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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="Wiki"> <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="Tartalomjegyzék" 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">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</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">Bevezető</div> </a> </li> <li id="toc-Képletek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Képletek"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Képletek</span> </div> </a> <button aria-controls="toc-Képletek-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>A(z) Képletek alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Képletek-sublist" class="vector-toc-list"> <li id="toc-Térfogat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Térfogat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Térfogat</span> </div> </a> <ul id="toc-Térfogat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Felszín" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Felszín"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Felszín</span> </div> </a> <ul id="toc-Felszín-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beírható_gömb_sugara" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Beírható_gömb_sugara"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Beírható gömb sugara</span> </div> </a> <ul id="toc-Beírható_gömb_sugara-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Egyenletek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Egyenletek"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Egyenletek</span> </div> </a> <ul id="toc-Egyenletek-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Az_egyenes_körkúp_mint_forgástest" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Az_egyenes_körkúp_mint_forgástest"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Az egyenes körkúp mint forgástest</span> </div> </a> <ul id="toc-Az_egyenes_körkúp_mint_forgástest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lineáris_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lineáris_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lineáris algebra</span> </div> </a> <ul id="toc-Lineáris_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_térfogat-_és_felszínképletek_bizonyítása" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_térfogat-_és_felszínképletek_bizonyítása"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>A térfogat- és felszínképletek bizonyítása</span> </div> </a> <button aria-controls="toc-A_térfogat-_és_felszínképletek_bizonyítása-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>A(z) A térfogat- és felszínképletek bizonyítása alszakasz kinyitása/becsukása</span> </button> <ul id="toc-A_térfogat-_és_felszínképletek_bizonyítása-sublist" class="vector-toc-list"> <li id="toc-Térfogat_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Térfogat_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Térfogat</span> </div> </a> <ul id="toc-Térfogat_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_kúppalást_felszíne" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_kúppalást_felszíne"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>A kúppalást felszíne</span> </div> </a> <ul id="toc-A_kúppalást_felszíne-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Források</span> </div> </a> <ul id="toc-Források-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Külső_hivatkozások" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Külső_hivatkozások"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Külső hivatkozások</span> </div> </a> <ul id="toc-Külső_hivatkozások-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Kúp</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="Ugrás egy más nyelvű szócikkre. Elérhető 103 nyelven" > <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-103" 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">103 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Cone" title="Cone – angol" lang="en" hreflang="en" data-title="Cone" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Ke%C3%ABl" title="Keël – afrikaans" lang="af" hreflang="af" data-title="Keël" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%BE%E1%8C%A3%E1%8C%A3" title="ሾጣጣ – amhara" lang="am" hreflang="am" data-title="ሾጣጣ" data-language-autonym="አማርኛ" data-language-local-name="amhara" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AE%D8%B1%D9%88%D8%B7" title="مخروط – arab" lang="ar" hreflang="ar" data-title="مخروط" data-language-autonym="العربية" data-language-local-name="arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Pullu" title="Pullu – ajmara" lang="ay" hreflang="ay" data-title="Pullu" data-language-autonym="Aymar aru" data-language-local-name="ajmara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Konus" title="Konus – azerbajdzsáni" lang="az" hreflang="az" data-title="Konus" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" 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/%DA%A9%D9%88%D9%86%D9%88%D8%B3" title="کونوس – South Azerbaijani" lang="azb" hreflang="azb" data-title="کونوس" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ban mw-list-item"><a href="https://ban.wikipedia.org/wiki/Kojong" title="Kojong – balinéz" lang="ban" hreflang="ban" data-title="Kojong" data-language-autonym="Basa Bali" data-language-local-name="balinéz" class="interlanguage-link-target"><span>Basa Bali</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – belarusz" lang="be" hreflang="be" data-title="Конус" data-language-autonym="Беларуская" data-language-local-name="belarusz" 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%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Конус" 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%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – bolgár" lang="bg" hreflang="bg" data-title="Конус" data-language-autonym="Български" data-language-local-name="bolgár" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kupa_(geometrija)" title="Kupa (geometrija) – bosnyák" lang="bs" hreflang="bs" data-title="Kupa (geometrija)" data-language-autonym="Bosanski" data-language-local-name="bosnyák" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Con" title="Con – katalán" lang="ca" hreflang="ca" data-title="Con" data-language-autonym="Català" data-language-local-name="katalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-chy mw-list-item"><a href="https://chy.wikipedia.org/wiki/Ts%C3%A9-%C3%A9%C5%A1k%C3%B4sa%27%C3%A9vetov%C3%A1to" title="Tsé-éškôsa'évetováto – csejen" lang="chy" hreflang="chy" data-title="Tsé-éškôsa'évetováto" data-language-autonym="Tsetsêhestâhese" data-language-local-name="csejen" class="interlanguage-link-target"><span>Tsetsêhestâhese</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%82%D9%88%D9%88%DA%86%DB%95%DA%A9" title="قووچەک – közép-ázsiai