CDs and Cavalieri’s Principle
I finally got around to shedding my library of CD cases (I know, I’m quite behind), but it got me thinking about Cavalieri’s Principle.
Cavalieri’s Principle essentially states that if two prisms have the property that all corresponding cross sections have the same area, then those prisms have the same volume.
For instance, here we have two stacks of CD cases. Every cross-section of each “prism” here is a single CD case. Since corresponding cross-sections always have equal area, Cavalieri’s Principle tells us that these prisms have equal volume, even though one of the stacks is oblique.
That the stacks have equal volume is made clearer with a simple transformation of the stack on the right.
Here’s another demonstration of Cavalieri’s Principle using the CDs themselves.
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4 Comments
Roy · April 8, 2011 at 12:25 pm
Would this also apply to cylinders?
MrHonner · April 8, 2011 at 12:51 pm
Yes. Cavalieri’s Principle doesn’t make any [unusual] demands of the cross-sectional shape; it’s all about the area.
For a visual demonstration with cylinders, check back next week!
Andy Huynh · April 8, 2011 at 1:02 pm
It doesn’t have to be slant either. It could be crooked but as long as the cross section stays the same, it has the same volume. The easiest way to do this at home is with a stack of coins.
MrHonner · April 8, 2011 at 2:47 pm
Or a stack of CD’s that you pulled out of their boxes. 🙂
You can also think of Cavalieri’s principle in two dimensions by comparing a rectangle and a parallelogram with equal bases and heights. Andy’s point holds here, too: you could also throw in a shape whose with the same base, but sides are [equally] curvy.