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New Concept: Cavalieri's Principle
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Knowing that two figures have the same base area and height is not enough to determine that they have the same volume. Right prisms with these criteria, however, have another characteristic: their cross-sections at every height are equal in area. Having this additional characteristic does mean that the volumes are equal, according to the following principle.
Cavalieri's principle:
Two 3-dimensional figures have the same volume if
(1) they have the same height, and
(2) the areas of their cross-sections at every height are equal.
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This principle is related to a theorem that can be proven in higher mathematics. Cavalieri's principle does not need to be proven in order to be understood, however. The attached resource presents an example of Cavalieri's principle that you may have seen before.
Refer to the attached resource, which demonstrates Cavalieri's principle using CD cases. Since right prisms have been the focus up to this point, it is important for students to understand that in general prisms of the same base area and height have the same volume, even if one or both are oblique.
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Resources
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RadEditor - HTML WYSIWYG Editor. MS Word-like content editing experience thanks to a rich set of formatting tools, dropdowns, dialogs, system modules and built-in spell-check.
| RadEditor's components - toolbar, content area, modes and modules |
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| Toolbar's wrapper | | | | | |
| Content area wrapper | |
| RadEditor's bottom area: Design, Html and Preview modes, Statistics module and resize handle. |
It contains RadEditor's Modes/views (HTML, Design and Preview), Statistics and Resizer
| Editor Mode buttons | Statistics module | Editor resizer |
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| RadEditor's Modules - special tools used to provide extra information such as Tag Inspector, Real Time HTML Viewer, Tag Properties and other. | |
| | | |
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RadEditor - HTML WYSIWYG Editor. MS Word-like content editing experience thanks to a rich set of formatting tools, dropdowns, dialogs, system modules and built-in spell-check.
| RadEditor's components - toolbar, content area, modes and modules |
| | | |
| Toolbar's wrapper | | | | | |
| Content area wrapper | |
| RadEditor's bottom area: Design, Html and Preview modes, Statistics module and resize handle. |
It contains RadEditor's Modes/views (HTML, Design and Preview), Statistics and Resizer
| Editor Mode buttons | Statistics module | Editor resizer |
| |
|
|
| RadEditor's Modules - special tools used to provide extra information such as Tag Inspector, Real Time HTML Viewer, Tag Properties and other. | |
| | | |
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