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Another Demonstration of Cavalieri's Principle
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These two pyramids have the same base area and same height. Also, the areas of their cross-sections at every height are equal. According to Cavalieri's principle, these two pyramids have the same volume.
Students who have already learned the volume formula for pyramids may comment that they do not need to know anything about the cross-sections to determine that the volumes of the pyramids are equal. This would be a great opportunity to clarify that both shapes share the same volume algorithm (1/3 x base area x height) because the areas of their cross-sections are equal at every height. Moreover, it can be shown that such is true of the cross-sections of any two pyramids of the same base area and height. This could make for a worthwhile, challenging task related to Common Core standard G-SRT.5 for students who can handle it. Otherwise, a less-detailed explanation will suffice:
"We've already seen that two prisms with the same base area and height will automatically have cross-sections with equal areas at every height, due to properties of prisms. Similarly, properties of pyramids ensure that areas of corresponding cross-sections will be the same whenever their base areas and heights are the same."
That Cavalieri's principle can apply to cylinders may also be noted. This idea, should it come up, can be reinforced using the attached resource (which provides a similar illustration to that of the prior resource, this time using actual CDs). However, pyramids are highlighted here in order to clarify that it is not necessary for the cross-sections within a shape to have the same area in order for Cavalieri's principle to apply—only that the corresponding cross-sections of the shapes are the same at any height. With this in mind, ask,
In what way are cross-sections of pyramids different than cross-sections of prisms?
If more focus is needed, you might ask,
In the case of a prism, how do cross-sections that are parallel to the base compare to the base?
Students should respond that they are congruent. If necessary, refer back to slide 10.
Is the same true of cross-sections of a pyramid that are parallel to the base?
Upon gathering a "no" response, assert that the cross-sections do not have to be congruent to the bases in order for Cavalieri's principle to apply. Use the following slide to check for understanding of situations where Cavalieri's principle can be applied to compare volumes.
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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 |
| | | |
| 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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