{"id":8739,"date":"2024-07-27T13:58:42","date_gmt":"2024-07-27T11:58:42","guid":{"rendered":"https:\/\/mathe-lernen.net\/?p=8739"},"modified":"2024-07-28T11:37:10","modified_gmt":"2024-07-28T09:37:10","slug":"zusammengesetzte-koerper","status":"publish","type":"post","link":"https:\/\/mathe-lernen.net\/?p=8739","title":{"rendered":"Zusammengesetzte K\u00f6rper"},"content":{"rendered":"\n        <div class=\"modelViewerBlock\" data-attributes='{&quot;align&quot;:&quot;center&quot;,&quot;uniqueId&quot;:&quot;model8736&quot;,&quot;currentViewer&quot;:&quot;O3DViewer&quot;,&quot;multiple&quot;:false,&quot;O3DVSettings&quot;:{&quot;isFullscreen&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;camera&quot;:null,&quot;mouseControl&quot;:true},&quot;environmentImage&quot;:null,&quot;lazyLoad&quot;:false,&quot;loading&quot;:&quot;auto&quot;,&quot;autoplay&quot;:false,&quot;shadow&quot;:&quot;1&quot;,&quot;autoRotate&quot;:false,&quot;zoomLevel&quot;:null,&quot;zoom&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;hotspotStyle&quot;:&quot;style-1&quot;,&quot;preload&quot;:&quot;auto&quot;,&quot;rotationPerSecond&quot;:&quot;30&quot;,&quot;mouseControl&quot;:true,&quot;lockXAxisRotation&quot;:false,&quot;lockYAxisRotation&quot;:false,&quot;fullscreen&quot;:true,&quot;zoomInOutBtn&quot;:false,&quot;cameraBtn&quot;:false,&quot;variant&quot;:false,&quot;loadingPercentage&quot;:false,&quot;progressBar&quot;:true,&quot;rotate&quot;:false,&quot;rotateDelay&quot;:3000,&quot;rotateAlongX&quot;:&quot;0&quot;,&quot;rotateAlongY&quot;:&quot;75&quot;,&quot;exposure&quot;:&quot;1&quot;,&quot;stylesheet&quot;:null,&quot;additional&quot;:{&quot;ID&quot;:&quot;&quot;,&quot;Class&quot;:&quot;&quot;,&quot;CSS&quot;:null},&quot;animation&quot;:false,&quot;woo&quot;:false,&quot;selectedAnimation&quot;:&quot;&quot;,&quot;placement&quot;:&quot;shortcode&quot;,&quot;styles&quot;:{&quot;width&quot;:&quot;100%&quot;,&quot;height&quot;:&quot;320px&quot;,&quot;bgColor&quot;:&quot;#474747&quot;,&quot;progressBarColor&quot;:null},&quot;model&quot;:{&quot;modelUrl&quot;:&quot;https:\\\/\\\/mathe-lernen.net\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/zus9.glb&quot;,&quot;poster&quot;:&quot;&quot;},&quot;models&quot;:[]}'>\n            <div class=\"bp3d_backup_view\" style=\"display: none;height:350px;\">\n                    <model-viewer camera-controls src=\"https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/zus9.glb\" style=\"height: 350px;\"><\/model-viewer>\n            <\/div>\n            <script>\n                setTimeout(() => {\n                    let backupModels = document.querySelectorAll('.bp3d_backup_view');\n                    if(backupModels.length > 0){\n                        backupModels.forEach(element => {\n                            if(element){\n                                element.style.display = 'block';\n                                setTimeout(() => {\n                                    let adminMessages = document.querySelectorAll('.bp3d_admin_message');\n                                    if(adminMessages.length > 0){\n                                        adminMessages.forEach(adminMessage => {\n                                            if(adminMessage){\n                                                adminMessage.style.display = 'block';\n                                            }\n                                        });\n                                    }\n                                }, 5000);\n                            }\n                        });\n                    }\n                }, 5000);\n            <\/script>\n        <\/div>\n\n\n\n\n\n<p>Wir unterscheiden <strong>Additionsk\u00f6rper<\/strong>, bei denen sich die Einzelvolumina zum Gesamtvolumen addieren lassen<br>und <strong>Subtraktionsk\u00f6rper<\/strong>, bei denen durch Schnitte, Bohrungen, Fr\u00e4sungen o.