kurd" lang="ckb" hreflang="ckb" data-title="قووچەک" data-language-autonym="کوردی" data-language-local-name="közép-ázsiai kurd" 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/Ku%C5%BEel" title="Kužel – cseh" lang="cs" hreflang="cs" data-title="Kužel" data-language-autonym="Čeština" data-language-local-name="cseh" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – csuvas" lang="cv" hreflang="cv" data-title="Конус" data-language-autonym="Чӑвашла" data-language-local-name="csuvas" 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/C%C3%B4n" title="Côn – walesi" lang="cy" hreflang="cy" data-title="Côn" data-language-autonym="Cymraeg" data-language-local-name="walesi" 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/Kegle_(geometri)" title="Kegle (geometri) – dán" lang="da" hreflang="da" data-title="Kegle (geometri)" data-language-autonym="Dansk" data-language-local-name="dán" 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/Kegel_(Geometrie)" title="Kegel (Geometrie) – német" lang="de" hreflang="de" data-title="Kegel (Geometrie)" data-language-autonym="Deutsch" data-language-local-name="német" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8E%CE%BD%CE%BF%CF%82" title="Κώνος – görög" lang="el" hreflang="el" data-title="Κώνος" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Konuso" title="Konuso – eszperantó" lang="eo" hreflang="eo" data-title="Konuso" data-language-autonym="Esperanto" data-language-local-name="eszperantó" 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/Cono_(geometr%C3%ADa)" title="Cono (geometría) – spanyol" lang="es" hreflang="es" data-title="Cono (geometría)" data-language-autonym="Español" data-language-local-name="spanyol" 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/Koonus" title="Koonus – észt" lang="et" hreflang="et" data-title="Koonus" data-language-autonym="Eesti" data-language-local-name="észt" 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/Kono" title="Kono – baszk" lang="eu" hreflang="eu" data-title="Kono" data-language-autonym="Euskara" data-language-local-name="baszk" 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%AE%D8%B1%D9%88%D8%B7" title="مخروط – perzsa" lang="fa" hreflang="fa" data-title="مخروط" data-language-autonym="فارسی" data-language-local-name="perzsa" 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/Kartio" title="Kartio – finn" lang="fi" hreflang="fi" data-title="Kartio" data-language-autonym="Suomi" data-language-local-name="finn" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/C%C3%B4ne_(g%C3%A9om%C3%A9trie)" title="Cône (géométrie) – francia" lang="fr" hreflang="fr" data-title="Cône (géométrie)" data-language-autonym="Français" data-language-local-name="francia" 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/Keegel" title="Keegel – északi fríz" lang="frr" hreflang="frr" data-title="Keegel" data-language-autonym="Nordfriisk" data-language-local-name="északi fríz" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/C%C3%B3n" title="Cón – ír" lang="ga" hreflang="ga" data-title="Cón" data-language-autonym="Gaeilge" data-language-local-name="ír" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%8C%90%E5%BD%A2" title="錐形 – gan kínai" lang="gan" hreflang="gan" data-title="錐形" data-language-autonym="贛語" data-language-local-name="gan kínai" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Konik" title="Konik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Konik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cono" title="Cono – gallego" lang="gl" hreflang="gl" data-title="Cono" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B6%E0%AA%82%E0%AA%95%E0%AB%81" title="શંકુ – gudzsaráti" lang="gu" hreflang="gu" data-title="શંકુ" data-language-autonym="ગુજરાતી" data-language-local-name="gudzsaráti" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A8%D7%95%D7%98" title="חרוט – héber" lang="he" hreflang="he" data-title="חרוט" data-language-autonym="עברית" data-language-local-name="héber" 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%B6%E0%A4%82%E0%A4%95%E0%A5%81" title="शंकु – hindi" lang="hi" hreflang="hi" data-title="शंकु" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Sto%C5%BEac" title="Stožac – horvát" lang="hr" hreflang="hr" data-title="Stožac" data-language-autonym="Hrvatski" data-language-local-name="horvát" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B6" title="Կոն – örmény" lang="hy" hreflang="hy" data-title="Կոն" data-language-autonym="Հայերեն" data-language-local-name="örmény" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%94%D5%B8%D5%B6" title="Քոն – Western Armenian" lang="hyw" hreflang="hyw" data-title="Քոն" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Cono_(geometria)" title="Cono (geometria) – interlingva" lang="ia" hreflang="ia" data-title="Cono (geometria)" data-language-autonym="Interlingua" data-language-local-name="interlingva" 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/Kerucut" title="Kerucut – indonéz" lang="id" hreflang="id" data-title="Kerucut" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéz" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Keila_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Keila (rúmfræði) – izlandi" lang="is" hreflang="is" data-title="Keila (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="izlandi" 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/Cono" title="Cono – olasz" lang="it" hreflang="it" data-title="Cono" data-language-autonym="Italiano" data-language-local-name="olasz" 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/%E5%86%86%E9%8C%90" title="円錐 – japán" lang="ja" hreflang="ja" data-title="円錐" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pasungan" title="Pasungan – jávai" lang="jv" hreflang="jv" data-title="Pasungan" data-language-autonym="Jawa" data-language-local-name="jávai" 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%99%E1%83%9D%E1%83%9C%E1%83%A3%E1%83%A1%E1%83%98" title="კონუსი – grúz" lang="ka" hreflang="ka" data-title="კონუსი" data-language-autonym="ქართული" data-language-local-name="grúz" 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/Akawsar_(tusnakt)" title="Akawsar (tusnakt) – kabije" lang="kab" hreflang="kab" data-title="Akawsar (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabije" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – kazah" lang="kk" hreflang="kk" data-title="Конус" data-language-autonym="Қазақша" data-language-local-name="kazah" 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%84%E1%9E%93" title="កោន – khmer" lang="km" hreflang="km" data-title="កោន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B6%E0%B2%82%E0%B2%95%E0%B3%81" title="ಶಂಕು – kannada" lang="kn" hreflang="kn" data-title="ಶಂಕು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%EB%BF%94" title="원뿔 – koreai" lang="ko" hreflang="ko" data-title="원뿔" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – kirgiz" lang="ky" hreflang="ky" data-title="Конус" data-language-autonym="Кыргызча" data-language-local-name="kirgiz" 