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What Cavalieri's Principle Does NOT Say
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Cavalieri's principle says that if these things are true...
| The figures have the same height. |
| The cross-sections of the figures at every height have the same area. |
then the figures have the same volume.
It does NOT say that if two figures have the same volume, then these things are true:
| The figures have the same height. |
| The cross-sections of the figures at every height have the same area. |
This could lead to an important discussion about the difference between "if" statements and "if and only if" statements, especially if the state framework happens to include this topic. Otherwise, conversations of this nature could be counted as addressing Mathematical Practice standard 6: Attend to Precision.
After making the following point about what Cavalieri's principle does not say, present the challenge below.
Cavalieri's principle does not necessarily apply to all figures, even if the figures have the same volume.
Prove it!
Draw and mark the measurements of two figures that have the same volume, but do not have one or both of the following:
- same height
- same cross-sectional area at every height
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Student responses will vary. If students struggle to make sense of the question, pointing out that choice A from the previous slide presents such a pair of figures may be helpful.
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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.
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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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