\u00e4. Teile vom Volumen eines K\u00f6rpers<br>verloren gehen, also subtrahiert werden. Kombinationen davon existieren ebenfalls.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"292\" src=\"https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2-1024x292.png\" alt=\"\" class=\"wp-image-8741\" srcset=\"https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2-1024x292.png 1024w, https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2-300x86.png 300w, https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2-768x219.png 768w, https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2-1536x438.png 1536w, https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2-200x57.png 200w, https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/image-2.png 1669w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<ol class=\"has-light-gray-background-color has-background wp-block-list\">\n<li>Einzelvolumen der Teilk\u00f6rper berechnen<\/li>\n\n\n\n<li>zum Gesamtvolumen addieren oder entsprechend subtrahieren<\/li>\n<\/ol>\n\n\n\n<p class=\"has-white-color has-primary-background-color has-text-color has-background has-link-color wp-elements-854784249978e8190941ebbc7665abed\">Oberfl\u00e4chenberechnung<\/p>\n\n\n\n<p>Die Oberfl\u00e4chenberechnung von zusammengesetzten K\u00f6rpern geht von dem Fakt aus, dass alles, was von au\u00dfen zu sehen oder einzuf\u00e4rben ist zur Oberfl\u00e4che des K\u00f6rpers geh\u00f6rt. Ausnahme bildet hier die Hohlkugel und vergleichbare K\u00f6rper mit einer zu betrachtenden \u201eInnenfl\u00e4che\u201c.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#91e4fe\"><strong>Hilfreich ist hier immer die Darstellung des K\u00f6rpers als &#8222;Netzskizze&#8220;!<\/strong><\/p>\n\n\n\n<p>Im Beispiel sieht man, dass die beiden K\u00f6rper \u00fcber kongruente Einzel-Fl\u00e4chen verf\u00fcgen, die ihre Oberfl\u00e4che bilden.<br><br>1 &#8211; Grundfl\u00e4che &#8211; Kreis(violett) , <br>2 &#8211; Zylinder-Mantel &#8211; Rechteck(blau) <br>3 &#8211; Kegelmantel &#8211; Kreissektor (orange)<\/p>\n\n\n\n<p>Addiert man diese 3 Fl\u00e4chen, entsteht so in diesem Fall der Oberfl\u00e4cheninhalt.<\/p>\n\n\n\n        <div class=\"modelViewerBlock\" data-attributes='{&quot;align&quot;:&quot;center&quot;,&quot;uniqueId&quot;:&quot;model8742&quot;,&quot;currentViewer&quot;:&quot;O3DViewer&quot;,&quot;multiple&quot;:false,&quot;O3DVSettings&quot;:{&quot;isFullscreen&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;camera&quot;:null,&quot;mouseControl&quot;:true},&quot;environmentImage&quot;:null,&quot;lazyLoad&quot;:false,&quot;loading&quot;:&quot;auto&quot;,&quot;autoplay&quot;:false,&quot;shadow&quot;:&quot;1&quot;,&quot;autoRotate&quot;:false,&quot;zoomLevel&quot;:null,&quot;zoom&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;hotspotStyle&quot;:&quot;style-1&quot;,&quot;preload&quot;:&quot;auto&quot;,&quot;rotationPerSecond&quot;:&quot;30&quot;,&quot;mouseControl&quot;:true,&quot;lockXAxisRotation&quot;:false,&quot;lockYAxisRotation&quot;:false,&quot;fullscreen&quot;:true,&quot;zoomInOutBtn&quot;:false,&quot;cameraBtn&quot;:false,&quot;variant&quot;:false,&quot;loadingPercentage&quot;:false,&quot;progressBar&quot;:true,&quot;rotate&quot;:false,&quot;rotateDelay&quot;:3000,&quot;rotateAlongX&quot;:&quot;0&quot;,&quot;rotateAlongY&quot;:&quot;75&quot;,&quot;exposure&quot;:&quot;1&quot;,&quot;stylesheet&quot;:null,&quot;additional&quot;:{&quot;ID&quot;:&quot;&quot;,&quot;Class&quot;:&quot;&quot;,&quot;CSS&quot;:null},&quot;animation&quot;:false,&quot;woo&quot;:false,&quot;selectedAnimation&quot;:&quot;&quot;,&quot;placement&quot;:&quot;shortcode&quot;,&quot;styles&quot;:{&quot;width&quot;:&quot;100%&quot;,&quot;height&quot;:&quot;320px&quot;,&quot;bgColor&quot;:&quot;#565656&quot;,&quot;progressBarColor&quot;:null},&quot;model&quot;:{&quot;modelUrl&quot;:&quot;https:\\\/\\\/mathe-lernen.net\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/zus9c.glb&quot;,&quot;poster&quot;:&quot;&quot;},&quot;models&quot;:[]}'>\n            <div class=\"bp3d_backup_view\" style=\"display: none;height:350px;\">\n                    <model-viewer camera-controls src=\"https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/zus9c.glb\" style=\"height: 350px;\"><\/model-viewer>\n            <\/div>\n            <script>\n                setTimeout(() => {\n                    let backupModels = document.querySelectorAll('.bp3d_backup_view');\n                    if(backupModels.length > 0){\n                        backupModels.forEach(element => {\n                            if(element){\n                                element.style.display = 'block';\n                                setTimeout(() => {\n                                    let adminMessages = document.querySelectorAll('.bp3d_admin_message');\n                                    if(adminMessages.length > 0){\n                                        adminMessages.forEach(adminMessage => {\n                                            if(adminMessage){\n                                                adminMessage.style.display = 'block';\n                                            }\n                                        });\n                                    }\n                                }, 5000);\n                            }\n                        });\n                    }\n                }, 5000);\n            <\/script>\n        <\/div>\n\n\n\n\n\n<p class=\"has-text-align-left\">\u00dcber jeden zusammengesetzten K\u00f6rper muss man individuell nachdenken, um die Oberfl\u00e4che, also die Art und Anzahl seiner Au\u00dfenfl\u00e4chen, zu ermitteln.<br>Ein einfaches Addieren oder Subtrahieren der Oberfl\u00e4chen der K\u00f6rper mit <br>\\( Ao _{gesamt} \\) = \\( Ao _{Zylinder} \\) \u00b1 \\(Ao_{Kegel} \\) hilft hier nicht oder nur bedingt weiter!<\/p>\n\n\n\n<p class=\"has-white-color has-primary-background-color has-text-color has-background has-link-color wp-elements-fc34c8a735361686009db25617f6dd66\">Klebefl\u00e4chen<\/p>\n\n\n\n<p>Achtung \u201e<strong>Klebefl\u00e4chen<\/strong>\u201c: Beim Zusammensetzen der K\u00f6rper \u2026verschwinden Fl\u00e4chen im Inneren und z\u00e4hlen somit nicht mehr zur &#8222;Oberfl\u00e4chen&#8220;, m\u00fcssen also eventuell abgezogen werden.<\/p>\n\n\n\n<p>Die 3 K\u00f6rper besitzen einzeln betrachtet 6 Fl\u00e4chen, &#8222;zusammengeklebt&#8220; sind nur noch 3 davon sichtbar.