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page-Àrea rootpage-Àrea skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Vés al contingut</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">amaga</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegació </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visiteu la pàgina principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Article_aleatori" title="Carrega una pàgina a l’atzar [x]" accesskey="x"><span>Article a l'atzar</span></a></li><li id="n-Articles-de-qualitat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat"><span>Articles de qualitat</span></a></li> </ul> </div> </div> <div id="p-Comunitat" class="vector-menu mw-portlet mw-portlet-Comunitat" > <div class="vector-menu-heading"> Comunitat </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Portal" title="Sobre el projecte, què podeu fer, on trobareu les coses"><span>Portal viquipedista</span></a></li><li id="n-Agenda-d'actes" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Trobades"><span>Agenda d'actes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Canvis_recents" title="Una llista dels canvis recents al wiki [r]" accesskey="r"><span>Canvis recents</span></a></li><li id="n-La-taverna" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:La_taverna"><span>La taverna</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Contacte"><span>Contacte</span></a></li><li id="n-Xat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Canals_IRC"><span>Xat</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Ajuda" title="El lloc per a saber més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Viquipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-ca.svg" style="width: 7.5em; height: 1.4375em;"> <img class="mw-logo-tagline" alt="l'Enciclopèdia Lliure" src="/static/images/mobile/copyright/wikipedia-tagline-ca.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Cerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cerca a la Viquipèdia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Cerca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca a Viquipèdia" aria-label="Cerca a Viquipèdia" autocapitalize="sentences" title="Cerca a la Viquipèdia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Cerca"> </div> <button class="cdx-button cdx-search-input__end-button">Cerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Eines personals"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparença</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca" class=""><span>Donatius</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=%C3%80rea" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=%C3%80rea" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Eines personals</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú d'usuari" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca"><span>Donatius</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crea_compte&returnto=%C3%80rea" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crea un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Registre_i_entrada&returnto=%C3%80rea" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Inicia la sessió</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contingut" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inici</div> </a> </li> <li id="toc-Història" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Història"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Història</span> </div> </a> <ul id="toc-Història-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definició_formal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definició_formal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definició formal</span> </div> </a> <ul id="toc-Definició_formal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unitats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unitats"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Unitats</span> </div> </a> <ul id="toc-Unitats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Àrees_bàsiques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Àrees_bàsiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Àrees bàsiques</span> </div> </a> <button aria-controls="toc-Àrees_bàsiques-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>Commuta la subsecció Àrees bàsiques</span> </button> <ul id="toc-Àrees_bàsiques-sublist" class="vector-toc-list"> <li id="toc-Àrea_d'un_triangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Àrea_d'un_triangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Àrea d'un triangle</span> </div> </a> <ul id="toc-Àrea_d'un_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Àrea_d'un_quadrilàter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Àrea_d'un_quadrilàter"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Àrea d'un quadrilàter</span> </div> </a> <ul id="toc-Àrea_d'un_quadrilàter-sublist" class="vector-toc-list"> <li id="toc-Rectangles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rectangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Rectangles</span> </div> </a> <ul id="toc-Rectangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mètode_de_la_dissecció" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mètode_de_la_dissecció"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Mètode de la dissecció</span> </div> </a> <ul id="toc-Mètode_de_la_dissecció-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altres_quadrilàters" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Altres_quadrilàters"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.3</span> <span>Altres quadrilàters</span> </div> </a> <ul id="toc-Altres_quadrilàters-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Àrea_del_cercle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Àrea_del_cercle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Àrea del cercle</span> </div> </a> <ul id="toc-Àrea_del_cercle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Àrea_superficial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Àrea_superficial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Àrea superficial</span> </div> </a> <ul id="toc-Àrea_superficial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Llista_de_fórmules" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Llista_de_fórmules"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Llista de fórmules</span> </div> </a> <ul id="toc-Llista_de_fórmules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fórmules_addicionals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fórmules_addicionals"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Fórmules addicionals</span> </div> </a> <button aria-controls="toc-Fórmules_addicionals-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>Commuta la subsecció Fórmules addicionals</span> </button> <ul id="toc-Fórmules_addicionals-sublist" class="vector-toc-list"> <li id="toc-Àrees_de_figures_de_dues_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Àrees_de_figures_de_dues_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Àrees de figures de dues dimensions</span> </div> </a> <ul id="toc-Àrees_de_figures_de_dues_dimensions-sublist" class="vector-toc-list"> <li id="toc-Triangle" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Triangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Triangle</span> </div> </a> <ul id="toc-Triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polígon_simple" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Polígon_simple"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Polígon simple</span> </div> </a> <ul id="toc-Polígon_simple-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Àrees_en_el_càlcul" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Àrees_en_el_càlcul"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Àrees en el càlcul</span> </div> </a> <ul id="toc-Àrees_en_el_càlcul-sublist" class="vector-toc-list"> <li id="toc-Àrea_entre_dues_funcions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Àrea_entre_dues_funcions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Àrea entre dues funcions</span> </div> </a> <ul id="toc-Àrea_entre_dues_funcions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Àrea_tancada_per_una_corba_paramètrica" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Àrea_tancada_per_una_corba_paramètrica"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>Àrea tancada per una corba paramètrica</span> </div> </a> <ul id="toc-Àrea_tancada_per_una_corba_paramètrica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Superfície_de_figures_tridimensionals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Superfície_de_figures_tridimensionals"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Superfície de figures tridimensionals</span> </div> </a> <ul id="toc-Superfície_de_figures_tridimensionals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaços_externs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enllaços_externs"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Enllaços externs</span> </div> </a> <ul id="toc-Enllaços_externs-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="Contingut" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Àrea</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="Vés a un article en una altra llengua. Disponible en 164 llengües" > <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-164" 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">164 llengües</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/Oppervlakte" title="Oppervlakte - afrikaans" lang="af" hreflang="af" data-title="Oppervlakte" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Fl%C3%A4cheninhalt" title="Flächeninhalt - alemany suís" lang="gsw" hreflang="gsw" data-title="Flächeninhalt" data-language-autonym="Alemannisch" data-language-local-name="alemany suís" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Aria" title="Aria - aragonès" lang="an" hreflang="an" data-title="Aria" data-language-autonym="Aragonés" data-language-local-name="aragonès" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल - angika" lang="anp" hreflang="anp" data-title="क्षेत्रफल" data-language-autonym="अंगिका" data-language-local-name="angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%A9" title="مساحة - àrab" lang="ar" hreflang="ar" data-title="مساحة" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%AB%DC%9B%DC%9D%DC%9A%DC%98%DC%AC%DC%90" title="ܫܛܝܚܘܬܐ - arameu" lang="arc" hreflang="arc" data-title="ܫܛܝܚܘܬܐ" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="arameu" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AA%D9%8A%D8%B3%D8%A7%D8%B9" title="تيساع - Moroccan Arabic" lang="ary" hreflang="ary" data-title="تيساع" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D9%87" title="مساحه - àrab egipci" lang="arz" hreflang="arz" data-title="مساحه" data-language-autonym="مصرى" data-language-local-name="àrab egipci" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A7%B0%E0%A6%AB%E0%A6%B2" title="ক্ষেত্ৰফল - assamès" lang="as" hreflang="as" data-title="ক্ষেত্ৰফল" data-language-autonym="অসমীয়া" data-language-local-name="assamès" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81rea_(xeometr%C3%ADa)" title="Área (xeometría) - asturià" lang="ast" hreflang="ast" data-title="Área (xeometría)" data-language-autonym="Asturianu" data-language-local-name="asturià" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-av mw-list-item"><a href="https://av.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0%D0%B4%D1%8C" title="Площадь - àvar" lang="av" hreflang="av" data-title="Площадь" data-language-autonym="Авар" data-language-local-name="àvar" class="interlanguage-link-target"><span>Авар</span></a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल - awadhi" lang="awa" hreflang="awa" data-title="क्षेत्रफल" data-language-autonym="अवधी" data-language-local-name="awadhi" class="interlanguage-link-target"><span>अवधी</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sah%C9%99_(%C3%B6l%C3%A7%C3%BC_parametri)" title="Sahə (ölçü parametri) - azerbaidjanès" lang="az" hreflang="az" data-title="Sahə (ölçü parametri)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%AA" title="مساحت - South Azerbaijani" lang="azb" hreflang="azb" data-title="مساحت" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B9%D2%99%D0%B0%D0%BD" title="Майҙан - baixkir" lang="ba" hreflang="ba" data-title="Майҙан" data-language-autonym="Башҡортса" data-language-local-name="baixkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Fl%C3%A4chn" title="Flächn - bavarès" lang="bar" hreflang="bar" data-title="Flächn" data-language-autonym="Boarisch" data-language-local-name="bavarès" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Pluots" title="Pluots - Samogitian" lang="sgs" hreflang="sgs" data-title="Pluots" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Hiwas" title="Hiwas - Central Bikol" lang="bcl" hreflang="bcl" data-title="Hiwas" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%88%D1%87%D0%B0" title="Плошча - belarús" lang="be" hreflang="be" data-title="Плошча" data-language-autonym="Беларуская" data-language-local-name="belarús" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%88%D1%87%D0%B0" title="Плошча - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Плошча" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89" title="Площ - búlgar" lang="bg" hreflang="bg" data-title="Площ" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल - Bhojpuri" lang="bh" hreflang="bh" data-title="क्षेत्रफल" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%AB%E0%A6%B2" title="ক্ষেত্রফল - bengalí" lang="bn" hreflang="bn" data-title="ক্ষেত্রফল" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Gorread" title="Gorread - bretó" lang="br" hreflang="br" data-title="Gorread" data-language-autonym="Brezhoneg" data-language-local-name="bretó" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina - bosnià" lang="bs" hreflang="bs" data-title="Površina" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-btm mw-list-item"><a href="https://btm.wikipedia.org/wiki/Bolak" title="Bolak - Batak Mandailing" lang="btm" hreflang="btm" data-title="Bolak" data-language-autonym="Batak Mandailing" data-language-local-name="Batak Mandailing" class="interlanguage-link-target"><span>Batak Mandailing</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Mi%C3%AAng-c%C3%A9k" title="Miêng-cék - Mindong" lang="cdo" hreflang="cdo" data-title="Miêng-cék" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B9%D0%B4%D0%B0" title="Майда - txetxè" lang="ce" hreflang="ce" data-title="Майда" data-language-autonym="Нохчийн" data-language-local-name="txetxè" class="interlanguage-link-target"><span>Нохчийн</span></a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Langyab" title="Langyab - cebuà" lang="ceb" hreflang="ceb" data-title="Langyab" data-language-autonym="Cebuano" data-language-local-name="cebuà" class="interlanguage-link-target"><span>Cebuano</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88%D8%A8%DB%95%D8%B1" title="ڕووبەر - kurd central" lang="ckb" hreflang="ckb" data-title="ڕووبەر" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Obsah" title="Obsah - txec" lang="cs" hreflang="cs" data-title="Obsah" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%A5" title="Пространиѥ - eslau eclesiàstic" lang="cu" hreflang="cu" data-title="Пространиѥ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="eslau eclesiàstic" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D1%82%C4%83%D0%BA" title="Лаптăк - txuvaix" lang="cv" hreflang="cv" data-title="Лаптăк" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Arwynebedd" title="Arwynebedd - gal·lès" lang="cy" hreflang="cy" data-title="Arwynebedd" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Areal" title="Areal - danès" lang="da" hreflang="da" data-title="Areal" data-language-autonym="Dansk" data-language-local-name="danès" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fl%C3%A4cheninhalt" title="Flächeninhalt - alemany" lang="de" hreflang="de" data-title="Flächeninhalt" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Erd" title="Erd - Zazaki" lang="diq" hreflang="diq" data-title="Erd" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Wop%C5%9Bimje%C5%9Be_p%C5%82oni" title="Wopśimjeśe płoni - baix sòrab" lang="dsb" hreflang="dsb" data-title="Wopśimjeśe płoni" data-language-autonym="Dolnoserbski" data-language-local-name="baix sòrab" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-dv mw-list-item"><a