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/Conus" title="Conus – latin" lang="la" hreflang="la" data-title="Conus" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Cono" title="Cono – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Cono" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/K%C5%ABgis" title="Kūgis – litván" lang="lt" hreflang="lt" data-title="Kūgis" data-language-autonym="Lietuvių" data-language-local-name="litván" 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/Konuss" title="Konuss – lett" lang="lv" hreflang="lv" data-title="Konuss" data-language-autonym="Latviešu" data-language-local-name="lett" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81%D1%81%D1%8C" title="Конуссь – moksán" lang="mdf" hreflang="mdf" data-title="Конуссь" data-language-autonym="Мокшень" data-language-local-name="moksán" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Kitsoloha" title="Kitsoloha – malgas" lang="mg" hreflang="mg" data-title="Kitsoloha" data-language-autonym="Malagasy" data-language-local-name="malgas" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – macedón" lang="mk" hreflang="mk" data-title="Конус" data-language-autonym="Македонски" data-language-local-name="macedón" 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%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%82%E0%B4%AA%E0%B4%BF%E0%B4%95" title="വൃത്തസ്തൂപിക – malajálam" lang="ml" hreflang="ml" data-title="വൃത്തസ്തൂപിക" data-language-autonym="മലയാളം" data-language-local-name="malajálam" 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%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – mongol" lang="mn" hreflang="mn" data-title="Конус" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kon" title="Kon – maláj" lang="ms" hreflang="ms" data-title="Kon" data-language-autonym="Bahasa Melayu" data-language-local-name="maláj" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kegel_(ruimtelijke_figuur)" title="Kegel (ruimtelijke figuur) – holland" lang="nl" hreflang="nl" data-title="Kegel (ruimtelijke figuur)" data-language-autonym="Nederlands" data-language-local-name="holland" 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/Kjegle" title="Kjegle – norvég (nynorsk)" lang="nn" hreflang="nn" data-title="Kjegle" data-language-autonym="Norsk nynorsk" data-language-local-name="norvég (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/Kjegle" title="Kjegle – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Kjegle" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (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/C%C3%B2n" title="Còn – okszitán" lang="oc" hreflang="oc" data-title="Còn" data-language-autonym="Occitan" data-language-local-name="okszitán" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Biilalee_(Koonii)" title="Biilalee (Koonii) – oromo" lang="om" hreflang="om" data-title="Biilalee (Koonii)" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sto%C5%BCek_(bry%C5%82a)" title="Stożek (bryła) – lengyel" lang="pl" hreflang="pl" data-title="Stożek (bryła)" data-language-autonym="Polski" data-language-local-name="lengyel" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/C%C3%B2no" title="Còno – Piedmontese" lang="pms" hreflang="pms" data-title="Còno" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%A8%D9%88%DA%A9%D8%B1" title="بوکر – pastu" lang="ps" hreflang="ps" data-title="بوکر" data-language-autonym="پښتو" data-language-local-name="pastu" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Cone" title="Cone – portugál" lang="pt" hreflang="pt" data-title="Cone" data-language-autonym="Português" data-language-local-name="portugál" 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/Chuqu" title="Chuqu – kecsua" lang="qu" hreflang="qu" data-title="Chuqu" data-language-autonym="Runa Simi" data-language-local-name="kecsua" 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/Con" title="Con – román" lang="ro" hreflang="ro" data-title="Con" data-language-autonym="Română" data-language-local-name="román" 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%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – orosz" lang="ru" hreflang="ru" data-title="Конус" data-language-autonym="Русский" data-language-local-name="orosz" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Conu" title="Conu – szicíliai" lang="scn" hreflang="scn" data-title="Conu" data-language-autonym="Sicilianu" data-language-local-name="szicíliai" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sto%C5%BEac" title="Stožac – szerbhorvát" lang="sh" hreflang="sh" data-title="Stožac" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="szerbhorvát" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%9A%E0%B6%AD%E0%B7%94%E0%B7%80" title="කේතුව – szingaléz" lang="si" hreflang="si" data-title="කේතුව" data-language-autonym="සිංහල" data-language-local-name="szingaléz" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Cone" title="Cone – Simple English" lang="en-simple" hreflang="en-simple" data-title="Cone" 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/Ku%C5%BEe%C4%BE" title="Kužeľ – szlovák" lang="sk" hreflang="sk" data-title="Kužeľ" data-language-autonym="Slovenčina" data-language-local-name="szlovák" 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/Sto%C5%BEec" title="Stožec – szlovén" lang="sl" hreflang="sl" data-title="Stožec" data-language-autonym="Slovenščina" data-language-local-name="szlovén" 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/Charaka" title="Charaka – sona" lang="sn" hreflang="sn" data-title="Charaka" data-language-autonym="ChiShona" data-language-local-name="sona" 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/Toobin" title="Toobin – szomáli" lang="so" hreflang="so" data-title="Toobin" data-language-autonym="Soomaaliga" data-language-local-name="szomáli" 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/Koni" title="Koni – albán" lang="sq" hreflang="sq" data-title="Koni" data-language-autonym="Shqip" data-language-local-name="albán" 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%9A%D1%83%D0%BF%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Купа (геометрија) – szerb" lang="sr" hreflang="sr" data-title="Купа (геометрија)" data-language-autonym="Српски / srpski" data-language-local-name="szerb" 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/Congcot" title="Congcot – szundanéz" lang="su" hreflang="su" data-title="Congcot" data-language-autonym="Sunda" data-language-local-name="szundanéz" 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/Kon" title="Kon – svéd" lang="sv" hreflang="sv" data-title="Kon" data-language-autonym="Svenska" data-language-local-name="svéd" 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/Pia" title="Pia – szuahéli" lang="sw" hreflang="sw" data-title="Pia" data-language-autonym="Kiswahili" data-language-local-name="szuahéli" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%82%E0%AE%AE%E0%AF%8D%E0%AE%AA%E0%AF%81" title="கூம்பு – tamil" lang="ta" hreflang="ta" data-title="கூம்பு" data-language-autonym="தமிழ்" data-language-local-name="tamil" 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%B6%E0%B0%82%E0%B0%95%E0%B1%81%E0%B0%B5%E0%B1%81" title="శంకువు – telugu" lang="te" hreflang="te" data-title="శంకువు" 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%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – tadzsik" lang="tg" hreflang="tg" data-title="Конус" data-language-autonym="Тоҷикӣ" data-language-local-name="tadzsik" 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%97%E0%B8%A3%E0%B8%87%E0%B8%81%E0%B8%A3%E0%B8%A7%E0%B8%A2" title="ทรงกรวย – thai" lang="th" hreflang="th" data-title="ทรงกรวย" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Balisuso" title="Balisuso – tagalog" lang="tl" hreflang="tl" data-title="Balisuso" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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/Koni" title="Koni – török" lang="tr" hreflang="tr" data-title="Koni" data-language-autonym="Türkçe" data-language-local-name="török" 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/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – tatár" lang="tt" hreflang="tt" data-title="Конус" data-language-autonym="Татарча / tatarça" data-language-local-name="tatár" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tyv mw-list-item"><a href="https://tyv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – tuvai" lang="tyv" hreflang="tyv" data-title="Конус" data-language-autonym="Тыва дыл" data-language-local-name="tuvai" class="interlanguage-link-target"><span>Тыва дыл</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81" title="Конус – ukrán" lang="uk" hreflang="uk" data-title="Конус" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Konus" title="Konus – üzbég" lang="uz" hreflang="uz" data-title="Konus" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="üzbég" 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/M%E1%BA%B7t_n%C3%B3n" title="Mặt nón – vietnámi" lang="vi" hreflang="vi" data-title="Mặt nón" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Cono_(heyometriya)" title="Cono (heyometriya) – varaó" lang="war" hreflang="war" data-title="Cono (heyometriya)" data-language-autonym="Winaray" data-language-local-name="varaó" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9C%86%E9%94%A5" title="圆锥 – wu kínai" lang="wuu" hreflang="wuu" data-title="圆锥" data-language-autonym="吴语" data-language-local-name="wu kínai" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%A0%D7%95%D7%A1" title="קאנוס – jiddis" lang="yi" hreflang="yi" data-title="קאנוס" data-language-autonym="ייִדיש" data-language-local-name="jiddis" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9C%86%E9%94%A5" title="圆锥 – kínai" lang="zh" hreflang="zh" data-title="圆锥" data-language-autonym="中文" data-language-local-name="kínai" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%93%E9%8C%90" title="圓錐 – kantoni" lang="yue" hreflang="yue" data-title="圓錐" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q42344#sitelinks-wikipedia" title="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások szerkesztése</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Névterek"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/K%C3%BAp" title="A lap megtekintése [c]" accesskey="c"><span>Szócikk</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Vita:K%C3%BAp" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Nyelvvariáns váltása" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button 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class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=K%C3%BAp&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eszközök" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eszközök</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </div> <div 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erre?</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Kapcsol%C3%B3d%C3%B3_v%C3%A1ltoztat%C3%A1sok/K%C3%BAp" rel="nofollow" title="Az erről a lapról hivatkozott lapok utolsó változtatásai [k]" accesskey="k"><span>Kapcsolódó változtatások</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Speci%C3%A1lis_lapok" title="Az összes speciális lap listája [q]" accesskey="q"><span>Speciális lapok</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=K%C3%BAp&oldid=27008628" title="Állandó hivatkozás ezen lap ezen változatához"><span>Hivatkozás erre a változatra</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=K%C3%BAp&action=info" title="További információk erről a lapról"><span>Lapinformációk</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Hivatkoz%C3%A1s&page=K%C3%BAp&id=27008628&wpFormIdentifier=titleform" title="Információk a lap idézésével kapcsolatban"><span>Hogyan hivatkozz erre a lapra?</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:UrlShortener&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FK%25C3%25BAp"><span>Rövidített URL készítése</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:QrCode&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FK%25C3%25BAp"><span>QR-kód letöltése</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Nyomtatás/exportálás </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:K%C3%B6nyv&bookcmd=book_creator&referer=K%C3%BAp"><span>Könyv készítése</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:DownloadAsPdf&page=K%C3%BAp&action=show-download-screen"><span>Letöltés PDF-ként</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=K%C3%BAp&printable=yes" title="A lap nyomtatható változata [p]" accesskey="p"><span>Nyomtatható változat</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Társprojektek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Cones" hreflang="en"><span>Wikimédia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q42344" title="Kapcsolt adattárelem [g]" accesskey="g"><span>Wikidata-adatlap</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Megjelenés</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">elrejtés</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-indicator-fr-review-status" class="mw-indicator"><indicator name="fr-review-status" class="mw-fr-review-status-indicator" id="mw-fr-revision-toggle"><span class="cdx-fr-css-icon-review--status--stable"></span><b>Ellenőrzött</b></indicator></div> </div> <div id="siteSub" class="noprint">A Wikipédiából, a szabad enciklopédiából</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Változat állapota</h2><p class="cdx-dialog__header__subtitle">Ez a lap egy ellenőrzött változata</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Speci%C3%A1lis:Badtitle/Message">ellenőrizve</a>: <i>2024. március 26.</i><p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/F%C3%A1jl:Cone_3d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cone_3d.png/250px-Cone_3d.