<\/p>\n\n\n\n        <div class=\"modelViewerBlock\" data-attributes='{&quot;align&quot;:&quot;center&quot;,&quot;uniqueId&quot;:&quot;model8753&quot;,&quot;currentViewer&quot;:&quot;O3DViewer&quot;,&quot;multiple&quot;:false,&quot;O3DVSettings&quot;:{&quot;isFullscreen&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;camera&quot;:null,&quot;mouseControl&quot;:true},&quot;environmentImage&quot;:null,&quot;lazyLoad&quot;:false,&quot;loading&quot;:&quot;auto&quot;,&quot;autoplay&quot;:false,&quot;shadow&quot;:&quot;1&quot;,&quot;autoRotate&quot;:false,&quot;zoomLevel&quot;:null,&quot;zoom&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;hotspotStyle&quot;:&quot;style-1&quot;,&quot;preload&quot;:&quot;auto&quot;,&quot;rotationPerSecond&quot;:&quot;30&quot;,&quot;mouseControl&quot;:true,&quot;lockXAxisRotation&quot;:false,&quot;lockYAxisRotation&quot;:false,&quot;fullscreen&quot;:true,&quot;zoomInOutBtn&quot;:false,&quot;cameraBtn&quot;:false,&quot;variant&quot;:false,&quot;loadingPercentage&quot;:false,&quot;progressBar&quot;:true,&quot;rotate&quot;:false,&quot;rotateDelay&quot;:3000,&quot;rotateAlongX&quot;:&quot;0&quot;,&quot;rotateAlongY&quot;:&quot;75&quot;,&quot;exposure&quot;:&quot;1&quot;,&quot;stylesheet&quot;:null,&quot;additional&quot;:{&quot;ID&quot;:&quot;&quot;,&quot;Class&quot;:&quot;&quot;,&quot;CSS&quot;:null},&quot;animation&quot;:false,&quot;woo&quot;:false,&quot;selectedAnimation&quot;:&quot;&quot;,&quot;placement&quot;:&quot;shortcode&quot;,&quot;styles&quot;:{&quot;width&quot;:&quot;100%&quot;,&quot;height&quot;:&quot;320px&quot;,&quot;bgColor&quot;:&quot;#545454&quot;,&quot;progressBarColor&quot;:null},&quot;model&quot;:{&quot;modelUrl&quot;:&quot;https:\\\/\\\/mathe-lernen.net\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/zus14xx.glb&quot;,&quot;poster&quot;:&quot;&quot;},&quot;models&quot;:[]}'>\n            <div class=\"bp3d_backup_view\" style=\"display: none;height:350px;\">\n                    <model-viewer camera-controls src=\"https:\/\/mathe-lernen.net\/wp-content\/uploads\/2024\/07\/zus14xx.glb\" style=\"height: 350px;\"><\/model-viewer>\n            <\/div>\n            <script>\n                setTimeout(() => {\n                    let backupModels = document.querySelectorAll('.bp3d_backup_view');\n                    if(backupModels.length > 0){\n                        backupModels.forEach(element => {\n                            if(element){\n                                element.style.display = 'block';\n                                setTimeout(() => {\n                                    let adminMessages = document.querySelectorAll('.bp3d_admin_message');\n                                    if(adminMessages.length > 0){\n                                        adminMessages.forEach(adminMessage => {\n                                            if(adminMessage){\n                                                adminMessage.style.display = 'block';\n                                            }\n                                        });\n                                    }\n                                }, 5000);\n                            }\n                        });\n                    }\n                }, 5000);\n            <\/script>\n        <\/div>\n\n\n\n\n\n<p class=\"has-white-color has-primary-background-color has-text-color has-background has-link-color wp-elements-7247316f8f6daafbe37ffd43fc2ad107\">Bohrungen, Fr\u00e4sungen, Schnitte<\/p>\n\n\n\n<p>\u201e<strong>Bohrungen<\/strong>\u201c, &#8222;<strong>Fr\u00e4sungen<\/strong>&#8220; und &#8222;<strong>K\u00f6rperschnitte<\/strong>&#8220; entfernen Volumen, k\u00f6nnen aber auch zus\u00e4tzliche Oberfl\u00e4che schaffen.