href="https://dv.wikipedia.org/wiki/%DE%87%DE%A6%DE%86%DE%A6%DE%89%DE%A8%DE%82%DE%B0" title="އަކަމިން - divehi" lang="dv" hreflang="dv" data-title="އަކަމިން" data-language-autonym="ދިވެހިބަސް" data-language-local-name="divehi" class="interlanguage-link-target"><span>ދިވެހިބަސް</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD" title="Εμβαδόν - grec" lang="el" hreflang="el" data-title="Εμβαδόν" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Area" title="Area - anglès" lang="en" hreflang="en" data-title="Area" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Areo" title="Areo - esperanto" lang="eo" hreflang="eo" data-title="Areo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81rea" title="Área - espanyol" lang="es" hreflang="es" data-title="Área" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pindala" title="Pindala - estonià" lang="et" hreflang="et" data-title="Pindala" data-language-autonym="Eesti" data-language-local-name="estonià" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Azalera" title="Azalera - basc" lang="eu" hreflang="eu" data-title="Azalera" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%AA" title="مساحت - persa" lang="fa" hreflang="fa" data-title="مساحت" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pinta-ala" title="Pinta-ala - finès" lang="fi" hreflang="fi" data-title="Pinta-ala" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Pindala" title="Pindala - Võro" lang="vro" hreflang="vro" data-title="Pindala" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Iwasewase" title="Iwasewase - fijià" lang="fj" hreflang="fj" data-title="Iwasewase" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijià" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/V%C3%ADdd" title="Vídd - feroès" lang="fo" hreflang="fo" data-title="Vídd" data-language-autonym="Føroyskt" data-language-local-name="feroès" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Aire_(g%C3%A9om%C3%A9trie)" title="Aire (géométrie) - francès" lang="fr" hreflang="fr" data-title="Aire (géométrie)" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Areaal_(Miat)" title="Areaal (Miat) - frisó septentrional" lang="frr" hreflang="frr" data-title="Areaal (Miat)" data-language-autonym="Nordfriisk" data-language-local-name="frisó septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Oerflak" title="Oerflak - frisó occidental" lang="fy" hreflang="fy" data-title="Oerflak" data-language-autonym="Frysk" data-language-local-name="frisó occidental" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%9D%A2%E7%A9%8D" title="面積 - xinès gan" lang="gan" hreflang="gan" data-title="面積" data-language-autonym="贛語" data-language-local-name="xinès gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Farsaingeachd" title="Farsaingeachd - gaèlic escocès" lang="gd" hreflang="gd" data-title="Farsaingeachd" data-language-autonym="Gàidhlig" data-language-local-name="gaèlic escocès" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81rea" title="Área - gallec" lang="gl" hreflang="gl" data-title="Área" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Tendaha" title="Tendaha - guaraní" lang="gn" hreflang="gn" data-title="Tendaha" data-language-autonym="Avañe'ẽ" data-language-local-name="guaraní" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%95%E0%AB%8D%E0%AA%B7%E0%AB%87%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%AB%E0%AA%B3" title="ક્ષેત્રફળ - gujarati" lang="gu" hreflang="gu" data-title="ક્ષેત્રફળ" data-language-autonym="ગુજરાતી" data-language-local-name="gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Eaghtyr" title="Eaghtyr - manx" lang="gv" hreflang="gv" data-title="Eaghtyr" data-language-autonym="Gaelg" data-language-local-name="manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Mien-chit" title="Mien-chit - xinès hakka" lang="hak" hreflang="hak" data-title="Mien-chit" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="xinès hakka" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/%CA%BBAlea" title="ʻAlea - hawaià" lang="haw" hreflang="haw" data-title="ʻAlea" data-language-autonym="Hawaiʻi" data-language-local-name="hawaià" class="interlanguage-link-target"><span>Hawaiʻi</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%98%D7%97" title="שטח - hebreu" lang="he" hreflang="he" data-title="שטח" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल - hindi" lang="hi" hreflang="hi" data-title="क्षेत्रफल" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Area" title="Area - hindi de Fiji" lang="hif" hreflang="hif" data-title="Area" data-language-autonym="Fiji Hindi" data-language-local-name="hindi de Fiji" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina - croat" lang="hr" hreflang="hr" data-title="Površina" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Wobsah_p%C5%99estrjenje" title="Wobsah přestrjenje - alt sòrab" lang="hsb" hreflang="hsb" data-title="Wobsah přestrjenje" data-language-autonym="Hornjoserbsce" data-language-local-name="alt sòrab" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ter%C3%BClet_(matematika)" title="Terület (matematika) - hongarès" lang="hu" hreflang="hu" data-title="Terület (matematika)" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%AF%D5%A5%D6%80%D5%A5%D5%BD" title="Մակերես - armeni" lang="hy" hreflang="hy" data-title="Մակերես" data-language-autonym="Հայերեն" data-language-local-name="armeni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Area" title="Area - interlingua" lang="ia" hreflang="ia" data-title="Area" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Luas" title="Luas - indonesi" lang="id" hreflang="id" data-title="Luas" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Kalawa" title="Kalawa - ilocano" lang="ilo" hreflang="ilo" data-title="Kalawa" data-language-autonym="Ilokano" data-language-local-name="ilocano" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Areo" title="Areo - ido" lang="io" hreflang="io" data-title="Areo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Flatarm%C3%A1l" title="Flatarmál - islandès" lang="is" hreflang="is" data-title="Flatarmál" data-language-autonym="Íslenska" data-language-local-name="islandès" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Area" title="Area - italià" lang="it" hreflang="it" data-title="Area" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9D%A2%E7%A9%8D" title="面積 - japonès" lang="ja" hreflang="ja" data-title="面積" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Ieria" title="Ieria - crioll anglès de Jamaica" lang="jam" hreflang="jam" data-title="Ieria" data-language-autonym="Patois" data-language-local-name="crioll anglès de Jamaica" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Jembar" title="Jembar - javanès" lang="jv" hreflang="jv" data-title="Jembar" data-language-autonym="Jawa" data-language-local-name="javanès" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%90%E1%83%A0%E1%83%97%E1%83%9D%E1%83%91%E1%83%98" title="ფართობი - georgià" lang="ka" hreflang="ka" data-title="ფართობი" data-language-autonym="ქართული" data-language-local-name="georgià" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tajumma" title="Tajumma - cabilenc" lang="kab" hreflang="kab" data-title="Tajumma" data-language-autonym="Taqbaylit" data-language-local-name="cabilenc" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%A9%D3%80%D1%8B%D0%BF%D3%80%D1%8D_%D0%B8%D0%BD%D0%B0%D0%B3%D1%8A" title="ЩӀыпӀэ инагъ - kabardí" lang="kbd" hreflang="kbd" data-title="ЩӀыпӀэ инагъ" data-language-autonym="Адыгэбзэ" data-language-local-name="kabardí" class="interlanguage-link-target"><span>Адыгэбзэ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%83%D0%B4%D0%B0%D0%BD_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Аудан (геометрия) - kazakh" lang="kk" hreflang="kk" data-title="Аудан (геометрия)" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%9B%E1%9E%B6%E1%9E%95%E1%9F%92%E1%9E%91%E1%9F%83" title="ក្រលាផ្ទៃ - khmer" lang="km" hreflang="km" data-title="ក្រលាផ្ទៃ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%84%93%EC%9D%B4" title="넓이 - coreà" lang="ko" hreflang="ko" data-title="넓이" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/R%C3%BBerd" title="Rûerd - kurd" lang="ku" hreflang="ku" data-title="Rûerd" data-language-autonym="Kurdî" data-language-local-name="kurd" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Enep_(fysegieth)" title="Enep (fysegieth) - còrnic" lang="kw" hreflang="kw" data-title="Enep (fysegieth)" data-language-autonym="Kernowek" data-language-local-name="còrnic" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%8F%D0%BD%D1%82" title="Аянт - kirguís" lang="ky" hreflang="ky" data-title="Аянт" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Area_(geometria)" title="Area (geometria) - llatí" lang="la" hreflang="la" data-title="Area (geometria)" data-language-autonym="Latina" data-language-local-name="llatí" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Fl%C3%A4ch" title="Fläch - luxemburguès" lang="lb" hreflang="lb" data-title="Fläch" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburguès" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Oppervlak" title="Oppervlak - limburguès" lang="li" hreflang="li" data-title="Oppervlak" data-language-autonym="Limburgs" data-language-local-name="limburguès" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Area" title="Area - lígur" lang="lij" hreflang="lij" data-title="Area" data-language-autonym="Ligure" data-language-local-name="lígur" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Area" title="Area - llombard" lang="lmo" hreflang="lmo" data-title="Area" data-language-autonym="Lombard" data-language-local-name="llombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Etando" title="Etando - lingala" lang="ln" hreflang="ln" data-title="Etando" data-language-autonym="Lingála" data-language-local-name="lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%99%E0%BA%B7%E0%BB%89%E0%BA%AD%E0%BA%97%E0%BA%B5%E0%BB%88" title="ເນື້ອທີ່ - laosià" lang="lo" hreflang="lo" data-title="ເນື້ອທີ່" data-language-autonym="ລາວ" data-language-local-name="laosià" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Plotas" title="Plotas - lituà" lang="lt" hreflang="lt" data-title="Plotas" data-language-autonym="Lietuvių" data-language-local-name="lituà" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Laukums" title="Laukums - letó" lang="lv" hreflang="lv" data-title="Laukums" data-language-autonym="Latviešu" data-language-local-name="letó" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल - maithili" lang="mai" hreflang="mai" data-title="क्षेत्रफल" data-language-autonym="मैथिली" data-language-local-name="maithili" class="interlanguage-link-target"><span>मैथिली</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Velarantany" title="Velarantany - malgaix" lang="mg" hreflang="mg" data-title="Velarantany" data-language-autonym="Malagasy" data-language-local-name="malgaix" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D0%B4%D1%8B%D0%BA" title="Кумдык - Eastern Mari" lang="mhr" hreflang="mhr" data-title="Кумдык" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%88%D1%82%D0%B8%D0%BD%D0%B0" title="Плоштина - macedoni" lang="mk" hreflang="mk" data-title="Плоштина" data-language-autonym="Македонски" data-language-local-name="macedoni" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%80%E0%B5%BC%E0%B4%A3%E0%B5%8D%E0%B4%A3%E0%B4%82" title="വിസ്തീർണ്ണം - malaiàlam" lang="ml" hreflang="ml" data-title="വിസ്തീർണ്ണം" data-language-autonym="മലയാളം" data-language-local-name="malaiàlam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BB%D0%B1%D0%B0%D0%B9" title="Талбай - mongol" lang="mn" hreflang="mn" data-title="Талбай" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%84%EA%AF%A5%EA%AF%9B%EA%AF%86%EA%AF%A5%EA%AF%8E%EA%AF%95" title="ꯄꯥꯛꯆꯥꯎꯕ - manipurí" lang="mni" hreflang="mni" data-title="ꯄꯥꯛꯆꯥꯎꯕ" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="manipurí" class="interlanguage-link-target"><span>ꯃꯤꯇꯩ ꯂꯣꯟ</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B3" title="क्षेत्रफळ - marathi" lang="mr" hreflang="mr" data-title="क्षेत्रफळ" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Luas" title="Luas - malai" lang="ms" hreflang="ms" data-title="Luas" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/%C3%81ria" title="Ária - mirandès" lang="mwl" hreflang="mwl" data-title="Ária" data-language-autonym="Mirandés" data-language-local-name="mirandès" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A7%E1%80%9B%E1%80%AD%E1%80%9A%E1%80%AC" title="ဧရိယာ - birmà" lang="my" hreflang="my" data-title="ဧရိယာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmà" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%DA%AF%D8%AA%DB%8C" title="گتی - mazanderani" lang="mzn" hreflang="mzn" data-title="گتی" data-language-autonym="مازِرونی" data-language-local-name="mazanderani" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Flach" title="Flach - baix alemany" lang="nds" hreflang="nds" data-title="Flach" data-language-autonym="Plattdüütsch" data-language-local-name="baix alemany" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Oppervlakte" title="Oppervlakte - baix saxó" lang="nds-NL" hreflang="nds-NL" data-title="Oppervlakte" data-language-autonym="Nedersaksies" data-language-local-name="baix saxó" class="interlanguage-link-target"><span>Nedersaksies</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल - nepalès" lang="ne" hreflang="ne" data-title="क्षेत्रफल" data-language-autonym="नेपाली" data-language-local-name="nepalès" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Oppervlakte" title="Oppervlakte - neerlandès" lang="nl" hreflang="nl" data-title="Oppervlakte" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Flatevidd" title="Flatevidd - noruec nynorsk" lang="nn" hreflang="nn" data-title="Flatevidd" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Areal" title="Areal - noruec bokmål" lang="nb" hreflang="nb" data-title="Areal" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Aira" title="Aira - occità" lang="oc" hreflang="oc" data-title="Aira" data-language-autonym="Occitan" data-language-local-name="occità" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%A4%C3%A6%D0%B7%D1%83%D0%B0%D1%82" title="Фæзуат - osseta" lang="os" hreflang="os" data-title="Фæзуат" data-language-autonym="Ирон" data-language-local-name="osseta" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%96%E0%A9%87%E0%A8%A4%E0%A8%B0%E0%A8%AB%E0%A8%B2" title="ਖੇਤਰਫਲ - panjabi" lang="pa" hreflang="pa" data-title="ਖੇਤਰਫਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Fl%C3%A4che" title="Fläche - alemany palatí" lang="pfl" hreflang="pfl" data-title="Fläche" data-language-autonym="Pälzisch" data-language-local-name="alemany palatí" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pole_powierzchni" title="Pole powierzchni - polonès" lang="pl" hreflang="pl" data-title="Pole powierzchni" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%D9%82%D8%A8%DB%81" title="رقبہ - Western Punjabi" lang="pnb" hreflang="pnb" data-title="رقبہ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81rea" title="Área - portuguès" lang="pt" hreflang="pt" data-title="Área" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Hallka_k%27iti_k%27anchar" title="Hallka k'iti k'anchar - quítxua" lang="qu" hreflang="qu" data-title="Hallka k'iti k'anchar" data-language-autonym="Runa Simi" data-language-local-name="quítxua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Arie" title="Arie - romanès" lang="ro" hreflang="ro" data-title="Arie" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0%D0%B4%D1%8C" title="Площадь - rus" lang="ru" hreflang="ru" data-title="Площадь" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2%E0%A4%AE%E0%A5%8D" title="क्षेत्रफलम् - sànscrit" lang="sa" hreflang="sa" data-title="क्षेत्रफलम्" data-language-autonym="संस्कृतम्" data-language-local-name="sànscrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%98%D1%8D%D0%BD" title="Иэн - iacut" lang="sah" hreflang="sah" data-title="Иэн" data-language-autonym="Саха тыла" data-language-local-name="iacut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/%C3%80ria_(supirfici)" title="Ària (supirfici) - sicilià" lang="scn" hreflang="scn" data-title="Ària (supirfici)" data-language-autonym="Sicilianu" data-language-local-name="sicilià" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Area" title="Area - escocès" lang="sco" hreflang="sco" data-title="Area" data-language-autonym="Scots" data-language-local-name="escocès" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%A7%D9%8A%D8%B1%D8%A7%D8%B6%D9%8A" title="ايراضي - sindi" lang="sd" hreflang="sd" data-title="ايراضي" data-language-autonym="سنڌي" data-language-local-name="sindi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina - serbocroat" lang="sh" hreflang="sh" data-title="Površina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Area" title="Area - Simple English" lang="en-simple" hreflang="en-simple" data-title="Area" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Plocha_(%C3%BAtvar)" title="Plocha (útvar) - eslovac" lang="sk" hreflang="sk" data-title="Plocha (útvar)" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina - eslovè" lang="sl" hreflang="sl" data-title="Površina" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nharaunda" title="Nharaunda - shona" lang="sn" hreflang="sn" data-title="Nharaunda" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Bed" title="Bed - somali" lang="so" hreflang="so" data-title="Bed" data-language-autonym="Soomaaliga" data-language-local-name="somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sip%C3%ABrfaqja" title="Sipërfaqja - albanès" lang="sq" hreflang="sq" data-title="Sipërfaqja" data-language-autonym="Shqip" data-language-local-name="albanès" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D0%B0" title="Површина - serbi" lang="sr" hreflang="sr" data-title="Површина" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Ar%C3%A9a" title="Aréa - sondanès" lang="su" hreflang="su" data-title="Aréa" data-language-autonym="Sunda" data-language-local-name="sondanès" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Area" title="Area - suec" lang="sv" hreflang="sv" data-title="Area" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Eneo" title="Eneo - suahili" lang="sw" hreflang="sw" data-title="Eneo" data-language-autonym="Kiswahili" data-language-local-name="suahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Plac_rozlygowa%C5%84o" title="Plac rozlygowańo - silesià" lang="szl" hreflang="szl" data-title="Plac rozlygowańo" data-language-autonym="Ślůnski" data-language-local-name="silesià" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B0%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%B3%E0%AE%B5%E0%AF%81" title="பரப்பளவு - tàmil" lang="ta" hreflang="ta" data-title="பரப்பளவு" data-language-autonym="தமிழ்" data-language-local-name="tàmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B0%BF%E0%B0%B8%E0%B1%8D%E0%B0%A4%E0%B1%80%E0%B0%B0%E0%B1%8D%E0%B0%A3%E0%B0%82" title="విస్తీర్ణం - telugu" lang="te" hreflang="te" data-title="విస్తీర్ణం" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%81%D0%BE%D2%B3%D0%B0%D1%82" title="Масоҳат - tadjik" lang="tg" hreflang="tg" data-title="Масоҳат" data-language-autonym="Тоҷикӣ" data-language-local-name="tadjik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B7%E0%B9%89%E0%B8%99%E0%B8%97%E0%B8%B5%E0%B9%88" title="พื้นที่ - tai" lang="th" hreflang="th" data-title="พื้นที่" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-ti mw-list-item"><a href="https://ti.wikipedia.org/wiki/%E1%8C%BD%E1%8D%8D%E1%88%93%E1%89%B5_%E1%88%98%E1%88%AC%E1%89%B5" title="ጽፍሓት መሬት - tigrinya" lang="ti" hreflang="ti" data-title="ጽፍሓት መሬት" data-language-autonym="ትግርኛ" data-language-local-name="tigrinya" class="interlanguage-link-target"><span>ትግርኛ</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sukat" title="Sukat - tagal" lang="tl" hreflang="tl" data-title="Sukat" data-language-autonym="Tagalog" data-language-local-name="tagal" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alan" title="Alan - turc" lang="tr" hreflang="tr" data-title="Alan" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/M%C3%A4ydan" title="Mäydan - tàtar" lang="tt" hreflang="tt" data-title="Mäydan" data-language-autonym="Татарча / tatarça" data-language-local-name="tàtar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0" title="Площа - ucraïnès" lang="uk" hreflang="uk" data-title="Площа" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%D9%82%D8%A8%DB%81" title="رقبہ - urdú" lang="ur" hreflang="ur" data-title="رقبہ" data-language-autonym="اردو" data-language-local-name="urdú" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yuza" title="Yuza - uzbek" lang="uz" hreflang="uz" data-title="Yuza" 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-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Di%E1%BB%87n_t%C3%ADch" title="Diện tích - vietnamita" lang="vi" hreflang="vi" data-title="Diện tích" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Ippervlak" title="Ippervlak - flamenc occidental" lang="vls" hreflang="vls" data-title="Ippervlak" data-language-autonym="West-Vlams" data-language-local-name="flamenc occidental" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Sitind%C3%AAye" title="Sitindêye - való" lang="wa" hreflang="wa" data-title="Sitindêye" data-language-autonym="Walon" data-language-local-name="való" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Kahaluag" title="Kahaluag - waray" lang="war" hreflang="war" data-title="Kahaluag" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Yaatuwaay" title="Yaatuwaay - wòlof" lang="wo" hreflang="wo" data-title="Yaatuwaay" data-language-autonym="Wolof" data-language-local-name="wòlof" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%9D%A2%E7%A7%AF" title="面积 - xinès wu" lang="wuu" hreflang="wuu" data-title="面积" data-language-autonym="吴语" data-language-local-name="xinès wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A4%E1%83%90%E1%83%A0%E1%83%97%E1%83%9D%E1%83%91%E1%83%98" title="ფართობი - mingrelià" lang="xmf" hreflang="xmf" data-title="ფართობი" data-language-autonym="მარგალური" data-language-local-name="mingrelià" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A9%D7%98%D7%97" title="שטח - ídix" lang="yi" hreflang="yi" data-title="שטח" data-language-autonym="ייִדיש" data-language-local-name="ídix" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%80%C3%A0l%C3%A0" title="Ààlà - ioruba" lang="yo" hreflang="yo" data-title="Ààlà" data-language-autonym="Yorùbá" data-language-local-name="ioruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zea mw-list-item"><a href="https://zea.wikipedia.org/wiki/Oppervlak" title="Oppervlak - zelandès" lang="zea" hreflang="zea" data-title="Oppervlak" data-language-autonym="Zeêuws" data-language-local-name="zelandès" class="interlanguage-link-target"><span>Zeêuws</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9D%A2%E7%A7%AF" title="面积 - xinès" lang="zh" hreflang="zh" data-title="面积" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Bi%C4%81n-chek" title="Biān-chek - xinès min del sud" lang="nan" hreflang="nan" data-title="Biān-chek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="xinès min del sud" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9D%A2%E7%A9%8D" title="面積 - cantonès" lang="yue" hreflang="yue" data-title="面積" data-language-autonym="粵語" data-language-local-name="cantonès" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div 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id="mw-indicator-ind-100" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Viquip%C3%A8dia:Llista_dels_1000_articles_fonamentals" title="Els 1.000 fonamentals de la Viquipèdia"><img alt="Els 1.000 fonamentals de la Viquipèdia" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Segell_1000_unificat_Viquip%C3%A8dia.svg/30px-Segell_1000_unificat_Viquip%C3%A8dia.svg.png" decoding="async" width="30" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Segell_1000_unificat_Viquip%C3%A8dia.svg/45px-Segell_1000_unificat_Viquip%C3%A8dia.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Segell_1000_unificat_Viquip%C3%A8dia.svg/60px-Segell_1000_unificat_Viquip%C3%A8dia.svg.png 2x" data-file-width="2408" data-file-height="2896" /></a></span></div></div> </div> <div id="siteSub" class="noprint">De la Viquipèdia, l'enciclopèdia lliure</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r34261971">.mw-parser-output .hatnote{width:100%;border-color:#77ccff;color:var(--color-base,#202122);background-color:#f5f5f5;margin-bottom:1em;font-style:italic}.mw-parser-output .hatnote i{font-style:normal}@media screen{html.skin-theme-clientpref-night .mw-parser-output .hatnote{color:var(--color-inverted,#fff);background-color:var(--background-color-inverted,#101418)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .hatnote{color:var(--color-inverted,#fff);background-color:var(--background-color-inverted,#101418)}}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <table class="hatnote" cellspacing="5"> <tbody><tr> <td style="width: 25px; vertical-align: top;"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/22px-Disambig_grey.svg.png" decoding="async" width="22" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/33px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/44px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> </td> <td>«<b>superfície</b>» redirigeix aquí. Vegeu-ne altres significats a «<a href="/wiki/Superf%C3%ADcie_(desambiguaci%C3%B3)" class="mw-disambig" title="Superfície (desambiguació)">Superfície (desambiguació)</a>». </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34261971"> <table class="hatnote" cellspacing="5"> <tbody><tr> <td style="width: 25px; vertical-align: top;"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/22px-Disambig_grey.svg.png" decoding="async" width="22" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/33px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/44px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> </td> <td>Per a altres significats, vegeu «<a href="/wiki/%C3%80rea_(unitat_de_superf%C3%ADcie)" title="Àrea (unitat de superfície)">Àrea (unitat de superfície)</a>». </td></tr></tbody></table> <table class="infobox" style="font-size:90%;width:25em"><caption style="font-weight:bold;background-color: #b0d1ad"><span style="float:left;"><span typeof="mw:File"><a href="/wiki/Fitxer:Noun_project_1842.svg" class="mw-file-description" title="Infotaula de magnitud física"><img alt="Infotaula de magnitud física" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Noun_project_1842.svg/25px-Noun_project_1842.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Noun_project_1842.svg/38px-Noun_project_1842.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Noun_project_1842.svg/50px-Noun_project_1842.svg.png 2x" data-file-width="100" data-file-height="100" /></a></span></span>Àrea</caption><tbody><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Símbol</th><td class="infobox-data">S <span class="penicon" data-bridge-edit-flow="single-best-value"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q11500?uselang=ca#P416" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Unitats</th><td class="infobox-data"><a href="/wiki/Metre_quadrat" title="Metre quadrat">metre quadrat</a> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q11500?uselang=ca#P8111" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Fórmula</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 A=\iint \mathrm {d} x\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\iint \mathrm {d} x\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e354fd697c9c03b056c2cabad034b6c53d86c6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.719ex; height:5.676ex;" alt="{\displaystyle A=\iint \mathrm {d} x\mathrm {d} y}"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q11500?uselang=ca#P2534" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Area.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area.svg/220px-Area.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area.svg/330px-Area.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area.svg/440px-Area.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>L'àrea combinada d'aquestes tres formes és d'entre quinze i setze <a href="/wiki/Quadrat_(pol%C3%ADgon)" title="Quadrat (polígon)">quadrats</a>.</figcaption></figure> <p>L'<b>àrea</b> és una <a href="/wiki/Quantitat" title="Quantitat">quantitat</a> que expressa l'extensió d'una <a href="/wiki/Superf%C3%ADcie" class="mw-redirect" title="Superfície">superfície</a> o <a href="/wiki/Forma_geom%C3%A8trica" title="Forma geomètrica">forma</a> de dues dimensions al <a href="/wiki/Pla_(geometria)" class="mw-redirect" title="Pla (geometria)">pla</a>. L'àrea es pot entendre com la quantitat de material que seria necessària per crear un model de la forma, o la quantitat de pintura necessària per cobrir la superfície amb una sola capa. És l'analogia en dues dimensions de la <a href="/wiki/Longitud" title="Longitud">longitud</a> d'una <a href="/wiki/Corba" title="Corba">corba</a> (concepte unidimensional) i del <a href="/wiki/Volum" title="Volum">volum</a> d'un <a href="/wiki/S%C3%B2lid" title="Sòlid">sòlid</a> (concepte tridimensional). </p><p>L'àrea d'una figura pot ser mesurada comparant la forma amb <a href="/wiki/Quadrat_(pol%C3%ADgon)" title="Quadrat (polígon)">quadrats</a> d'una mida fixa. En el <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">Sistema Internacional d'Unitats</a> (SI), la unitat estàndard d'àrea és el <a href="/wiki/Metre_quadrat" title="Metre quadrat">metre quadrat</a> (m²), que és l'àrea d'un quadrat els <a href="/wiki/Costats" class="mw-redirect" title="Costats">costats</a> del qual mesuren un metre de llargada.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Una forma amb una àrea de tres metres quadrats tindria la mateixa àrea que tres d'aquests quadrats. En <a href="/wiki/Matem%C3%A0tiques" title="Matemàtiques">matemàtiques</a>, el <a href="/wiki/Quadrat_unitari" class="mw-redirect" title="Quadrat unitari">quadrat unitari</a> es defineix com el que té una àrea igual a u. Pel que fa a la <a href="/wiki/Notaci%C3%B3_matem%C3%A0tica" title="Notació matemàtica">notació</a>, si l'àrea correspon a una superfície plana se sol denotar com <i>A</i>, i si correspon a una superfície tridimensional se sol denominar <i>S</i>.<sup id="cite_ref-crc1763_2-0" class="reference"><a href="#cite_note-crc1763-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Hi ha moltes <a href="/wiki/F%C3%B3rmula" title="Fórmula">fórmules</a> conegudes per determinar les àrees de formes simples com <a href="/wiki/Triangle" title="Triangle">triangles</a>, <a href="/wiki/Rectangle" title="Rectangle">rectangles</a> i <a href="/wiki/Cercle" title="Cercle">cercles</a>. Fent ús d'aquestes fórmules es pot determinar l'àrea de qualsevol polígon <a href="/wiki/Triangulaci%C3%B3_d%27un_pol%C3%ADgon" title="Triangulació d'un polígon">dividint el polígon en triangles</a>.<sup id="cite_ref-bkos_3-0" class="reference"><a href="#cite_note-bkos-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Per formes amb costats corbats se sol necessitar el <a href="/wiki/C%C3%A0lcul_infinitesimal" title="Càlcul infinitesimal">càlcul</a> per trobar l'àrea; de fet, el problema de determinar l'àrea de figures planes fou una gran motivació pel <a href="/wiki/Hist%C3%B2ria_del_c%C3%A0lcul" title="Història del càlcul">desenvolupament històric del càlcul</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Per una forma sòlida com una <a href="/wiki/Esfera" title="Esfera">esfera</a>, un <a href="/wiki/Con" title="Con">con</a> o un <a href="/wiki/Cilindre" title="Cilindre">cilindre</a> l'àrea de la seva superfície externa s'anomena <a href="/wiki/%C3%80rea_superficial" class="mw-redirect" title="Àrea superficial">àrea superficial</a>. Les fórmules per les àrees superficials foren trobades ja pels <a href="/wiki/Matem%C3%A0tiques_de_l%27antiga_Gr%C3%A8cia" title="Matemàtiques de l'antiga Grècia">grecs antics</a>, però esbrinar l'àrea de sòlids més complicats sol necessitar l'ús del <a href="/wiki/C%C3%A0lcul_multivariable" title="Càlcul multivariable">càlcul amb múltiples variables</a>. </p><p>L'àrea té un paper important en les matemàtiques modernes. En addició a la seva òbvia importància en <a href="/wiki/Geometria" title="Geometria">geometria</a> i càlcul, l'àrea està relacionada amb la definició dels <a href="/wiki/Determinant_(matem%C3%A0tiques)" title="Determinant (matemàtiques)">determinants</a> en <a href="/wiki/%C3%80lgebra_lineal" title="Àlgebra lineal">àlgebra lineal</a> i és una propietat bàsica de superfícies en <a href="/wiki/Geometria_diferencial" title="Geometria diferencial">geometria diferencial</a>.<sup id="cite_ref-doCarmo_5-0" class="reference"><a href="#cite_note-doCarmo-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> En <a href="/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica">anàlisi matemàtica</a>, l'àrea d'un subconjunt del pla es defineix amb la <a href="/wiki/Mesura_de_Lebesgue" title="Mesura de Lebesgue">mesura de Lebesgue</a>,<sup id="cite_ref-Rudin_6-0" class="reference"><a href="#cite_note-Rudin-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> tot i que no tot subconjunt és mesurable. En general, en matemàtiques avançades l'àrea es percep com un cas especial del <a href="/wiki/Volum" title="Volum">volum</a> en regions de dues dimensions. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Història"><span id="Hist.C3.B2ria"></span>Història</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=1" title="Modifica la secció: Història"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La idea que l'àrea és la mesura que proporciona la mida de la regió dins d'una <a href="/wiki/Figura_geom%C3%A8trica" title="Figura geomètrica">figura geomètrica</a> prové de l'antiguitat. A l'<a href="/wiki/Antic_Egipte" title="Antic Egipte">antic Egipte</a>, després de la crescuda anual del riu <a href="/wiki/Nil" title="Nil">Nil</a> inundant els camps, sorgeix la necessitat de calcular l'àrea de cada parcel·la agrícola per restablir els seus límits; per a solucionar això, els egipcis van inventar la <a href="/wiki/Geometria" title="Geometria">geometria</a>, segons <a href="/wiki/Her%C3%B2dot" title="Heròdot">Heròdot</a>.<sup id="cite_ref-Heròdot_7-0" class="reference"><a href="#cite_note-Heròdot-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>El mètode de calcular l'àrea d'un <a href="/wiki/Pol%C3%ADgon" title="Polígon">polígon</a> com a suma d'àrees triangulars és un mètode que va ser proposat per primera vegada pel savi grec <a href="/wiki/Antif%C3%B2" class="mw-redirect" title="Antifò">Antifò</a> cap a l'any 430 aC. Trobar l'àrea d'una figura corba comporta més dificultat; el <a href="/wiki/M%C3%A8tode_d%27exhausti%C3%B3" title="Mètode d'exhaustió">mètode d'exhaustió</a> consisteix a inscriure i circumscriure polígons a la figura geomètrica, augmentar el nombre de costats d'aquests polígons i trobar l'àrea buscada. Amb aquest mètode, <a href="/wiki/%C3%88udox_de_Cnidos" class="mw-redirect" title="Èudox de Cnidos">Èudox de Cnidos</a> va aconseguir trobar la fórmula per a calcular l'àrea d'un cercle. Aquest sistema va ser utilitzat més tard per <a href="/wiki/Arquimedes" title="Arquimedes">Arquimedes</a> per a resoldre altres problemes similars.<sup id="cite_ref-problemaarea_8-0" class="reference"><a href="#cite_note-problemaarea-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definició_formal"><span id="Definici.C3.B3_formal"></span>Definició formal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=2" title="Modifica la secció: Definició formal"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una aproximació a la definició del que significa <i>àrea</i> és l'ús d'<a href="/wiki/Axioma" title="Axioma">axiomes</a>. Per exemple, es pot definir l'àrea com una funció <i>a</i> d'una col·lecció <i>M</i> d'un tipus especial de figures planes respecte al conjunt de <a href="/wiki/Nombre_real" title="Nombre real">nombres reals</a> tal que satisfà les següents propietats: </p> <ul><li>Per tot <i>S</i> en <i>M</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 a(S)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(S)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acdc00f97dd6e98df722f8f6c1ebf550b3b323ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.799ex; height:2.843ex;" alt="{\displaystyle a(S)\geq 0}"></span>.</li> <li>Si <i>S</i> i <i>T</i> estan a <i>M</i> llavors també hi estan <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 S\cup T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∪</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cup T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777b06b6d7d85a73d1a2f6039f42440386cba91c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.718ex; height:2.176ex;" alt="{\displaystyle S\cup T}"></span> 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 S\cap T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∩</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cap T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b41651b2cef48e2abb7f669e455ca91c23559c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.718ex; height:2.176ex;" alt="{\displaystyle S\cap T}"></span> i, a més, <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(S\cup T)=a(S)+a(T)-a(S\cap T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>∪</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>∩</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(S\cup T)=a(S)+a(T)-a(S\cap T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5682219ab0493fd07b52ceae5e3325bdbb710172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.507ex; height:2.843ex;" alt="{\displaystyle a(S\cup T)=a(S)+a(T)-a(S\cap T)}"></span>.</li> <li>Si <i>S</i> i <i>T</i> estan a <i>M</i> amb <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 S\subset T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bf7ab27d85f6a7220e58f2fee59077aaefc36f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.234ex; height:2.176ex;" alt="{\displaystyle S\subset T}"></span> llavors <i>T</i> − <i>S</i> és a <i>M</i> 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 a(T-S)=a(T)-a(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(T-S)=a(T)-a(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e101080b2b22313b48bf427a07863b6d1bd745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.168ex; height:2.843ex;" alt="{\displaystyle a(T-S)=a(T)-a(S)}"></span>.</li> <li>Si un conjunt <i>S</i> és a <i>M</i> i <i>S</i> és congruent amb <i>T</i> llavors <i>T</i> està també a <i>M</i> 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 a(S)=a(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(S)=a(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b41a7536692d871cdc692de11354b934a3ab17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.312ex; height:2.843ex;" alt="{\displaystyle a(S)=a(T)}"></span>.</li> <li>Tot rectangle <i>R</i> és a <i>M</i>. Si el rectangle té longitud <i>h</i> i amplada <i>k</i> llavors <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(R)=hk}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(R)=hk}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36be1d1af2e5a7bdfb7b260a7e9d0a7b25c76654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.452ex; height:2.843ex;" alt="{\displaystyle a(R)=hk}"></span>.</li> <li>Sigui <i>Q</i> un conjunt tancat entre dues regions etapa <i>S</i> i <i>T</i>. Una regió etapa està formada per una unió finita de rectangles adjacents que reposen sobre la mateixa base, és a dir <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 S\subset Q\subset T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>Q</mi> <mo>⊂</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset Q\subset T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78c1cb3108bc988134b685e59198886ba53f9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.171ex; height:2.509ex;" alt="{\displaystyle S\subset Q\subset T}"></span>. Si hi ha un únic nombre <i>c</i> tal que <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(S)\leq c\leq a(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>c</mi> <mo>≤</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(S)\leq c\leq a(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b256adefb39e71ee35f0a0ed044f6ba8038ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.417ex; height:2.843ex;" alt="{\displaystyle a(S)\leq c\leq a(T)}"></span> per totes les regions etapa <i>S</i> i <i>T</i>, llavors <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(Q)=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(Q)=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1feb0d9cc00cf98e9e95450cc632aeeb04ff7d1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.983ex; height:2.843ex;" alt="{\displaystyle a(Q)=c}"></span>.</li></ul> <p>Es pot provar que una funció d'àrea com aquesta existeix.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Unitats">Unitats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=3" title="Modifica la secció: Unitats"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r30997230">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Articles principals: <a href="/wiki/Unitat_de_superf%C3%ADcie" title="Unitat de superfície">Unitat de superfície</a> i <a href="/wiki/Unitats_de_superf%C3%ADcie" class="mw-redirect" title="Unitats de superfície">unitats de superfície</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:SquareMeterQuadrat.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SquareMeterQuadrat.JPG/220px-SquareMeterQuadrat.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SquareMeterQuadrat.JPG/330px-SquareMeterQuadrat.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SquareMeterQuadrat.JPG/440px-SquareMeterQuadrat.JPG 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption>Un <a href="/wiki/Metre_quadrat" title="Metre quadrat">metre quadrat</a> fet de tubs de PVC.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Area_conversion_-_square_mm_in_a_square_cm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Area_conversion_-_square_mm_in_a_square_cm.png/220px-Area_conversion_-_square_mm_in_a_square_cm.png" decoding="async" width="220" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Area_conversion_-_square_mm_in_a_square_cm.png/330px-Area_conversion_-_square_mm_in_a_square_cm.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Area_conversion_-_square_mm_in_a_square_cm.png/440px-Area_conversion_-_square_mm_in_a_square_cm.png 2x" data-file-width="831" data-file-height="699" /></a><figcaption>Tot i que hi ha 10 mm en 1 cm, hi ha 100 mm² en 1 cm².</figcaption></figure> <p>La <a href="/wiki/Unitat_de_mesura" title="Unitat de mesura">unitat de mesura</a> del <a href="/wiki/Sistema_Internacional" class="mw-redirect" title="Sistema Internacional">Sistema Internacional</a> per a mesurar l'àrea és el <a href="/wiki/Metre_quadrat" title="Metre quadrat">metre quadrat</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> una <a href="/wiki/Unitats_derivades_del_SI" title="Unitats derivades del SI">unitat derivada</a> definida a partir del <a href="/wiki/Metre" title="Metre">metre</a>. </p><p>Cada unitat de longitud té la seva corresponent unitat derivada de superfície; per als principals <a href="/wiki/Prefixos_del_SI" class="mw-redirect" title="Prefixos del SI">prefixos del Sistema Internacional</a> s'obtenen les següents unitats. </p> <ul><li><a href="/wiki/Quil%C3%B2metre_quadrat" title="Quilòmetre quadrat">quilòmetre quadrat</a> (km²) = 1.000.000 m²</li> <li>decímetre quadrat (dm²) = 0,01 m²</li> <li>centímetre quadrat (cm²) = 0,0001 m²</li> <li>mil·límetre quadrat (mm²) = 0,000001 m²</li></ul> <p>Una altra unitat no reconeguda pel Sistema Internacional, però molt utilitzada és l'<a href="/wiki/Hect%C3%A0rea" title="Hectàrea">hectàrea</a>, que és equivalent a un hectòmetre quadrat (1 hm² = 10.000 m²) i a 100 <a href="/wiki/%C3%80rea_(unitat_de_superf%C3%ADcie)" title="Àrea (unitat de superfície)">àrees</a> (1 àrea = 100 m²). </p><p>Hi ha d'altres unitats no oficials que s'usen en certs àmbits com el <a href="/wiki/Barn" title="Barn">barn</a>, d'ordre molt petit i que s'usa en l'àmbit nuclear. En el món agrari s'havien utilitzat moltes unitats, sovint pròpies d'un indret determinat, com per exemple la <a href="/wiki/Fanecada" title="Fanecada">fanecada</a>, la <a href="/wiki/Tafulla" title="Tafulla">tafulla</a> o la <a href="/wiki/Vessana" title="Vessana">vessana</a>. En el <a href="/wiki/Sistema_anglosax%C3%B3" class="mw-redirect" title="Sistema anglosaxó">Sistema Imperial d'Unitats</a> (el sistema anglosaxó), la unitat base és la <a href="/wiki/Iarda_quadrada" title="Iarda quadrada">iarda quadrada</a>, que equival a 0,83612736 m²; d'ella se'n deriven la <a href="/w/index.php?title=Polzada_quadrada&action=edit&redlink=1" class="new" title="Polzada quadrada (encara no existeix)">polzada quadrada</a>, el <a href="/w/index.php?title=Peu_quadrat&action=edit&redlink=1" class="new" title="Peu quadrat (encara no existeix)">peu quadrat</a> i l'<a href="/wiki/Acre_(unitat_de_superf%C3%ADcie)" title="Acre (unitat de superfície)">acre</a>, entre d'altres. </p> <div class="mw-heading mw-heading2"><h2 id="Àrees_bàsiques"><span id=".C3.80rees_b.C3.A0siques"></span>Àrees bàsiques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=4" title="Modifica la secció: Àrees bàsiques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Àrea_d'un_triangle"><span id=".C3.80rea_d.27un_triangle"></span>Àrea d'un triangle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=5" title="Modifica la secció: Àrea d'un triangle"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'àrea d'un <a href="/wiki/Triangle" title="Triangle">triangle</a> és igual al semiproducte entre la longitud d'una base i l'altura relativa a aquesta:<sup id="cite_ref-spiegel_abellanas_11-0" class="reference"><a href="#cite_note-spiegel_abellanas-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A={\frac {b\cdot h}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>h</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {b\cdot h}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcca4fc2882400f01bbf33b2dfa758d8942ba26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.693ex; height:5.343ex;" alt="{\displaystyle A={\frac {b\cdot h}{2}}}"></span> </p> </blockquote> <dl><dd>on <i>b</i> és la base del triangle i <i>h</i> és l'altura corresponent a la base. (es pot considerar qualsevol costat com a base)</dd></dl> <p>Si el <a href="/wiki/Triangle_rectangle" title="Triangle rectangle">triangle és rectangle</a>, l'altura coincideix amb un dels <a href="/wiki/Catet" title="Catet">catets</a>, amb el que l'àrea és igual al semiproducte dels catets: </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={\frac {a\cdot b}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {a\cdot b}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c48fa15935cd42e88f9d692579f8b94b7d2a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.584ex; height:5.343ex;" alt="{\displaystyle A={\frac {a\cdot b}{2}}}"></span></dd> <dd>on <i>a</i> i <i>b</i> són els catets.</dd></dl> <p>Si es coneix la longitud dels costats, es pot aplicar la <a href="/wiki/F%C3%B3rmula_d%27Her%C3%B3" title="Fórmula d'Heró">fórmula d'Heró</a>:<sup id="cite_ref-crc69_12-0" class="reference"><a href="#cite_note-crc69-12"><span class="cite-bracket">[</span>12<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 A={\sqrt {s(s-a)(s-b)(s-c)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2040da62a4f48c9f502e3f38e44133524401c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.71ex; height:4.843ex;" alt="{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}}"></span></dd></dl> <dl><dd>on <i>a</i>, <i>b</i>, <i>c</i> són els valors de les longituds dels seus costats, <i>s</i> = ½ (<i>a</i> + <i>b</i> + <i>c</i>) és el semi<a href="/wiki/Per%C3%ADmetre" title="Perímetre">perímetre</a> del triangle.</dd></dl> <p>Si el triangle és <a href="/wiki/Triangle_equil%C3%A0ter" title="Triangle equilàter">equilàter</a>, l'àrea és igual a un quart del quadrat d'un costat per l'<a href="/wiki/Arrel_quadrada_de_3" title="Arrel quadrada de 3">arrel quadrada de 3</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={\frac {{\sqrt {3}}\cdot a^{2}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {{\sqrt {3}}\cdot a^{2}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458b1e2fa62308ee9807d41841aa78adee9cd5d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.739ex; height:5.843ex;" alt="{\displaystyle A={\frac {{\sqrt {3}}\cdot a^{2}}{4}}}"></span></dd></dl> <dl><dd>on <i>a</i> és un costat del triangle.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Àrea_d'un_quadrilàter"><span id=".C3.80rea_d.27un_quadril.C3.A0ter"></span>Àrea d'un quadrilàter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=6" title="Modifica la secció: Àrea d'un quadrilàter"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Rectangles">Rectangles</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=7" title="Modifica la secció: Rectangles"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:RectangleLengthWidth.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/RectangleLengthWidth.svg/220px-RectangleLengthWidth.svg.png" decoding="async" width="220" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/RectangleLengthWidth.svg/330px-RectangleLengthWidth.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/RectangleLengthWidth.svg/440px-RectangleLengthWidth.svg.png 2x" data-file-width="220" data-file-height="170" /></a><figcaption>L'àrea d'aquest rectangle és <i>lw</i>.