Zylinder Pyramide Kegel/Der Satz von Cavalieri: Unterschied zwischen den Versionen

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Wenn es dir schwer fällt, dir das Ganze richtig vorzustellen, nimm dir vorne am Pult zwei der Bierdeckelstapel und stelle die einzelnen Situationen damit nach.}}
Wenn es dir schwer fällt, dir das Ganze richtig vorzustellen, nimm dir vorne am Pult zwei der Bierdeckelstapel und stelle die einzelnen Situationen damit nach.}}
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[https://ggbm.at/mpQGZVpe GeoGebra Applet "Volumenvergleich von Zylindern (1) + (2)"]


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showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "false" />
<br>
 
[https://ggbm.at/W9hf8qwn GeoGebra Applet "Volumenvergleich von Zylindern (3)"]
 
<br>


<br><br><br>
<span style="color:blue">'''Zurück zur Ausgangsfrage:'''</span> <br>
<span style="color:blue">'''Zurück zur Ausgangsfrage:'''</span> <br>


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}}
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" 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[https://ggbm.at/r6yKgffu GeoGebra Applet "Volumenvergleich von Prismen mit unterschiedlicher Grundflächenform"]
 
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{{Lösung versteckt|1=Es kommt nicht auf die Form der Grundfläche, sondern auf den Grundflächen'''inhalt''' an!<br>
{{Lösung versteckt|1=Es kommt nicht auf die Form der Grundfläche, sondern auf den Grundflächen'''inhalt''' an!<br>

Version vom 17. Oktober 2016, 12:54 Uhr

Vorlage:Lernpfad Inhalt und Drumherum

Zur Person

Bonaventura Cavalieri.jpeg Bonaventura Francesco Cavalieri (1598-1647) war ein italienischer Mathematiker und Astronom.

Im "Satz von Cavalieri" (auch "Prinzip von Cavalieri" genannt) geht es um die Volumengleichheit zweier Körper.


Erarbeitung des Satzes von Cavalieri


Zylinder gerade geschwungen.jpg

Peter: "Gib mir das rechte Glas, da passt mehr rein! Ich hab so einen Durst!"

Sandra: "So ein Quatsch! In die Gläser passt doch gleich viel!"



Vorlage:Arbeiten

Achtung

Zusätzliches Anschauungsmaterial zum Anfassen:

Wenn es dir schwer fällt, dir das Ganze richtig vorzustellen, nimm dir vorne am Pult zwei der Bierdeckelstapel und stelle die einzelnen Situationen damit nach.



GeoGebra Applet "Volumenvergleich von Zylindern (1) + (2)"


GeoGebra Applet "Volumenvergleich von Zylindern (3)"


Zurück zur Ausgangsfrage:

Wer hat nun Recht? Peter oder Sandra? Begründe deine Antwort!



Vorlage:Arbeiten

Du hast gerade Sachverhalte herausgearbeitet, welche Bonaventura Cavalieri in seinem berühmten (grundlegenden) Satz formuliert hat. Du kannst den Satz in deinem Schulbuch (Lambacher Schweizer, Ausgabe 2010) auf S. 22 oder Fokus Mathematik, Ausgabe 2016 auf S. 44 nachlesen und deine Aufzeichnungen - wenn nötig - ergänzen oder berichtigen!



Vorlage:Arbeiten


GeoGebra Applet "Volumenvergleich von Prismen mit unterschiedlicher Grundflächenform"



Es kommt nicht auf die Form der Grundfläche, sondern auf den Grundflächeninhalt an!

Bsp.: Ein Prisma mit quadratischer Grundflächen und ein Prisma mit dreieckiger Grundfläche haben das gleiche Volumen, wenn ihre Grundflächeninhalte, ihre Höhe und die zur Grundfläche parallelen Schnittflächen in gleicher Höhe gleich groß sind.




Übungsaufgaben


Vorlage:Arbeiten

Das schiefe Prisma besitzt das gleiche Volumen wie ein senkrechtes Prisma mit den angegebenen Maßen. Also:
Der geschwungene Zylinder besitzt das gleiche Volumen wie ein senkrechter Zylinder mit den angegebenen Maßen. Also:

Zu deiner Lösung gehört auch der Rechenweg!



Vorlage:Arbeiten