Main>Tanjaweis |
Main>Tanjaweis |
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| <ggb_applet width="1584" height="717" version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "false" />
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| | [https://ggbm.at/fuqnuF99 GeoGebra Applet "Gerade und schiefe Kegel"] |
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| ==Mantelfläche und Mantelflächeninhalt== | | ==Mantelfläche und Mantelflächeninhalt== |
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| <ggb_applet width="1586" height="717" version="4.0" 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" 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| | [https://ggbm.at/bPeH6yeB GeoGebra Applet "Volumenvergleich Pyramide und Kegel"] |
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Version vom 17. Oktober 2016, 13:34 Uhr
Vorlage:Lernpfad Inhalt und Drumherum
Der Kegel - Eine kleine Einführung
In der vorherigen Lerneinheit hast du die Pyramide mit einem beliebigen Vieleck als Grundfläche kennengelernt.
Ersetzt man nun das Vieleck der Grundfläche durch einen Kreis, so erhält man einen verwandten Spitzkörper: den Kegel!
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. . . . ![Turmspitze.jpg](/images/thumb/b/b6/Turmspitze.jpg/240px-Turmspitze.jpg)
Ob Eistüte, Pylonen oder Turmspitzen, man findet sehr häufig kegelförmige Objekte in unserer Lebenswelt.
Eigenschaften des Kegels
Vorlage:Arbeiten
GeoGebra Applet "Gerade und schiefe Kegel"
Mantelfläche und Mantelflächeninhalt
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Vorlage:Kasten grün
Oberfläche und Oberflächeninhalt
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Volumen des Kegels
Vorlage:Arbeiten
Vorlage:Arbeiten
GeoGebra Applet "Volumenvergleich Pyramide und Kegel"
Übungsaufgaben: Berechnungen rund um den Kegel
Vorlage:Arbeiten
Der Trichter ist ein Kegel. Zur Berechnung des Volumens benötigen wir den Radius r und die Höhe h des Kegels.
Die Bogenlänge b des Kreisausschnitts mit Radius s berechnet sich durch:
Die Bogenlänge b entspricht dem Umfang des Grundkreises des Kegels mit Radius r, also
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Die Höhe h wird über den Satz von Pythagoras berechnet (oben in der Abbildung kannst du das benötigte rechtwinklige Dreieck erkennen!):
(Hier könnte man jetzt noch teilweise die Wurzel ziehen! Also
)
Nun kann das Kegelvolumen berechnet werden:
Der Trichter hat ein Volumen von ungefähr 877,61 cm³, also weniger als ein Liter!
Vorlage:Arbeiten