</figcaption></figure> <p>La fórmula d'àrea més bàsica és la de l'àrea del <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>. Donat un rectangle de longituds <i>l</i> i <i>w</i>, la fórmula de l'àrea és: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A\;=\;lw}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>l</mi> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;lw}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb020572f99d2b5586772f0c07a702e4b6168c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.489ex; height:2.176ex;" alt="{\displaystyle A\;=\;lw}"></span> </p> </blockquote> <p>És a dir, l'àrea del rectangle és l'amplada multiplicada per l'alçada. Un cas especial és el quadrat, els costats del qual són iguals, de longitud <i>s</i>; la seva fórmula és, doncs: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A\;=\;s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;s^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5830f645f1ec37055fd6f664cf4c60b9796ea026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.277ex; height:2.676ex;" alt="{\displaystyle A\;=\;s^{2}}"></span> </p> </blockquote> <p>La fórmula de l'àrea del rectangle sorgeix directament de les propietats bàsiques de l'àrea i, de vegades, es pren com a definició o <a href="/wiki/Axioma" title="Axioma">axioma</a>. Per altra banda, si la <a href="/wiki/Geometria" title="Geometria">geometria</a> hagués estat desenvolupada abans que l'<a href="/wiki/Aritm%C3%A8tica" title="Aritmètica">aritmètica</a>, aquesta fórmula podria haver estat usada per definir la <a href="/wiki/Multiplicaci%C3%B3" title="Multiplicació">multiplicació</a> de <a href="/wiki/Nombre_real" title="Nombre real">nombres reals</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Mètode_de_la_dissecció"><span id="M.C3.A8tode_de_la_dissecci.C3.B3"></span>Mètode de la dissecció</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=8" title="Modifica la secció: Mètode de la dissecció"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:ParallelogramArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/220px-ParallelogramArea.svg.png" decoding="async" width="220" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/330px-ParallelogramArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/440px-ParallelogramArea.svg.png 2x" data-file-width="204" data-file-height="268" /></a><figcaption>Figures d'igual àrea.</figcaption></figure> <p>Moltes fórmules simples d'àrea s'aconsegueixen mitjançant el <a href="/w/index.php?title=M%C3%A8tode_de_la_dissecci%C3%B3&action=edit&redlink=1" class="new" title="Mètode de la dissecció (encara no existeix)">mètode de la dissecció</a>, que consisteix a partir la forma en peces la suma de les àrees de les quals sigui l'àrea de la forma original. </p><p>Per exemple, qualsevol <a href="/wiki/Paral%C2%B7lelogram" title="Paral·lelogram">paral·lelogram</a> pot ser dividit en un <a href="/wiki/Trapezoide" title="Trapezoide">trapezoide</a> i un <a href="/wiki/Triangle_rectangle" title="Triangle rectangle">triangle rectangle</a>, tal com es mostra a la figura de l'esquerra. Si el triangle es mou a l'altra banda del trapezoide, llavors la figura resultant és un rectangle; d'aquí es conclou que l'àrea del paral·lelogram és la mateixa que la d'aquest rectangle: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A\;=\;bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>b</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13faf3a7f244f0c47199d281bb41d1c841b3e52e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.468ex; height:2.176ex;" alt="{\displaystyle A\;=\;bh}"></span> </p> </blockquote> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:TriangleArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/TriangleArea.svg/220px-TriangleArea.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/TriangleArea.svg/330px-TriangleArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/TriangleArea.svg/440px-TriangleArea.svg.png 2x" data-file-width="220" data-file-height="160" /></a><figcaption>Dos triangles congruents.</figcaption></figure> <p>De totes maneres, el mateix paral·lelogram també es pot dividir tallant per la <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> obtenint dos triangles <a href="/wiki/Angles_congruents" class="mw-redirect" title="Angles congruents">congruents</a>, tal com es mostra a la figura. L'àrea de cada triangle és la meitat de l'àrea del paral·lelogram: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A\;=\;{\frac {1}{2}}bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>b</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;{\frac {1}{2}}bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2e03862a5d6bd47dd9bfb032a564b0ea8122cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.467ex; height:5.176ex;" alt="{\displaystyle A\;=\;{\frac {1}{2}}bh}"></span> </p> </blockquote> <p>Els mateixos arguments es poden utilitzar per trobar les fórmules del trapezoide, del rombe i de polígons més complexos. </p> <div class="mw-heading mw-heading4"><h4 id="Altres_quadrilàters"><span id="Altres_quadril.C3.A0ters"></span>Altres quadrilàters</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=9" title="Modifica la secció: Altres quadrilàters"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Tetragon_measures.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Tetragon_measures.svg/220px-Tetragon_measures.svg.png" decoding="async" width="220" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Tetragon_measures.svg/330px-Tetragon_measures.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Tetragon_measures.svg/440px-Tetragon_measures.svg.png 2x" data-file-width="324" data-file-height="360" /></a><figcaption>Trapezoide.</figcaption></figure> <p>L'àrea del <a href="/wiki/Trapezoide" title="Trapezoide">trapezoide</a> o de qualsevol <a href="/wiki/Quadril%C3%A0ter" title="Quadrilàter">quadrilàter</a> és igual al semiproducte de les seves diagonals pel <a href="/wiki/Sinus" title="Sinus">sinus</a> de l'angle que formen: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A={\frac {{\overline {AC}}\cdot {\overline {BD}}\cdot \sin \theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {{\overline {AC}}\cdot {\overline {BD}}\cdot \sin \theta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c59036ac101d1701f211db5a927df080dad20b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.865ex; height:6.176ex;" alt="{\displaystyle A={\frac {{\overline {AC}}\cdot {\overline {BD}}\cdot \sin \theta }{2}}}"></span> </p> </blockquote> <p>L'àrea també es pot obtenir mitjançant <a href="/wiki/Triangulaci%C3%B3_d%27un_pol%C3%ADgon" title="Triangulació d'un polígon">triangulació</a>: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A={\frac {a\cdot d\cdot \sin \alpha +b\cdot c\cdot \sin \gamma }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>d</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {a\cdot d\cdot \sin \alpha +b\cdot c\cdot \sin \gamma }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d55d005fa088a2c136d7a328cad7a7459febd8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.92ex; height:5.343ex;" alt="{\displaystyle A={\frac {a\cdot d\cdot \sin \alpha +b\cdot c\cdot \sin \gamma }{2}}}"></span> </p> </blockquote> <dl><dd>Essent: <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 \alpha \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652e1fd9c3a2ca00e1a517783cdbb0e18e4181f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle \alpha \,}"></span> l'angle comprès entre els costats <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> <mspace width="thinmathspace" /> </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/1d73aa5354c24942dab5316be466465a9d171510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,}"></span> 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 d\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f203b686a923982358b9274fb508753ac31b996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle d\,}"></span>.</dd> <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 \gamma \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da7961fee8269d576e5d06e838bf8695fc5179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.649ex; height:2.176ex;" alt="{\displaystyle \gamma \,}"></span> l'angle comprès entre els costats <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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1bcf19f4ec75b1d2cc0be001e58a314fb0a940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.385ex; height:2.176ex;" alt="{\displaystyle b\,}"></span> 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 c\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8573e7d95140b0d4068258d8162e189563baee6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.394ex; height:1.676ex;" alt="{\displaystyle c\,}"></span>.</dd></dl></dd></dl> <p><br /> El <a href="/wiki/Rombe" title="Rombe">rombe</a> és un paral·lelogram en el qual els 4 costats són iguals però els angles són iguals dos a dos. La seva àrea ve donada pel semiproducte de les seves dues diagonals: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A={\frac {{\overline {AC}}\cdot {\overline {BD}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {{\overline {AC}}\cdot {\overline {BD}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90fd4ebd7e95fb08d8171086319938fcf52620f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.852ex; height:6.176ex;" alt="{\displaystyle A={\frac {{\overline {AC}}\cdot {\overline {BD}}}{2}}}"></span> </p> </blockquote> <p>El <a href="/wiki/Romboide" title="Romboide">romboide</a> té la seva àrea donada pel producte d'un dels seus costats i la seva <a href="/wiki/Altura_dimensional" class="mw-redirect" title="Altura dimensional">altura</a> respectiva: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A=b\cdot h\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>h</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=b\cdot h\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55194182f084f83d761e6b25f6b479b6470f9540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.244ex; height:2.176ex;" alt="{\displaystyle A=b\cdot h\,}"></span> </p> </blockquote> <p>El <a href="/wiki/Trapezi_(geometria)" class="mw-redirect" title="Trapezi (geometria)">trapezi</a>, el qual té dos costats oposats paral·lels entre si i dos costats no paral·lels, té una àrea que ve donada per la mitjana aritmètica dels seus costats paral·lels multiplicada per la distància entre les seves (altura): </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A={\frac {a+b}{2}}\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <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> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {a+b}{2}}\cdot h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88acc4bc72f097ef27daf84e7773366abe9b27a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.764ex; height:5.343ex;" alt="{\displaystyle A={\frac {a+b}{2}}\cdot h}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Àrea_del_cercle"><span id=".C3.80rea_del_cercle"></span>Àrea del cercle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=10" title="Modifica la secció: Àrea del cercle"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:CircleArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/220px-CircleArea.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/330px-CircleArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/440px-CircleArea.svg.png 2x" data-file-width="240" data-file-height="260" /></a><figcaption>Un cercle pot ser dividit en <a href="/wiki/Sector_circular" title="Sector circular">sectors circulars</a> que, un cop reorganitzats, formen aproximadament un paral·lelogram.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/%C3%80rea_del_cercle" title="Àrea del cercle">Àrea del cercle</a></div> <p>Donat un <a href="/wiki/Cercle" title="Cercle">cercle</a> de radi <i>r</i> és possible partir el cercle en <a href="/wiki/Sector_circular" title="Sector circular">sectors circulars</a>. Cada sector és aproximadament triangular, de manera que tots poder ser col·locats de manera que formin aproximadament un <a href="/wiki/Paral%C2%B7lelogram" title="Paral·lelogram">paral·lelogram</a>. L'alçada d'aquest és <i>r</i> i l'amplada és la meitat de la <a href="/wiki/Circumfer%C3%A8ncia" title="Circumferència">circumferència</a>, és a dir, <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 \pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7447766f482372c761858f993f1432bd3671b0dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:1.676ex;" alt="{\displaystyle \pi r}"></span>. L'àrea total és, doncs, <i>r</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 \pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7447766f482372c761858f993f1432bd3671b0dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:1.676ex;" alt="{\displaystyle \pi r}"></span>, o <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 \pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd37db3982ad4e1157dcf8ddbfb280e7bae3b192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.676ex;" alt="{\displaystyle \pi r^{2}}"></span>:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<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 A\;=\;\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac8536281bd02eb042ba2de6888412c00d02f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.567ex; height:2.676ex;" alt="{\displaystyle A\;=\;\pi r^{2}}"></span></dd></dl> <p>Tot i que el mètode de dissecció usat amb aquesta fórmula és tan sols aproximat, l'error esdevé més petit quan el cercle es divideix en més sectors. En el <a href="/wiki/L%C3%ADmit_(matem%C3%A0tiques)" class="mw-redirect" title="Límit (matemàtiques)">límit</a>, l'àrea del paral·lelogram aproximat és exactament <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 \pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd37db3982ad4e1157dcf8ddbfb280e7bae3b192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.676ex;" alt="{\displaystyle \pi r^{2}}"></span>, l'àrea del cercle. </p><p>Aquest argument és, de fet, una simple aplicació de les idees del <a href="/wiki/C%C3%A0lcul_infinitesimal" title="Càlcul infinitesimal">càlcul</a>. En temps antics, el <a href="/w/index.php?title=M%C3%A8tode_de_l%27exhausti%C3%B3&action=edit&redlink=1" class="new" title="Mètode de l'exhaustió (encara no existeix)">mètode de l'exhaustió</a> es feia servir de manera similar per trobar l'àrea del cercle; aquest mètode és considerat el precursor del <a href="/wiki/C%C3%A0lcul_integral" class="mw-redirect" title="Càlcul integral">càlcul integral</a>. Usant mètodes moderns, l'àrea del cercle es pot trobar mitjançant una <a href="/wiki/Integral_definida" class="mw-redirect" title="Integral definida">integral definida</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\;=\;\int _{-r}^{r}2{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;\int _{-r}^{r}2{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1a68d9f85e8bbcd8c9d6d1753c553e02a0e040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.634ex; height:6.009ex;" alt="{\displaystyle A\;=\;\int _{-r}^{r}2{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}}"></span></dd></dl> <p>Una altra manera per trobar l'àrea del cercle és inscrivint-hi un triangle i anar augmentant progressivament els costats del polígon inscrit fins a l'<a href="/wiki/Infinit" title="Infinit">infinit</a>. Si es consideren <i>n</i> punts <i>A</i><sub>1</sub>, <i>A</i>₂, ... <i>A</i><sub><i>n</i></sub> col·locats de manera regular sobre un cercle de centre <i>O</i> i radi <i>R</i> s'obté un <a href="/wiki/Pol%C3%ADgon_regular" class="mw-redirect" title="Polígon regular">polígon regular</a> de <i>n</i> costats constituït per <i>n</i> triangles isòsceles de la mateixa àrea <i>OA</i><sub>1</sub><i>A</i>₂<i>, </i>OA<i>₂</i>A<i>₃</i>, etc. L'àrea del polígon regular inscrit és, llavors, <i>n</i> vegades la d'aquest triangle. Si l'altura de cadascun dels triangles és <i>h</i><sub><i>n</i></sub>, l'àrea de cada triangle és <span style="white-space:nowrap;"><span class="frac nowrap"><sup>1</sup>⁄<sub>2</sub></span><i>h</i><sub><i>n</i></sub> × <i>A</i><sub>1</sub><i>A</i>₂</span>. Multiplicant-ho per <i>n</i> vegades, l'àrea del polígon resultant és, doncs, la meitat de l'altura <i>h</i><sub><i>n</i></sub> multiplicada pel <a href="/wiki/Per%C3%ADmetre" title="Perímetre">perímetre</a> del polígon. Ara bé, com que el nombre <i>n</i> de punts tendeix a l'infinit, l'altura <i>h</i><sub><i>n</i></sub> tendeix a <i>R</i> i el perímetre del polígon tendeix al perímetre del cercle, és a dir, 2π<i>R</i>, la qual cosa dona el resultat esperat de l'àrea del cercle, que és <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=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f7b7f93f93e7ba7bebb97efbe88e181ce332e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}"></span>. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Pi_archi_approx_inter.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Pi_archi_approx_inter.svg/660px-Pi_archi_approx_inter.svg.png" decoding="async" width="660" height="329" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Pi_archi_approx_inter.svg/990px-Pi_archi_approx_inter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Pi_archi_approx_inter.svg/1320px-Pi_archi_approx_inter.svg.png 2x" data-file-width="442" data-file-height="220" /></a><figcaption>Aproximacions successives al cercle a partir de polígons regulars inscrits en ell, amb <i>n</i> (nombre de costats) variant de 3 a 10.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Àrea_superficial"><span id=".C3.80rea_superficial"></span>Àrea superficial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=11" title="Modifica la secció: Àrea superficial"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Archimedes_sphere_and_cylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/220px-Archimedes_sphere_and_cylinder.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/330px-Archimedes_sphere_and_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/440px-Archimedes_sphere_and_cylinder.svg.png 2x" data-file-width="426" data-file-height="461" /></a><figcaption><a href="/wiki/Arquimedes" title="Arquimedes">Arquimedes</a> demostrà que l'àrea superficial i el <a href="/wiki/Volum" title="Volum">volum</a> d'una <a href="/wiki/Esfera" title="Esfera">esfera</a> és exactament 2/3 de l'àrea i del volum del <a href="/wiki/Cilindre" title="Cilindre">cilindre</a> que l'envolta.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/%C3%80rea_superficial" class="mw-redirect" title="Àrea superficial">Àrea superficial</a></div> <p>La majoria de les fórmules bàsiques per a obtenir l'àrea de la superfície d'un objecte tridimensional es basen en obtenir figures planes a partir de les seves cares i sumar l'àrea del conjunt. Per exemple, la superfície d'un <a href="/wiki/Cilindre" title="Cilindre">cilindre</a> (o qualsevol <a href="/wiki/Prisma_(geometria)" title="Prisma (geometria)">prisma</a>) s'obtindria tallant-lo i "aplanant" les seves cares: en el cas del cilindre s'obtindria un rectangle i dos <a href="/wiki/Cercle" title="Cercle">cercles</a>, figures de les quals se'n pot calcular la superfície fàcilment. De la mateixa manera, en el cas d'un <a href="/wiki/Con" title="Con">con</a>, en tallar-lo s'obté un <a href="/wiki/Sector_circular" title="Sector circular">sector circular</a> i un cercle. </p><p>El càlcul de l'àrea de la superfície d'una <a href="/wiki/Esfera" title="Esfera">esfera</a> és més difícil perquè l'esfera no té cares que es puguin aplanar (la seva <a href="/wiki/Curvatura_gaussiana" title="Curvatura gaussiana">curvatura gaussiana</a> és <a href="/wiki/Zero" title="Zero">zero</a>). La fórmula per a l'àrea de la superfície d'una esfera la va publicar per primer cop <a href="/wiki/Arquimedes" title="Arquimedes">Arquimedes</a> a la seva obra <i><a href="/w/index.php?title=De_l%27esfera_i_el_cilindre&action=edit&redlink=1" class="new" title="De l'esfera i el cilindre (encara no existeix)">De l'esfera i el cilindre</a></i>. La fórmula és: </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=4\pi \cdot r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mo>⋅</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi \cdot r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0679a17cb4473f0e7acce008b692744eb764de7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.118ex; height:2.676ex;" alt="{\displaystyle A=4\pi \cdot r^{2}}"></span></dd></dl> <p>on <i>r</i> és el radi de l'esfera. Igual que en la fórmula per l'àrea del cercle, qualsevol derivació d'aquesta usa inherentment mètodes similars al <a href="/wiki/C%C3%A0lcul_infinitesimal" title="Càlcul infinitesimal">càlcul</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Llista_de_fórmules"><span id="Llista_de_f.C3.B3rmules"></span>Llista de fórmules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=12" title="Modifica la secció: Llista de fórmules"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A continuació s'indiquen les fórmules per calcular la superfície de les figures més corrents. </p> <table class="wikitable"> <tbody><tr> <th>Forma </th> <th>Fórmula </th> <th>Variables </th></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> regular (<a href="/wiki/Triangle_equil%C3%A0ter" title="Triangle equilàter">triangle equilàter</a>) </td> <td><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 {\tfrac {1}{4}}{\sqrt {3}}s^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}{\sqrt {3}}s^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a22079ce523e5d26d76a458d45abcfac8720b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:7.288ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}{\sqrt {3}}s^{2}\,\!}"></span> </td> <td><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> és la longitud d'un costat del triangle. </td></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> </td> <td><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 {\sqrt {s(s-a)(s-b)(s-c)}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec878c707ac333aef6c7bfbd52205746ccb1b8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:24.256ex; height:4.843ex;" alt="{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}\,\!}"></span> </td> <td><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> és la meitat del <a href="/wiki/Per%C3%ADmetre" title="Perímetre">perímetre</a>, <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> 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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> són les longituds de cada costat. </td></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> </td> <td><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 {\tfrac {1}{2}}ab\sin(C)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}ab\sin(C)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2fc0a9c894a67a4d09e337f1e23f2a5b70e56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:11.091ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}ab\sin(C)\,\!}"></span> </td> <td><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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> són dos costats qualssevol 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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> és l'angle format entre ells. </td></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> </td> <td><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 {\tfrac {1}{2}}bh\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>b</mi> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}bh\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ccb3113c20ba5cf8e4aabae07eb4bdd2f1c80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:4.382ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}bh\,\!}"></span> </td> <td><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> són la <a href="/wiki/Base_(geometria)" title="Base (geometria)">base</a> i l'<a href="/wiki/Altura_(geometria)" title="Altura (geometria)">altura</a> (mesurada perpendicularment a la base) respectivament. </td></tr> <tr> <td><a href="/wiki/Quadrat_(geometria)" class="mw-redirect" title="Quadrat (geometria)">Quadrat</a> </td> <td><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 s^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469233a9b12544cb5bbbf4a75641e83f2794e30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.532ex; height:2.676ex;" alt="{\displaystyle s^{2}\,\!}"></span> </td> <td><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> és la longitud d'un costat del quadrat. </td></tr> <tr> <td><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a> </td> <td><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 lw\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>w</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle lw\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec852f35b0b20128e8ce419331f142ae74e99dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.745ex; height:2.176ex;" alt="{\displaystyle lw\,\!}"></span> </td> <td><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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> 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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> són les longituds dels costats del rectangle. </td></tr> <tr> <td><a href="/wiki/Rombe" title="Rombe">Rombe</a> </td> <td><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 {\tfrac {1}{2}}ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4164108e2658d2333ea13feee6a0cb0be43db7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.885ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}ab}"></span> </td> <td><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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> són les longituds de les dues <a href="/wiki/Diagonal" title="Diagonal">diagonals</a> del rombe. </td></tr> <tr> <td><a href="/wiki/Paral%C2%B7lelogram" title="Paral·lelogram">Paral·lelogram</a> </td> <td><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 bh\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bh\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df8554a2f57fafe9f2083a45b83c2214e10ec1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.724ex; height:2.176ex;" alt="{\displaystyle bh\,\!}"></span> </td> <td><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> és la longitud de la base 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> és l'altura perpendicular a la base. </td></tr> <tr> <td><a href="/wiki/Trapezi" title="Trapezi">Trapezi</a> </td> <td><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 {\tfrac {1}{2}}(a+b)h\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}(a+b)h\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90f10073ea05c60cd09b55fdb9a3f76e18693eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:10.261ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(a+b)h\,\!}"></span> </td> <td><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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> són els costats paral·lels 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> la distància (altura) entre aquests costats. </td></tr> <tr> <td><a href="/wiki/Hex%C3%A0gon" title="Hexàgon">Hexàgon</a> regular </td> <td><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 {\tfrac {3}{2}}{\sqrt {3}}s^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{2}}{\sqrt {3}}s^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a90512b78b33df84a3d81c62812d28e6a9d64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:7.288ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{2}}{\sqrt {3}}s^{2}\,\!}"></span> </td> <td><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> és la longitud d'un costat de l'hexàgon. </td></tr> <tr> <td><a href="/wiki/Oct%C3%B2gon" class="mw-redirect" title="Octògon">Octògon</a> regular </td> <td><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 2\left(1+{\sqrt {2}}\right)s^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left(1+{\sqrt {2}}\right)s^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61724e42c88efcd6b36261f78fcc6681e9e909a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:13.699ex; height:3.343ex;" alt="{\displaystyle 2\left(1+{\sqrt {2}}\right)s^{2}\,\!}"></span> </td> <td><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> és la longitud d'un costat de l'octògon. </td></tr> <tr> <td rowspan="2"><a href="/wiki/Pol%C3%ADgon_regular" class="mw-redirect" title="Polígon regular">Polígon regular</a> </td> <td><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 {ns^{2}}{4\cdot \tan(\pi /n)}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {ns^{2}}{4\cdot \tan(\pi /n)}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf4ab98412b20e5172aff46f7f2a7e432ef9d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-right: -0.387ex; width:13.123ex; height:6.509ex;" alt="{\displaystyle {\frac {ns^{2}}{4\cdot \tan(\pi /n)}}\,\!}"></span> </td> <td><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> és la longitud del costat 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> és el nombre de costats.<sup id="cite_ref-crc69_12-1" class="reference"><a href="#cite_note-crc69-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><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 {\tfrac {1}{2}}ap\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}ap\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1deb1344bf47fb9acfcf8a2115db81ab0a9e3455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:4.444ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}ap\,\!}"></span> </td> <td><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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> és l'<a href="/wiki/Apotema" title="Apotema">apotema</a> (el radi d'un cercle inscrit en el polígon) 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> és el <a href="/wiki/Per%C3%ADmetre" title="Perímetre">perímetre</a> del polígon. </td></tr> <tr> <td><a href="/wiki/Cercle" title="Cercle">Cercle</a> </td> <td><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 \pi r^{2}\ {\text{o bé}}\ {\frac {\pi d^{2}}{4}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>o bé</mtext> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}\ {\text{o bé}}\ {\frac {\pi d^{2}}{4}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ee9eb071e956c4591a208c7aeafad172b74725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:13.876ex; height:5.676ex;" alt="{\displaystyle \pi r^{2}\ {\text{o bé}}\ {\frac {\pi d^{2}}{4}}\,\!}"></span> </td> <td><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> és el radi 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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> el <a href="/wiki/Di%C3%A0metre" title="Diàmetre">diàmetre</a> </td></tr> <tr> <td><a href="/wiki/Sector_circular" title="Sector circular">Sector circular</a> </td> <td><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 {\tfrac {1}{2}}r^{2}\theta \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}r^{2}\theta \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23c2b04baa52d6209de63a87b11afd59abebf2e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:5.239ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}r^{2}\theta \,\!}"></span> </td> <td><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> 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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> són el radi i l'angle (en <a href="/wiki/Radian" title="Radian">radians</a>), respectivament. </td></tr> <tr> <td><a href="/wiki/El%C2%B7lipse" title="El·lipse">El·lipse</a> </td> <td><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 \pi ab\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ab\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c6066b3bd3ced4101ed6fae94b62ea4480e4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.946ex; height:2.176ex;" alt="{\displaystyle \pi ab\,\!}"></span> </td> <td><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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> són el <a href="/wiki/Semieix_major" title="Semieix major">semieix major</a> i el <a href="/wiki/Semieix_menor" title="Semieix menor">semieix menor</a>, respectivament. </td></tr> <tr> <td>Superfície total d'un <a href="/wiki/Cilindre" title="Cilindre">cilindre</a> </td> <td><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 2\pi r(r+h)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r(r+h)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c9869517b8ff53c209d2ff1586b10b7916773f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:10.968ex; height:2.843ex;" alt="{\displaystyle 2\pi r(r+h)\,\!}"></span> </td> <td><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> 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> són el radi i l'altura, respectivament. </td></tr> <tr> <td>Superfície lateral d'un cilindre </td> <td><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 2\pi rh\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi rh\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd91694dd8cf138a6ebb51e81d0b68b9e1a3b795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.269ex; height:2.176ex;" alt="{\displaystyle 2\pi rh\,\!