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cone_3d.png/375px-Cone_3d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cone_3d.png/500px-Cone_3d.png 2x" data-file-width="913" data-file-height="458" /></a><figcaption>Egyenes és ferde kúp</figcaption></figure> <p>A matematikában a <b>kúp</b> (idegen szóval <b>kónusz</b>) <a href="/wiki/G%C3%BAla" title="Gúla">gúlaszerű</a> térbeli test. A kúp alapja egy tetszőleges <a href="/wiki/Kateg%C3%B3ria:S%C3%ADkidomok" title="Kategória:Síkidomok">síkidom</a>, <a href="/w/index.php?title=M%C3%A9rtani_pal%C3%A1st&action=edit&redlink=1" class="new" title="Mértani palást (a lap nem létezik)">palástját</a> a csúcsot az alap határpontjaival összekötő egyenes szakaszok, az <i>alkotók</i> uniója alkotja. Megkülönböztethetünk <i>egyenes</i> és <i>ferde</i> kúpokat aszerint, hogy a csúcs merőleges vetülete az alapra egybeesik-e az alap középpontjával, ha utóbbi értelmezett. Kúp alatt leggyakrabban az egyenes, kör alapú kúpokat értik. A kúpot az alapjával párhuzamos síkkal elmetszve <a href="/w/index.php?title=Csonkak%C3%BAp&action=edit&redlink=1" class="new" title="Csonkakúp (a lap nem létezik)">csonkakúpot</a> kapunk. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Képletek"><span id="K.C3.A9pletek"></span>Képletek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=1" title="Szakasz szerkesztése: Képletek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A kúpoknak létezik térfogata és felszíne.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Térfogat"><span id="T.C3.A9rfogat"></span>Térfogat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=2" title="Szakasz szerkesztése: Térfogat"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jelölje <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> a kúp alapjának a területét, s legyen <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> a magassága. Ekkor a <a href="/wiki/T%C3%A9rfogat" title="Térfogat">térfogat</a> az alábbiak szerint számítható: </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 V={\frac {1}{3}}bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>b</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6590788f6fc58cda1e55df120c87c0e7472a3fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.221ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}bh}"></span></dd></dl> <p>Speciálisan, ha a kúp kör alapú, akkor <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>-rel jelölve a <a href="/wiki/K%C3%B6r_(geometria)" title="Kör (geometria)">kör</a> sugarát, így részletezhető a formula: </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 V={\frac {1}{3}}\pi r^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}\pi r^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8148cf955aa0660882254207e18ef9bf92463d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.658ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}\pi r^{2}h}"></span></dd></dl> <p>A másik esetben, ha az alap elliptikus, akkor pedig az <a href="/wiki/Ellipszis_(g%C3%B6rbe)" title="Ellipszis (görbe)">ellipszis</a> sugarait <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span> 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 r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"></span> szimbólumokkal jelölve a következőképpen: </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 V={\frac {1}{3}}\pi r_{1}r_{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}\pi r_{1}r_{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f59657678e658d66bde825dc826f177f9672c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.761ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}\pi r_{1}r_{2}h}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Felszín"><span id="Felsz.C3.ADn"></span>Felszín</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=3" title="Szakasz szerkesztése: Felszín"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A kúp felszíne az alap és a palást területének összege. Az egyenes, köralapú kúp esetében erre adható egyszerű képlet: </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}+\pi ra}"> <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> <mo>+</mo> <mi>π</mi> <mi>r</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}+\pi ra}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7fd2d349dba3c5a97f263136b0c25e1af9227b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.727ex; height:2.843ex;" alt="{\displaystyle A=\pi r^{2}+\pi ra}"></span></dd></dl> <p>ahol <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> a kúp egy alkotójának hossza, képlete: </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 {r^{2}+h^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {r^{2}+h^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41e3b2535eee34887603d7f98b967ee0ff9d6ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.989ex; height:3.509ex;" alt="{\displaystyle a={\sqrt {r^{2}+h^{2}}}}"></span></dd></dl> <p>Ez a <a href="/wiki/Pitagorasz-t%C3%A9tel" title="Pitagorasz-tétel">Pitagorasz-tétel</a> egyenes következménye. </p> <div class="mw-heading mw-heading3"><h3 id="Beírható_gömb_sugara"><span id="Be.C3.ADrhat.C3.B3_g.C3.B6mb_sugara"></span>Beírható gömb sugara</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=4" title="Szakasz szerkesztése: Beírható gömb sugara"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az egyenes körkúpba írható gömb ρ sugarának képlete: </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 \rho ={\frac {3V}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>V</mi> </mrow> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {3V}{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf97c1a8ebcda3a54dc84b04d5ae383dca190f17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.086ex; height:5.343ex;" alt="{\displaystyle \rho ={\frac {3V}{A}}}"></span></dd></dl> <p>ahol A jelöli a kúp felszínét, V pedig a térfogatát.