<\/p>\n\n\n\n<p>Hier ist zu sehen, wie neue Fl\u00e4chen entstehen (orange) und k\u00f6rpereigene Fl\u00e4chen entfernt (gr\u00fcn, blau) werden .<\/p>\n\n\n\n        <div class=\"modelViewerBlock\" data-attributes='{&quot;align&quot;:&quot;center&quot;,&quot;uniqueId&quot;:&quot;model8750&quot;,&quot;currentViewer&quot;:&quot;O3DViewer&quot;,&quot;multiple&quot;:false,&quot;O3DVSettings&quot;:{&quot;isFullscreen&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;camera&quot;:null,&quot;mouseControl&quot;:true},&quot;environmentImage&quot;:null,&quot;lazyLoad&quot;:false,&quot;loading&quot;:&quot;auto&quot;,&quot;autoplay&quot;:false,&quot;shadow&quot;:&quot;1&quot;,&quot;autoRotate&quot;:false,&quot;zoomLevel&quot;:null,&quot;zoom&quot;:true,&quot;isPagination&quot;:false,&quot;isNavigation&quot;:false,&quot;hotspotStyle&quot;:&quot;style-1&quot;,&quot;preload&quot;:&quot;auto&quot;,&quot;rotationPerSecond&quot;:&quot;30&quot;,&quot;mouseControl&quot;:true,&quot;lockXAxisRotation&quot;:false,&quot;lockYAxisRotation&quot;:false,&quot;fullscreen&quot;:true,&quot;zoomInOutBtn&quot;:false,&quot;cameraBtn&quot;:false,&quot;variant&quot;:false,&quot;loadingPercentage&quot;:false,&quot;progressBar&quot;:true,&quot;rotate&quot;:false,&quot;rotateDelay&quot;:3000,&quot;rotateAlongX&quot;:&quot;0&quot;,&quot;rotateAlongY&quot;:&quot;75&quot;,&quot;exposure&quot;:&quot;1&quot;,&quot;stylesheet&quot;:null,&quot;additional&quot;:{&quot;ID&quot;:&quot;&quot;,&quot;Class&quot;:&quot;&quot;,&quot;CSS&quot;:null},&quot;animation&quot;:false,&quot;woo&quot;:false,&quot;selectedAnimation&quot;:&quot;&quot;,&quot;placement&quot;:&quot;shortcode&quot;,&quot;styles&quot;:{&quot;width&quot;:&quot;100%&quot;,&quot;height&quot;:&quot;320px&quot;,&quot;bgColor&quot;:&quot;#686868&quot;,&quot;progressBarColor&quot;:null},&quot;model&quot;:{&quot;modelUrl&quot;:&quot;https:\\\/\\\/mathe-lernen.net\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/zus4abc.glb&quot;,&quot;poster&quot;:&quot;&quot;},&quot;models&quot;:[]}'>\n 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if(adminMessage){\n                                                adminMessage.style.display = 'block';\n                                            }\n                                        });\n                                    }\n                                }, 5000);\n                            }\n                        });\n                    }\n                }, 5000);\n            <\/script>\n        <\/div>\n\n\n<div class=\"pvc_clear\"><\/div><p id=\"pvc_stats_8739\" class=\"pvc_stats all  \" data-element-id=\"8739\" style=\"\"><i class=\"pvc-stats-icon small\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 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Schnitte, Bohrungen, Fr\u00e4sungen o.\u00e4. Teile vom Volumen eines K\u00f6rpersverloren gehen, also subtrahiert werden. Kombinationen davon existieren ebenfalls. Oberfl\u00e4chenberechnung Die Oberfl\u00e4chenberechnung von zusammengesetzten K\u00f6rpern geht von dem Fakt aus, dass alles, was von au\u00dfen zu sehen oder einzuf\u00e4rben ist zur Oberfl\u00e4che des K\u00f6rpers geh\u00f6rt. 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