}"></span> </td> <td><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> 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> són el radi i l'altura, respectivament. </td></tr> <tr> <td>Superfície total d'un <a href="/wiki/Con" title="Con">con</a> </td> <td><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 \pi r(r+l)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r(r+l)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eda1165be2830dc4647cd6d3ce53b7aa6d345ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.159ex; height:2.843ex;" alt="{\displaystyle \pi r(r+l)\,\!}"></span> </td> <td><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> 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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> són el radi i la <a href="/wiki/Generatriu" title="Generatriu">generatriu</a>, respectivament. </td></tr> <tr> <td>Superfície lateral d'un con </td> <td><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 \pi rl\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> <mi>l</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi rl\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c18837800b9e6ae0c2174a5d61ff9332358a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.461ex; height:2.176ex;" alt="{\displaystyle \pi rl\,\!}"></span> </td> <td><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> 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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> són el radi i la <a href="/wiki/Generatriu" title="Generatriu">generatriu</a>, respectivament. </td></tr> <tr> <td>Superfície total d'una <a href="/wiki/Esfera" title="Esfera">esfera</a> </td> <td><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 4\pi r^{2}\ {\text{o bé}}\ \pi d^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>o bé</mtext> </mrow> <mtext> </mtext> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}\ {\text{o bé}}\ \pi d^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0a4251976df6dba9a317bfeed63949ec78d887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:14.202ex; height:3.509ex;" alt="{\displaystyle 4\pi r^{2}\ {\text{o bé}}\ \pi d^{2}\,\!}"></span> </td> <td><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> 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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> són el radi i el diàmetre, respectivament. </td></tr> <tr> <td>Superfície total d'un <a href="/wiki/El%C2%B7lipsoide" title="El·lipsoide">el·lipsoide</a> </td> <td> </td> <td>Vegeu l'article. </td></tr> <tr> <td>Superfície total d'una <a href="/wiki/Pir%C3%A0mide_(geometria)" class="mw-redirect" title="Piràmide (geometria)">piràmide</a> </td> <td><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+{\frac {PL}{2}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>L</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B+{\frac {PL}{2}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50c5fca427dc232041d2cd4587b58aada992a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:9.156ex; height:5.176ex;" alt="{\displaystyle B+{\frac {PL}{2}}\,\!}"></span> </td> <td><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> és l'àrea de la base, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> és el <a href="/wiki/Per%C3%ADmetre" title="Perímetre">perímetre</a> de la base 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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> és la generatriu. </td></tr> <tr> <td>Conversió d'un quadrat a àrea circular </td> <td><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 {4}{\pi }}A\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>π</mi> </mfrac> </mrow> <mi>A</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{\pi }}A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f1d5a0046409bcf949fb15a3056dc5322acd3b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:4.298ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{\pi }}A\,\!}"></span> </td> <td><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> és l'àrea del quadrat, en unitats quadrades. </td></tr> <tr> <td>Conversió del cercle a àrea quadrada </td> <td><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 {1}{4}}C\pi \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>C</mi> <mi>π</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}C\pi \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7061a7c39e2f00ffcab0614e9aeca29daebae31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:5.484ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}C\pi \,\!}"></span> </td> <td><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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> és l'àrea del cercle, en unitats circulars. </td></tr></tbody></table> <p>L'àrea dels polígons irregulars es pot calcular usant la <a href="/w/index.php?title=F%C3%B3rmula_de_Surveyor&action=edit&redlink=1" class="new" title="Fórmula de Surveyor (encara no existeix)">fórmula de Surveyor</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Fórmules_addicionals"><span id="F.C3.B3rmules_addicionals"></span>Fórmules addicionals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=13" title="Modifica la secció: Fórmules addicionals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Àrees_de_figures_de_dues_dimensions"><span id=".C3.80rees_de_figures_de_dues_dimensions"></span>Àrees de figures de dues dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=14" title="Modifica la secció: Àrees de figures de dues dimensions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Triangle">Triangle</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=15" title="Modifica la secció: Triangle"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per calcular l'àrea d'un triangle es pot usar, a part de la fórmula més corrent, la següent fórmula: </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\;=\;{\tfrac {1}{2}}Bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>B</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;{\tfrac {1}{2}}Bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aea6cbd687cac346fad0dd7f1abc660da4c3a23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.893ex; height:3.509ex;" alt="{\displaystyle A\;=\;{\tfrac {1}{2}}Bh}"></span></dd></dl> <p>On <i>B</i> és qualsevol costat i <i>h</i> és la distància de la línia on reposa <i>B</i> fins al <a href="/wiki/V%C3%A8rtex_(geometria)" title="Vèrtex (geometria)">vèrtex</a> oposat del triangle. Aquesta fórmula es pot usar si l'altura <i>h</i> és coneguda. Si es coneixen les longituds dels tres costats llavors es pot usar la <a href="/wiki/F%C3%B3rmula_d%27Her%C3%B3" title="Fórmula d'Heró">fórmula d'Heró</a>, tal com s'ha descrit anteriorment: </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 {\sqrt {s(s-a)(s-b)(s-c)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7e3feb227eeda3c19b052c797ee19a01a6c467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.869ex; height:4.843ex;" alt="{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}"></span></dd></dl> <p>On <i>a</i>, <i>b</i>, <i>c</i> són els costats del triangle 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 s={\tfrac {1}{2}}(a+b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}(a+b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2c4193212526a50585182f301e85e2f1cdfde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.571ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"></span>, és a dir, la meitat del <a href="/wiki/Per%C3%ADmetre" title="Perímetre">perímetre</a>). Si es coneixen un angle i els dos costats que el formen, llavors: </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 {\tfrac {1}{2}}ab\sin(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}ab\sin(C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3719986bc49ddbdd159b5920e995f1c79f0b6c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.704ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}ab\sin(C)}"></span></dd></dl> <p>On <i>C</i> és l'angle donat i <i>a</i> i <i>b</i> són els costats que el formen. Si el triangle es troba en un pla de coordenades, es pot usar una matriu i se simplifica en el <i>valor absolut de (x<sub>1</sub>y₂+ x₂y₃+ x₃y<sub>1</sub> - x₂y<sub>1</sub>- x₃y₂- x<sub>1</sub>y₃), tot això dividit per 2</i>. Aquesta fórmula també es coneix com a <a href="/w/index.php?title=F%C3%B3rmula_del_cord%C3%B3&action=edit&redlink=1" class="new" title="Fórmula del cordó (encara no existeix)">fórmula del cordó</a> i és una manera fàcil i ràpida de resoldre l'àrea d'un triangle del qual se'n saben els tres punts localitzats en el <a href="/wiki/Pla_cartesi%C3%A0" class="mw-redirect" title="Pla cartesià">pla cartesià</a>. Aquesta fórmula també es pot usar per trobar àrees d'altres polígons quan se'n coneixen els seus vèrtexs. Una altra aproximació pel triangle situat en un pla amb coordenades es pot dur a terme utilitzant <a href="/wiki/C%C3%A0lcul_infinitesimal" title="Càlcul infinitesimal">càlcul infinitesimal</a> per trobar l'àrea. </p> <div class="mw-heading mw-heading4"><h4 id="Polígon_simple"><span id="Pol.C3.ADgon_simple"></span>Polígon simple</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=16" title="Modifica la secció: Polígon simple"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un <a href="/wiki/Pol%C3%ADgon_simple" title="Polígon simple">polígon simple</a> construït en una xarxa de punts equidistants i tal que tots els vèrtexs del polígon són punts d'aquesta xarxa, llavors </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\;=\;i+{\frac {b}{2}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;i+{\frac {b}{2}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08da42f26bda96f4985b83661c99d39f4e30fb75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.776ex; height:5.343ex;" alt="{\displaystyle A\;=\;i+{\frac {b}{2}}-1}"></span></dd></dl> <p>On <i>i</i> és el nombre de punts de la xarxa dins el polígon i <i>b</i> és el nombre de punts fora del polígon. Aquesta resultat és conegut amb el nom de <a href="/wiki/Teorema_de_Pick" title="Teorema de Pick">teorema de Pick</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Àrees_en_el_càlcul"><span id=".C3.80rees_en_el_c.C3.A0lcul"></span>Àrees en el càlcul</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=17" title="Modifica la secció: Àrees en el càlcul"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Areabetweentwographs.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Areabetweentwographs.svg/220px-Areabetweentwographs.svg.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Areabetweentwographs.svg/330px-Areabetweentwographs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Areabetweentwographs.svg/440px-Areabetweentwographs.svg.png 2x" data-file-width="1000" data-file-height="900" /></a><figcaption>L'àrea entre dos gràfics pot ser avaluada calculant la diferència entre les <a href="/wiki/Integral" class="mw-redirect" title="Integral">integrals</a> de les dues funcions.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Àrea_entre_dues_funcions"><span id=".C3.80rea_entre_dues_funcions"></span>Àrea entre dues funcions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=18" title="Modifica la secció: Àrea entre dues funcions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una forma per trobar l'àrea delimitada entre dues <a href="/wiki/Funci%C3%B3_matem%C3%A0tica" class="mw-redirect" title="Funció matemàtica">funcions</a> és utilitzant el <a href="/wiki/Integraci%C3%B3" title="Integració">càlcul integral</a>. L'àrea compresa entre les corbes <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> 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 g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"></span> (amb <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 g(x)<f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)<f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2cbfb58d7a268933acb53e25430486ae9ea45c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle g(x)<f(x)}"></span>) en l'interval <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,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> és: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A(a,b)=\int _{a}^{b}|f(x)-g(x)|dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(a,b)=\int _{a}^{b}|f(x)-g(x)|dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a7107f1afaacca7c341db037db1310633572b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.053ex; height:6.343ex;" alt="{\displaystyle A(a,b)=\int _{a}^{b}|f(x)-g(x)|dx}"></span> </p> </blockquote> <div class="NavFrame" style="background-color: transparent; width:67%;margin-bottom:0px;float:left; border-radius:4px"> <div class="NavPic" style="display: none;"></div> <div class="NavHead" align="center" style="background-color: transparent;border-radius:4px;">Exemple d'àrea entre dues funcions</div> <div class="NavContent" align="left" style="padding:7px;">Si es vol trobar l'àrea delimitada entre la funció <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(x)=4-x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=4-x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed91cb47056d94beaafd7d78226581b52f0cec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.903ex; height:3.176ex;" alt="{\displaystyle f(x)=4-x^{2}}"></span> i l'eix x (<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 g(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9bb64a9a8234edd8fb1c622dddfab478719776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.843ex;" alt="{\displaystyle g(x)=0}"></span>) a l'interval <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 [-2,2]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-2,2]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94b820404eca2a458cb2c7d8c24be85fffccf90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-2,2]}"></span> s'utilitza l'equació anterior i, avaluant-la, s'obté el següent: <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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 A(-2,2)=\int _{-2}^{2}|4-x^{2}-0|dx=2\int _{0}^{2}4-x^{2}dx=2\left[8-\left({\frac {2^{3}-0}{3}}\right)\right]={\frac {32}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>4</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mn>8</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>0</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(-2,2)=\int _{-2}^{2}|4-x^{2}-0|dx=2\int _{0}^{2}4-x^{2}dx=2\left[8-\left({\frac {2^{3}-0}{3}}\right)\right]={\frac {32}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842be32fdbb63e90b93ff7261cbc76d9b1cdbb0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.387ex; height:6.343ex;" alt="{\displaystyle A(-2,2)=\int _{-2}^{2}|4-x^{2}-0|dx=2\int _{0}^{2}4-x^{2}dx=2\left[8-\left({\frac {2^{3}-0}{3}}\right)\right]={\frac {32}{3}}}"></span> </p> </blockquote> D'aquí es conclou que l'àrea delimitada és <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 {32}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {32}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0c521bac8c28d0cca31f4e96d1368a88bb7856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {32}{3}}}"></span>.