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Egyenletek">Egyenletek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=5" title="Szakasz szerkesztése: Egyenletek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/F%C3%A1jl:Cone.jpg" class="mw-file-description" title="Kúp paramétervonalakkal"><img alt="Kúp paramétervonalakkal" src="//upload.wikimedia.org/wikipedia/commons/c/c8/Cone.jpg" decoding="async" width="240" height="276" class="mw-file-element" data-file-width="240" data-file-height="276" /></a><figcaption>Kúp paramétervonalakkal</figcaption></figure> <p>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 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> magasságú 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 \vartheta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϑ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00eaf197c35bbfa391b9477490a4af955416837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }"></span> fél nyílásszögű kúp, aminek forgástengelye 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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> tengely, csúcsa az origó, így paraméterezhető: </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 S(s,t,u)=\left(u\mathrm {tg} s\cos t,u\mathrm {tg} s\sin t,u\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">g</mi> </mrow> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">g</mi> </mrow> <mi>s</mi> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(s,t,u)=\left(u\mathrm {tg} s\cos t,u\mathrm {tg} s\sin t,u\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98dcdcda3fd6187ad805f6c353f22070b9dcaaef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.11ex; height:2.843ex;" alt="{\displaystyle S(s,t,u)=\left(u\mathrm {tg} s\cos t,u\mathrm {tg} s\sin t,u\right)}"></span></dd></dl> <p>ahol <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,t,u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s,t,u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be86046a144c712651e85eb2168015bf76ced0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.328ex; height:2.343ex;" alt="{\displaystyle s,t,u}"></span> rendre 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 [0,\vartheta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ϑ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\vartheta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19cb44ab63c07fd5ea98a8131b95a46bffa097d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.122ex; height:2.843ex;" alt="{\displaystyle [0,\vartheta )}"></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 [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec72cfde732f42822df3cbbe175b7465887eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle [0,2\pi )}"></span>, 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 [0,h]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>h</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,h]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbede90a8f7ff59267c875f09e715c896ce7a51a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle [0,h]}"></span> intervallumokba esik. </p><p>Ugyanez a test implicit az </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 \{S(x,y,z)\leq 0,z\geq 0,z\leq h\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>0</mn> <mo>,</mo> <mi>z</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi>z</mi> <mo>≤</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{S(x,y,z)\leq 0,z\geq 0,z\leq h\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4abf60907a0e8ef3de83c67f1575c7acb8c218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.478ex; height:2.843ex;" alt="{\displaystyle \{S(x,y,z)\leq 0,z\geq 0,z\leq h\}}"></span></dd></dl> <p>egyenlőtlenségekkel adható meg, ahol </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 S(x,y,z)=(x^{2}+y^{2})(\cos \vartheta )^{2}-z^{2}(\sin \vartheta )^{2}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϑ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϑ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(x,y,z)=(x^{2}+y^{2})(\cos \vartheta )^{2}-z^{2}(\sin \vartheta )^{2}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becc272e89413bc1fb697e8a6137f5dce494f0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.531ex; height:3.176ex;" alt="{\displaystyle S(x,y,z)=(x^{2}+y^{2})(\cos \vartheta )^{2}-z^{2}(\sin \vartheta )^{2}.\,}"></span></dd></dl> <p>Általánosabban 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 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> vektorral párhuzamos forgástengelyű origó csúcsú körkúp, aminek fél nyílásszöge <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 \vartheta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϑ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00eaf197c35bbfa391b9477490a4af955416837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }"></span> az <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(u)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc77708730718a005c51a00ec7b42b248d75653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.899ex; height:2.843ex;" alt="{\displaystyle S(u)=0}"></span> vektoregyenlettel adható meg, ahol </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 S(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \vartheta )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>⋅</mo> <mi>d</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>⋅</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϑ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \vartheta )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b749e63f899bb69d2ebe5053419689b32206d0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.469ex; height:3.176ex;" alt="{\displaystyle S(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \vartheta )^{2}}"></span>   vagy   <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(u)=u\cdot d-|d||u|\cos \vartheta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo>⋅</mo> <mi>d</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϑ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u)=u\cdot d-|d||u|\cos \vartheta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9326d34eca1fe23015ec16a8e7dbbbc796a748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.194ex; height:2.843ex;" alt="{\displaystyle S(u)=u\cdot d-|d||u|\cos \vartheta }"></span></dd></dl> <p>ahol <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 u=(x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=(x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859cf3150346d60ba03189cefdd0290d4c17861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.879ex; height:2.843ex;" alt="{\displaystyle u=(x,y,z)}"></span>, 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 u\cdot d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>⋅</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\cdot d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ac43224e648965087c181c6edde954ab52a524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.225ex; height:2.176ex;" alt="{\displaystyle u\cdot d}"></span> <a href="/wiki/Skal%C3%A1rszorzat" class="mw-redirect" title="Skalárszorzat">skalárszorzat</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Az_egyenes_körkúp_mint_forgástest"><span id="Az_egyenes_k.C3.B6rk.C3.BAp_mint_forg.C3.A1stest"></span>Az egyenes körkúp mint forgástest</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=6" title="Szakasz szerkesztése: Az egyenes körkúp mint forgástest"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az egyenes körkúp forgástestként is generálható egy <i>AB</i> szakaszt elforgatva annak pontosan egy végpontján áthaladó egyenes körül. Ebben az esetben az <i>AB</i> szakaszt nevezik a kúp <b>alkotójának</b> is. Ekkor fennáll az alábbi egyenlőség: </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/F%C3%A1jl:Generating_a_cone_as_a_rotation_body.