Peninsula" title="Iberian Peninsula">Iberian Peninsula</a> would let him reach the eastern coasts of <a href="/wiki/Asia" title="Asia">Asia</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> However, the 1492 enactment of that voyage <a href="/wiki/Voyages_of_Christopher_Columbus" title="Voyages of Christopher Columbus"> brought his fleet to the Americas</a>. The <a href="/wiki/Magellan_expedition" title="Magellan expedition">Magellan expedition</a> (1519–1522), which was the first <a href="/wiki/Circumnavigation" title="Circumnavigation">circumnavigation</a> of the World, soundly demonstrated the sphericity of the Earth,<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> and affirmed the original measurement of 40,000 km (25,000 mi) by Eratosthenes. </p><p>Around 1690, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> argued that Earth was closer to an <a href="/wiki/Spheroid#Oblate_spheroids" title="Spheroid"> oblate spheroid</a> than to a sphere. However, around 1730, <a href="/wiki/Jacques_Cassini" title="Jacques Cassini">Jacques Cassini</a> argued for a <a href="/wiki/Spheroid#Prolate_spheroids" title="Spheroid"> prolate spheroid</a> instead, due to different interpretations of the <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> involved.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> To settle the matter, the <a href="/wiki/French_Geodesic_Mission" class="mw-redirect" title="French Geodesic Mission">French Geodesic Mission</a> (1735–1739) measured one degree of <a href="/wiki/Latitude" title="Latitude">latitude</a> at two locations, one near the <a href="/wiki/Arctic_Circle" title="Arctic Circle">Arctic Circle</a> and the other near the <a href="/wiki/Equator" title="Equator">equator</a>. The expedition found that Newton's conjecture was correct:<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> the Earth is flattened at the <a href="/wiki/Geographical_pole" title="Geographical pole">poles</a> due to rotation's <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=28" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 24em;"> <ul><li><a href="/wiki/Earth%27s_circumference" title="Earth's circumference">Earth's circumference</a></li> <li><a href="/wiki/Earth_mass" title="Earth mass">Earth mass</a></li> <li><a href="/wiki/Effective_Earth_radius" class="mw-redirect" title="Effective Earth radius">Effective Earth radius</a></li> <li><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></li> <li><a href="/wiki/Geographical_distance" title="Geographical distance">Geographical distance</a></li> <li><a href="/wiki/Osculating_sphere" class="mw-redirect" title="Osculating sphere">Osculating sphere</a></li> <li><a href="/wiki/History_of_geodesy" title="History of geodesy">History of geodesy</a></li> <li><a href="/wiki/Planetary_radius" class="mw-redirect" title="Planetary radius">Planetary radius</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=29" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">For details see <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">figure of the Earth</a>, <a href="/wiki/Geoid" title="Geoid">geoid</a>, and <a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">There is no single center to the geoid; it varies according to local <a href="/wiki/Geodetic_system" class="mw-redirect" title="Geodetic system">geodetic</a> conditions.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">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.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">The value of the radius is completely dependent upon the latitude in the case of an ellipsoid model, and nearly so on the geoid.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">This follows from the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">International Astronomical Union</a> <a href="/wiki/2006_definition_of_planet" class="mw-redirect" title="2006 definition of planet">definition</a> rule (2): a planet assumes a shape due to <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">hydrostatic equilibrium</a> where <a href="/wiki/Gravity" title="Gravity">gravity</a> and <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal forces</a> are nearly balanced.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-curvprim-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-curvprim_17-0">^</a></b></span> <span class="reference-text">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.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><span class="texhtml mvar" style="font-style:italic;">N</span> 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.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-IAU_XXIX-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-IAU_XXIX_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IAU_XXIX_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IAU_XXIX_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-IAU_XXIX_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-IAU_XXIX_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-IAU_XXIX_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMamajekPrsaTorresHarmanec2015" class="citation arxiv cs1">Mamajek, E. 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In <a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a> (ed.). <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. Vol. 8 (11th ed.). Cambridge University Press. pp. <span class="nowrap">801–</span>813.