</div> <div style="clear:both;"></div> </div> <p><br /> El <a href="/wiki/Volum" title="Volum">volum</a> tancat entre dues funcions també es redueix al càlcul d'una <a href="/wiki/Integral" class="mw-redirect" title="Integral">integral</a>, de manera similar al pla. </p><p>Si la funció és de la forma <i>r</i> = <i>r</i>(θ) (expressada en <a href="/wiki/Coordenades_polars" title="Coordenades polars">coordenades polars</a>, llavors l'àrea és:<sup id="cite_ref-crc69_12-2" class="reference"><a href="#cite_note-crc69-12"><span class="cite-bracket">[</span>12<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 {1 \over 2}\int _{0}^{2\pi }r^{2}\,d\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over 2}\int _{0}^{2\pi }r^{2}\,d\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b93a94ffacb3c463c9db2fc496fd3620bdc9d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.029ex; height:6.176ex;" alt="{\displaystyle {1 \over 2}\int _{0}^{2\pi }r^{2}\,d\theta }"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Àrea_tancada_per_una_corba_paramètrica"><span id=".C3.80rea_tancada_per_una_corba_param.C3.A8trica"></span>Àrea tancada per una corba paramètrica</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=19" title="Modifica la secció: Àrea tancada per una corba paramètrica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Vegeu també: <a href="/wiki/Teorema_de_Green" title="Teorema de Green">Teorema de Green</a></div> <p>L'àrea tancada per una <a href="/w/index.php?title=Corba_param%C3%A8trica&action=edit&redlink=1" class="new" title="Corba paramètrica (encara no existeix)">corba paramètrica</a> <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 \scriptstyle {{\vec {u}}(t)=(x(t),y(t))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {{\vec {u}}(t)=(x(t),y(t))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f7ee83d873913b3886fd6b166ac7cc888bf4c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.332ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {{\vec {u}}(t)=(x(t),y(t))}}"></span> amb punts finals <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 \scriptstyle {{\vec {u}}(t_{0})={\vec {u}}(t_{1})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {{\vec {u}}(t_{0})={\vec {u}}(t_{1})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4b74388df04df20f04a237236bc8d4084c5e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.568ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {{\vec {u}}(t_{0})={\vec {u}}(t_{1})}}"></span> ve donada per les <a href="/wiki/Integral_de_l%C3%ADnia" class="mw-redirect" title="Integral de línia">integrals de línia</a> següents:<sup id="cite_ref-crc1763_2-1" class="reference"><a href="#cite_note-crc1763-2"><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 \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940fad5094a1ce067667a15559feac98c0f02d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.129ex; height:6.509ex;" alt="{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}"></span></dd></dl> <p>O bé la component <i>z</i> de: </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 {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed6a3cccc5098c44df7ff36a18ec47b50d3a8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.837ex; height:6.509ex;" alt="{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Superfície_de_figures_tridimensionals"><span id="Superf.C3.ADcie_de_figures_tridimensionals"></span>Superfície de figures tridimensionals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=20" title="Modifica la secció: Superfície de figures tridimensionals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La fórmula general per l'àrea superficial d'un gràfic d'una funció diferenciable contínua <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 z=f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}"></span>, on <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 \scriptstyle {(x,y)\in D\subset \mathbb {R} ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>D</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {(x,y)\in D\subset \mathbb {R} ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49edc675c8ea3d498d99cbd91b8040354d65df49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.248ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {(x,y)\in D\subset \mathbb {R} ^{2}}}"></span> 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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> és una regió en el pla <i>xy</i> de frontera suau, és: </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=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1ccc930568eb5c289a5a2e6a29c774b6138e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.713ex; height:7.676ex;" alt="{\displaystyle A=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}"></span></dd></dl> <p>Una fórmula encara més genèrica de l'àrea d'un gràfic d'una <a href="/wiki/Superf%C3%ADcie_param%C3%A8trica" title="Superfície paramètrica">superfície paramètrica</a> en la forma vectorial <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 \scriptstyle {\mathbf {r} =\mathbf {r} (u,v)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\mathbf {r} =\mathbf {r} (u,v)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79ec4541910be3b8bcd9171e578ae6f95d1db75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.311ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\mathbf {r} =\mathbf {r} (u,v)}}"></span>, on <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 \scriptstyle {\mathbf {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\mathbf {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880c25f05edbe458b7276725685a435ea0bb1dc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.779ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\mathbf {r} }}"></span> és una funció vectorial diferenciable contínua de <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 \scriptstyle {(u,v)\in D\subset \mathbb {R} ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>D</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {(u,v)\in D\subset \mathbb {R} ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333cc485af89c333a189d247888fa8b44ddc6bd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.228ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {(u,v)\in D\subset \mathbb {R} ^{2}}}"></span>, és:<sup id="cite_ref-doCarmo_5-1" class="reference"><a href="#cite_note-doCarmo-5"><span class="cite-bracket">[</span>5<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 A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9341a7cc20b6b411267a19d6a4a5aa75d4a2e8ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.291ex; height:5.843ex;" alt="{\displaystyle A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv}"></span></dd></dl> <p>Algunes superfícies aparentment simples poden mostrar algunes propietats molt interessants: per exemple, el gràfic de la funció <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 y=1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=1/x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1bdbfe1a617026ade72035dc95488195b293590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.909ex; height:2.843ex;" alt="{\displaystyle y=1/x}"></span> revolucionant al voltant de l'eix <i>x</i> per <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 x\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca3ced43f1713577888a8a7ade2d0aaf8354a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 1}"></span> dona una superfície anomenada <a href="/wiki/Banya_de_Gabriel" title="Banya de Gabriel">banya de Gabriel</a> que té <a href="/wiki/Volum" title="Volum">volum</a> finit però superfície infinita.<sup id="cite_ref-crc1763_2-2" class="reference"><a href="#cite_note-crc1763-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=21" title="Modifica la secció: Bibliografia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="Spiegel"><span style="font-variant: small-caps;">Spiegel</span>, Murray R.; <span style="font-variant: small-caps;">Abellanas</span>, Lorenzo. McGraw-Hill. <i>Fórmules i taules de matemàtica aplicada</i>, 1992. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/84-7615-197-7" title="Especial:Fonts bibliogràfiques/84-7615-197-7">ISBN 84-7615-197-7</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=F%C3%B3rmules+i+taules+de+matem%C3%A0tica+aplicada&rft.aulast=Spiegel&rft.aufirst=Murray+R.&rft.date=1992&rft.place=Aravaca+%28Madrid%29&rft.isbn=84-7615-197-7"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="crc"><span style="font-variant: small-caps;">Weisstein</span>, Eric W. Chapman&Hall. <i>CRC Concise Encyclopedia of Mathematics</i> (en anglès), 1999. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-8493-9640-9" title="Especial:Fonts bibliogràfiques/0-8493-9640-9">ISBN 0-8493-9640-9</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CRC+Concise+Encyclopedia+of+Mathematics&rft.aulast=Weisstein&rft.aufirst=Eric+W&rft.date=1999&rft.isbn=0-8493-9640-9"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=22" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="http://www.bipm.org/en/CGPM/db/11/12/">Resolution 12 of the 11th meeting of the CGPM (1960)</a>» (en anglès).  Bureau International des Poids et Mesures, 1960.</span></span> </li> <li id="cite_note-crc1763-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-crc1763_2-0">2,0</a></sup> <sup><a href="#cite_ref-crc1763_2-1">2,1</a></sup> <sup><a href="#cite_ref-crc1763_2-2">2,2</a></sup></span> <span class="reference-text"><a href="#crc">Concise Encyclopedia of Mathematics</a>: p. 1763</span> </li> <li id="cite_note-bkos-3"><span class="mw-cite-backlink"><a href="#cite_ref-bkos_3-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal">Mark de Berg, Marc van Kreveld, Mark Overmars i Otfried Schwarzkopf. <i>Computational Geometry</i>. 2a revisada.  Springer-Verlag, 2000. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/3-540-65620-0" title="Especial:Fonts bibliogràfiques/3-540-65620-0">ISBN 3-540-65620-0</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Geometry&rft.au=Mark+de+Berg%2C+Marc+van+Kreveld%2C+Mark+Overmars+i+Otfried+Schwarzkopf&rft.date=2000&rft.edition=2a+revisada&rft.pub=Springer-Verlag&rft.isbn=3-540-65620-0"><span style="display: none;"> </span></span> Capítol 3, Polygon Triangulation: pàg.45–61.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFBoyer1959"><a href="/wiki/Carl_Boyer" class="mw-redirect" title="Carl Boyer"><span style="font-variant: small-caps;">Boyer</span>, Carl B.</a> <i>A History of the Calculus and Its Conceptual Development</i>.  Dover, 1959. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/486606094" title="Especial:Fonts bibliogràfiques/486606094">ISBN 486606094</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Calculus+and+Its+Conceptual+Development&rft.aulast=Boyer&rft.aufirst=Carl+B.&rft.date=1959&rft.pub=Dover&rft.isbn=486606094"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-doCarmo-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-doCarmo_5-0">5,0</a></sup> <sup><a href="#cite_ref-doCarmo_5-1">5,1</a></sup></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFdo_Carmo1976"><a href="/wiki/Manfredo_do_Carmo" title="Manfredo do Carmo"><span style="font-variant: small-caps;">do Carmo</span>, Manfredo</a> <i>Differential Geometry of Curves and Surfaces</i>. Prentice-Hall, 1976, p. 98.</span></span> </li> <li id="cite_note-Rudin-6"><span class="mw-cite-backlink"><a href="#cite_ref-Rudin_6-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFRudin1966"><a href="/wiki/Walter_Rudin" title="Walter Rudin"><span style="font-variant: small-caps;">Rudin</span>, Walter</a>. McGraw-Hill. <i>Real and Complex Analysis</i>, 1966. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-07-100276-6" title="Especial:Fonts bibliogràfiques/0-07-100276-6">ISBN 0-07-100276-6</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Complex+Analysis&rft.aulast=Rudin&rft.aufirst=Walter&rft.date=1966&rft.isbn=0-07-100276-6"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Heròdot-7"><span class="mw-cite-backlink"><a href="#cite_ref-Heròdot_7-0">↑</a></span> <span class="reference-text">Heròdot, <i>Històries</i>. Llibre II.</span> </li> <li id="cite_note-problemaarea-8"><span class="mw-cite-backlink"><a href="#cite_ref-problemaarea_8-0">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="http://fca.unl.edu.ar">El problema de l'àrea</a>» (en castellà).</span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Vegeu, per exemple, <i>Elementary Geometry from an Advanced Standpoint</i> d'Edwin Moise.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://web.archive.org/web/20120716204202/http://www1.bipm.org/en/si/derived_units/2-2-1.html">Derived units expressed in terms of base units</a>» (en anglès).  <a href="/wiki/Oficina_Internacional_de_Pesos_i_Mesures" title="Oficina Internacional de Pesos i Mesures">BIPM</a>. Arxivat de l'<a rel="nofollow" class="external text" href="http://www.bipm.org/en/si/derived_units/2-2-1.html">original</a> el 2012-07-16. [Consulta: 4 juny 2011].</span></span> </li> <li id="cite_note-spiegel_abellanas-11"><span class="mw-cite-backlink"><a href="#cite_ref-spiegel_abellanas_11-0">↑</a></span> <span class="reference-text"><a href="#Spiegel">Spiegel i Abellanas, 1992</a>: p.9</span> </li> <li id="cite_note-crc69-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-crc69_12-0">12,0</a></sup> <sup><a href="#cite_ref-crc69_12-1">12,1</a></sup> <sup><a href="#cite_ref-crc69_12-2">12,2</a></sup></span> <span class="reference-text"><a href="#crc">Concise Encyclopedia of Mathematics</a>: p. 69</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><a href="#Spiegel">Spiegel i Abellanas, 1992</a>: p. 10</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFBraden1986"><span style="font-variant: small-caps;">Braden</span>, Bart «<a rel="nofollow" class="external text" href="https://web.archive.org/web/20031105063724/http://www.maa.org/pubs/Calc_articles/ma063.pdf">The Surveyor’s Area Formula</a>» (en anglès). <i>The College Mathematics Journal</i>, volum 17, núm. 4, 9-1986, pàg. 326–337. Arxivat de l'<a rel="nofollow" class="external text" href="http://www.maa.org/pubs/Calc_articles/ma063.pdf">original</a> el 2003-11-05 [Consulta: 10 juny 2011].</span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Enllaços_externs"><span id="Enlla.C3.A7os_externs"></span>Enllaços externs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%80rea&action=edit&section=23" title="Modifica la secció: Enllaços externs"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r33663753">.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:#f9f9f9;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}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Area" class="extiw" title="commons:Category:Area">Àrea</a></b></i></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.sengpielaudio.com/calculator-cross-section.htm">Conversió del diàmetre d'un cable a la secció circular i viceversa</a> <style data-mw-deduplicate="TemplateStyles:r33711417">.mw-parser-output .languageicon{font-size:0.95em;color:#555;background-color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .languageicon{background-color:inherit;color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .languageicon{background-color:inherit;color:white}}</style><span class="languageicon" title="En anglès">(anglès)</span></li> <li><span class="citació mathworld" id="Referència-Mathworld-Àrea"><a href="/wiki/Eric_W._Weisstein" class="mw-redirect" title="Eric W. Weisstein"><span style="font-variant:small-caps; font-variant-caps: small-caps;">Weisstein</span>, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Area.html">«Àrea»</a> a <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (en anglès).</span></li></ul> <p><br /> </p> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_d%27autoritats" title="Control d'autoritats">Registres d'autoritat</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Biblioth%C3%A8que_nationale_de_France" class="mw-redirect" title="Bibliothèque nationale de France">BNF</a> <span class="uid"> (<a rel="nofollow" class="external text" href="http://catalogue.bnf.fr/ark:/12148/cb12172891w">1</a>)</span></li> <li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid"> (<a rel="nofollow" class="external text" href="http://d-nb.info/gnd/4193807-0">1</a>)</span></li> <li><a href="/wiki/LCCN" class="mw-redirect" title="LCCN">LCCN</a> <span class="uid"> (<a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85006984">1</a>)</span></li></ul> </div></td></tr></tbody></table></div> <p><span style="display: none;" class="interProject"><a href="https://ca.wiktionary.org/wiki/%C3%A0rea" class="extiw" title="wikt:àrea">Viccionari</a></span> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Àrea&oldid=33090103">https://ca.wikipedia.org/w/index.php?title=Àrea&oldid=33090103</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categories</a>: <ul><li><a href="/wiki/Categoria:Magnituds_f%C3%ADsiques" title="Categoria:Magnituds físiques">Magnituds físiques</a></li><li><a href="/wiki/Categoria:Topografia" title="Categoria:Topografia">Topografia</a></li><li><a href="/wiki/Categoria:Geometria" title="Categoria:Geometria">Geometria</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categories ocultes: <ul><li><a href="/wiki/Categoria:P%C3%A0gines_amb_enlla%C3%A7_commonscat_des_de_Wikidata" title="Categoria:Pàgines amb enllaç commonscat des de Wikidata">Pàgines amb enllaç commonscat des de Wikidata</a></li><li><a href="/wiki/Categoria:1.000_articles_fonamentals" title="Categoria:1.000 articles fonamentals">1.000 articles fonamentals</a></li><li><a href="/wiki/Categoria:Ci%C3%A8ncia_(Els_1000_de_META)" title="Categoria:Ciència (Els 1000 de META)">Ciència (Els 1000 de META)</a></li><li><a href="/wiki/Categoria:Control_d%27autoritats" title="Categoria:Control d'autoritats">Control d'autoritats</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 11 feb 2024 a les 11:20.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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