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Generating_a_cone_as_a_rotation_body.png/220px-Generating_a_cone_as_a_rotation_body.png" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Generating_a_cone_as_a_rotation_body.png/330px-Generating_a_cone_as_a_rotation_body.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Generating_a_cone_as_a_rotation_body.png/440px-Generating_a_cone_as_a_rotation_body.png 2x" data-file-width="931" data-file-height="663" /></a><figcaption>Az egyenes körkúp konstrukciója <a href="/wiki/Forg%C3%A1stest" title="Forgástest">forgástestként</a></figcaption></figure> <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 |AB|^{2}=h^{2}+r^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>B</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |AB|^{2}=h^{2}+r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66aa5c6a5bc92ee420f11562cda6f5c70128a83b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.677ex; height:3.343ex;" alt="{\displaystyle |AB|^{2}=h^{2}+r^{2}\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Lineáris_algebra"><span id="Line.C3.A1ris_algebra"></span>Lineáris algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=7" title="Szakasz szerkesztése: Lineáris algebra"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra">lineáris algebrában</a> vektorok egy halmaza kúp, ha zárt a <a href="/wiki/Negat%C3%ADv_%C3%A9s_nemnegat%C3%ADv_sz%C3%A1mok" title="Negatív és nemnegatív számok">nemnegatív számmal</a> való szorzásra. </p><p>Egy kúp végesen generált, ha minden pontja előáll véges sok vektor <a href="/wiki/Line%C3%A1ris_kombin%C3%A1ci%C3%B3" title="Lineáris kombináció">lineáris kombinációjaként</a>. Egy kúp metszetkúp, ha előáll véges sok féltér metszeteként. Ebből azonnal következik, hogy metszetkúp mindig <a href="/wiki/Konvex_halmaz" title="Konvex halmaz">konvex</a>. Megmutatható, hogy metszetkúp mindig generált kúp, továbbá ha egy végesen generált kúp konvex, akkor metszetkúp. </p> <div class="mw-heading mw-heading2"><h2 id="A_térfogat-_és_felszínképletek_bizonyítása"><span id="A_t.C3.A9rfogat-_.C3.A9s_felsz.C3.ADnk.C3.A9pletek_bizony.C3.ADt.C3.A1sa"></span>A térfogat- és felszínképletek bizonyítása</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=8" title="Szakasz szerkesztése: A térfogat- és felszínképletek bizonyítása"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Térfogat_2"><span id="T.C3.A9rfogat_2"></span>Térfogat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=9" title="Szakasz szerkesztése: Térfogat"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az elemi geometriában gyakran a Cavalieri-elvet használják: veszünk egy ugyanakkora alapterületű és magasságú <a href="/wiki/G%C3%BAla" title="Gúla">gúlát</a>. Az alappal párhuzamosan szeletelve a két testet középpontos hasonlósággal adódik, hogy az ugyanolyan magasságú szeletek területe egyenlő. Ezért a két test térfogata egyenlő. </p><p>A <i>T</i> alapterületű és <i>h</i> magasságú gúla térfogata </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 V={\frac {1}{3}}\cdot T\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>T</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}\cdot T\cdot h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33d962b1197daa004df0f53e5297e24f230bd59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.218ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}\cdot T\cdot h}"></span></dd></dl> <p>Ez alapján a kúp térfogata </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 V={\frac {1}{3}}\cdot r^{2}\cdot \pi \cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}\cdot r^{2}\cdot \pi \cdot h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64b9d3c7f92aeb124a8d5309449beb20e9ddb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.695ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}\cdot r^{2}\cdot \pi \cdot h}"></span>.</dd></dl> <p>A kúp alapterülete növekvő oldalszámú sokszögekkel is közelíthető. </p><p>Egy másik bizonyítás az <a href="/wiki/Integr%C3%A1lsz%C3%A1m%C3%ADt%C3%A1s" class="mw-redirect" title="Integrálszámítás">integrálszámítást</a> hívja segítségül. A <a href="/wiki/Koordin%C3%A1ta-rendszer" title="Koordináta-rendszer">derékszögű koordináta-rendszerben</a> a kúp csúcsát az origóba, és az alapkör középpontját a (<i>h</i>,0) pontba teszi. Ezután a kúpot, mint végtelen sok lapos, <i>dx</i> magasságú <a href="/wiki/Henger" title="Henger">hengerből</a> összetett forgástestet tekinti. A párhuzamos szelők tételével: </p><p>Egy infinitezimális henger sugara: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{Z}(h)={\frac {r}{h}}\cdot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>h</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{Z}(h)={\frac {r}{h}}\cdot x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad40ce171135eac7104f35d9c21c3c895ba03f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.9ex; height:4.843ex;" alt="{\displaystyle r_{Z}(h)={\frac {r}{h}}\cdot x}"></span></dd></dl> <p>Egy infinitezimális henger térfogata: </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 \left({\frac {r}{h}}\cdot x\right)^{2}\cdot \pi \cdot dx={\frac {r^{2}}{h^{2}}}\cdot \pi \cdot x^{2}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>h</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {r}{h}}\cdot x\right)^{2}\cdot \pi \cdot dx={\frac {r^{2}}{h^{2}}}\cdot \pi \cdot x^{2}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25bf5efba24246031cc5198d9431158c34d94331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.584ex; height:6.009ex;" alt="{\displaystyle \left({\frac {r}{h}}\cdot x\right)^{2}\cdot \pi \cdot dx={\frac {r^{2}}{h^{2}}}\cdot \pi \cdot x^{2}\,dx}"></span></dd></dl> <p>A forgáskúp térfogata megegyezik ezeknek a hengereknek a térfogatösszegével. Ezt határozott integrállal számítja ki, ahol a határok 0 és <i>h</i>: </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 V=\int _{0}^{h}{\frac {r^{2}}{h^{2}}}\cdot \pi \cdot x^{2}\,dx={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \int _{0}^{h}x^{2}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{0}^{h}{\frac {r^{2}}{h^{2}}}\cdot \pi \cdot x^{2}\,dx={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \int _{0}^{h}x^{2}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f787a9d731e6bbb873b6d0dd925984d97bec189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.227ex; height:6.343ex;" alt="{\displaystyle V=\int _{0}^{h}{\frac {r^{2}}{h^{2}}}\cdot \pi \cdot x^{2}\,dx={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \int _{0}^{h}x^{2}\,dx}"></span></dd></dl> <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 V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \left[{\frac {x^{3}}{3}}\right]_{0}^{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <msubsup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \left[{\frac {x^{3}}{3}}\right]_{0}^{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7377374fee9c62c25942f9a1641b6fe1b9fcff38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.369ex; height:6.843ex;" alt="{\displaystyle