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4. Rund um den Kegel: Unterschied zwischen den Versionen

aus ZUM-Wiki, dem Wiki für Lehr- und Lerninhalte auf ZUM.de
Wechseln zu: Navigation, Suche
(Oberfläche und Oberflächeninhalt)
(Volumen des Kegels)
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{{Arbeiten|NUMMER=6|ARBEIT=
 
{{Arbeiten|NUMMER=6|ARBEIT=
 
'''Herleitung des Kegelvolumens'''<br>
 
'''Herleitung des Kegelvolumens'''<br>
Beweise, dass ein Kegel und eine Pyramide mit gleichem Grundflächeninhalt und gleicher Höhe auch das gleiche Volumen besitzen! <br>
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Beweise, dass ein Kegel und eine Pyramide mit gleichem Grundflächeninhalt und gleicher Höhe auch gleiches Volumen besitzen! <br>
Schaue dir dazu zunächst das Geogebra-Applet an...
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Nutze dazu auch das folgende Geogebra-Applet, bei dem du dich im ersten Schritt anschaulich von der Richtigkeit der Aussage überzeugen kannst. Schreibe anschließend einen allgemeingültigen Beweis auf.
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:{{Lösung versteckt|1=
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Der Beweis kann analog zu dem Beweis aus Aufgabe 5 der Lerneinheit "Rund um die Pyramide" geführt werden (Volumenvergleich zweier Pyramiden mit gleichem Grundflächeninhalt und gleicher Höhe)! '''Stichwort: Zentrische Streckung!'''
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}}
 
}}
 
}}
 
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" 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Version vom 15. November 2012, 19:14 Uhr

Inhaltsverzeichnis

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!


Eistüte umgedreht.jpg . . . .Kegel Pylon.jpg. . . . DSC04737 Istanbul - La Moschea Blu - Minareti - Foto G. Dall'Orto 29-5-2006.jpg. . . . Turmspitze.jpg

Ob Eistüte, Pylonen oder Turmspitzen, man findet sehr häufig kegelförmige Objekte in unserer Lebenswelt.



Eigenschaften des Kegels


Vorlage:Arbeiten







Mantelfläche und Mantelflächeninhalt


Vorlage:Arbeiten

Vorlage:Arbeiten

Vorlage:Arbeiten



Oberfläche und Oberflächeninhalt

Vorlage:Arbeiten



Volumen des Kegels


Vorlage:Arbeiten

Vorlage:Arbeiten