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Earth%2C+Figure+of+the&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=%3Cspan+class%3D%22nowrap%22%3E801-%3C%2Fspan%3E813&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rft.aulast=Clarke&rft.aufirst=Alexander+Ross&rft.au=Helmert%2C+Friedrich+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/biography/Eratosthenes">"Eratosthenes, the Greek Scientist"</a>. <i>Britannica.com</i>. 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Eratosthenes%2C+the+Greek+Scientist&rft.btitle=Britannica.com&rft.date=2016&rft_id=https%3A%2F%2Fwww.britannica.com%2Fbiography%2FEratosthenes&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Posidonius, <a rel="nofollow" class="external text" href="http://www.attalus.org/translate/poseidonius.html#202.K">fragment 202</a></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Cleomedes (<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1975QJRAS..16..152F">in Fragment 202</a>) 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), <i>Another Look at Eratosthenes' and Posidonius' Determinations of the Earth's Circumference</i>, Ql. J. of the Royal Astron. Soc., Vol. 16, p. 152.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThurston1994" class="citation book cs1">Thurston, Hugh (1994). <i>Early astronomy</i>. New York: Springer-Verlag New York. p. 138. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94107-X" title="Special:BookSources/0-387-94107-X"><bdi>0-387-94107-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Early+astronomy&rft.place=New+York&rft.pages=138&rft.pub=Springer-Verlag+New+York&rft.date=1994&rft.isbn=0-387-94107-X&rft.aulast=Thurston&rft.aufirst=Hugh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://projects.iq.harvard.edu/predictionx/almagest-ptolemy-elizabeth">"Almagest – Ptolemy (Elizabeth)"</a>. <i>projects.iq.harvard.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=projects.iq.harvard.edu&rft.atitle=Almagest+%E2%80%93+Ptolemy+%28Elizabeth%29&rft_id=https%3A%2F%2Fprojects.iq.harvard.edu%2Fpredictionx%2Falmagest-ptolemy-elizabeth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><a href="/wiki/John_Freely" title="John Freely">John Freely</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MfhjAAAAQBAJ"><i>Before Galileo: The Birth of Modern Science in Medieval Europe</i></a> (2013), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1468308501" title="Special:BookSources/978-1468308501">978-1468308501</a></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNancy_Smiler_Levinson2001" class="citation book cs1">Nancy Smiler Levinson (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1PbBzjBuW8IC&pg=PA39"><i>Magellan and the First Voyage Around the World</i></a>. Houghton Mifflin Harcourt. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-395-98773-5" title="Special:BookSources/978-0-395-98773-5"><bdi>978-0-395-98773-5</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">31 July</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Magellan+and+the+First+Voyage+Around+the+World&rft.pub=Houghton+Mifflin+Harcourt&rft.date=2001&rft.isbn=978-0-395-98773-5&rft.au=Nancy+Smiler+Levinson&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1PbBzjBuW8IC%26pg%3DPA39&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCassini1738" class="citation book cs1 cs1-prop-foreign-lang-source">Cassini, Jacques (1738). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180127004440/https://bibnum.obspm.fr/items/show/41044#?c=0&m=0&s=0&cv=0&z=-0.0556%2C-0.339%2C1.1111%2C1.9323"><i>Méthode de déterminer si la terre est sphérique ou non</i></a> (in French). Archived from <a rel="nofollow" class="external text" href="https://bibnum.obspm.fr/items/show/41044#?c=0&m=0&s=0&cv=0&z=-0.0556%2C-0.339%2C1.1111%2C1.9323">the original</a> on 2018-01-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=M%C3%A9thode+de+d%C3%A9terminer+si+la+terre+est+sph%C3%A9rique+ou+non&rft.date=1738&rft.aulast=Cassini&rft.aufirst=Jacques&rft_id=https%3A%2F%2Fbibnum.obspm.fr%2Fitems%2Fshow%2F41044%23%3Fc%3D0%26m%3D0%26s%3D0%26cv%3D0%26z%3D-0.0556%252C-0.339%252C1.1111%252C1.9323&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLevallois1986" class="citation web cs1">Levallois, Jean-Jacques (1986). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k5470853s">"La Vie des sciences"</a>. <i>Gallica</i>. pp. <span class="nowrap">277–</span>284, 288<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-05-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Gallica&rft.atitle=La+Vie+des+sciences&rft.pages=%3Cspan+class%3D%22nowrap%22%3E277-%3C%2Fspan%3E284%2C+288&rft.date=1986&rft.aulast=Levallois&rft.aufirst=Jean-Jacques&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k5470853s&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Earth_radius&action=edit&section=31" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikisource-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 <i>Encyclopædia Britannica</i></a> article "<span style="font-weight:bold;"><a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Earth,_Figure_of_the" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Earth, Figure of the">Earth, Figure of the</a></span>".</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMerrifield2010" class="citation web cs1">Merrifield, Michael R. 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Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=MATH+RENDER+ERROR+The+Earth%27s+Radius+%28and+exoplanets%29&rft.date=2010&rft.aulast=Merrifield&rft.aufirst=Michael+R.&rft_id=http%3A%2F%2Fwww.sixtysymbols.com%2Fvideos%2Fearthradius.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEarth+radius" class="Z3988"></span></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a 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