V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \left[{\frac {x^{3}}{3}}\right]_{0}^{h}}"></span></dd></dl> <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 V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \left({\frac {h^{3}}{3}}-{\frac {0^{3}}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \left({\frac {h^{3}}{3}}-{\frac {0^{3}}{3}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc98090dcb83ae88b198d5323a50c7f3b2b5d416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.059ex; height:6.343ex;" alt="{\displaystyle V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot \left({\frac {h^{3}}{3}}-{\frac {0^{3}}{3}}\right)}"></span></dd></dl> <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 V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot {\frac {h^{3}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot {\frac {h^{3}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e9db98880e71e6c03cdc66ed133ae84a72103c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.744ex; height:6.009ex;" alt="{\displaystyle V={\frac {r^{2}\cdot \pi }{h^{2}}}\cdot {\frac {h^{3}}{3}}}"></span></dd></dl> <p>Így jut az ismert </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 V={\frac {r^{2}\cdot \pi \cdot h}{3}}={\frac {1}{3}}\cdot r^{2}\cdot \pi \cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <mi>h</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {r^{2}\cdot \pi \cdot h}{3}}={\frac {1}{3}}\cdot r^{2}\cdot \pi \cdot h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2cbf4ea92f7acf53bc86de3436cc501441a05b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.762ex; height:5.676ex;" alt="{\displaystyle V={\frac {r^{2}\cdot \pi \cdot h}{3}}={\frac {1}{3}}\cdot r^{2}\cdot \pi \cdot h}"></span></dd></dl> <p>képlethez. </p> <div class="mw-heading mw-heading3"><h3 id="A_kúppalást_felszíne"><span id="A_k.C3.BAppal.C3.A1st_felsz.C3.ADne"></span>A kúppalást felszíne</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=10" title="Szakasz szerkesztése: A kúppalást felszíne"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az egyenes körkúp palástja görbült, de kiteríthető körcikké. Ennek sugara megegyezik a kúp alkotójának hosszával <i>(a)</i>. A körcikk α középponti szöge arányegyenlettel számítható: a középponti szög úgy aránylik a teljesszöghöz, mint az alapkör 2π<i>r</i> kerülete az <i>a</i> sugarú kör teljes kerületéhez: </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 \alpha :360^{\circ }=(2\pi r):(2\pi a)=r:a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>:</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :360^{\circ }=(2\pi r):(2\pi a)=r:a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc6eb0df40821ebad666eb4d159ad0c67f3afdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.202ex; height:2.843ex;" alt="{\displaystyle \alpha :360^{\circ }=(2\pi r):(2\pi a)=r:a}"></span></dd></dl> <p>ahol <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 {r^{2}+h^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {r^{2}+h^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41e3b2535eee34887603d7f98b967ee0ff9d6ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.989ex; height:3.509ex;" alt="{\displaystyle a={\sqrt {r^{2}+h^{2}}}}"></span> a kúp alkotója és a körcikk sugara. </p><p>A kúppalást felszíne eszerint a körcikk területképletéből adódóan </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_{P}={\frac {\alpha }{360^{\circ }}}\cdot a^{2}\cdot \pi ={\frac {r}{a}}\cdot a^{2}\cdot \pi =ra\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>a</mi> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>π</mi> <mo>=</mo> <mi>r</mi> <mi>a</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{P}={\frac {\alpha }{360^{\circ }}}\cdot a^{2}\cdot \pi ={\frac {r}{a}}\cdot a^{2}\cdot \pi =ra\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/166d3157c3366b52f0429a8732cf90eac0c97bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.508ex; height:4.843ex;" alt="{\displaystyle A_{P}={\frac {\alpha }{360^{\circ }}}\cdot a^{2}\cdot \pi ={\frac {r}{a}}\cdot a^{2}\cdot \pi =ra\pi }"></span></dd></dl> <p><br style="clear: both;" /> </p> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=11" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="book citation" style="font-style:normal" id="Reference-Hajós-1979"><a href="/wiki/Haj%C3%B3s_Gy%C3%B6rgy_(matematikus)" title="Hajós György (matematikus)">Hajós, György</a>. <i>Bevezetés a geometriába</i>, 6. kiadás, Budapest: Tankönyvkiadó (1979). <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9631747360" title="Speciális:Könyvforrások/9631747360">ISBN 9631747360</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bevezet%C3%A9s+a+geometri%C3%A1ba&rft.aulast=Haj%C3%B3s&rft.aufirst=Gy%C3%B6rgy&rft.date=1979&rft.edition=6.+kiad%C3%A1s&rft.pub=Tank%C3%B6nyvkiad%C3%B3&rft.place=Budapest&rft.isbn=9631747360"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Strohmajer János: Geometriai példatár II. Tankönyvkiadó, Budapest, 1982. 21. oldal 38-as feladat.</span> </li> </ol></div></div><div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"></div></div> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=12" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cs.elte.hu/~frank/jegyzet/opkut/ulin.2008.pdf">Frank András: Operációkutatás</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathsisfun.com/geometry/cone.html">Spinning Cone</a> from <a href="/w/index.php?title=Math_Is_Fun&action=edit&redlink=1" class="new" title="Math Is Fun (a lap nem létezik)">Math Is Fun</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090908060446/http://korthalsaltes.com/cone.htm">Paper model cone</a></li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/library/drmath/view/55017.html">Lateral surface area of an oblique cone</a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/GeneralizedCone.html">Generalized Cone</a> from <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com">Wolfram MathWorld</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Külső_hivatkozások"><span id="K.C3.BCls.C5.91_hivatkoz.C3.A1sok"></span>Külső hivatkozások</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=K%C3%BAp&action=edit&section=13" title="Szakasz szerkesztése: Külső hivatkozások"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation"><a href="/w/index.php?title=Eric_W._Weisstein&action=edit&redlink=1" class="new" title="Eric W. Weisstein (a lap nem létezik)">Weisstein, Eric W.</a>: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Cone.html"><i>Kúp</i></a> (angol nyelven). <a href="/wiki/MathWorld" title="MathWorld">Wolfram MathWorld</a></span></li> <li><span class="citation"><a href="/w/index.php?title=Eric_W._Weisstein&action=edit&redlink=1" class="new" title="Eric W. 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