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Given a sphere with radius r, a cone with radius r and height 2r, and a cylinder with radius r and height 2r, the sum of the volume of the cone and sphere is equal to the volume of the cylinder. If we look at the volume formulas, this is obvious. However, any ordinary person without mathematical training probably wouldn't find this intuitive.

I recall reading in a museum exhibit that before proving anything, Archimedes was able to slice up the sphere and cone and fit the pieces together into the cylinder--all in his mind. Can someone explain how one can slice up the shapes to do that?

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    Volumes of cones and cylinders depenc on more than just the radius, so I'm having trouble making sense out of your first sentence.2012-02-07
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    http://www.cut-the-knot.org/pythagoras/Archimedes.shtml2012-02-07
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    @Gerry: I expect that the cone and cylinder have height equal to their base radius and to the radius of the sphere.2012-02-07
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    @Gerry: I believe Archimedes worked under the assumption that $h = 2r$.2012-02-07
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    Would you happen to know Cavalieri's principle, by any chance?2012-02-07
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    [Very related](http://math.stackexchange.com/questions/164/why-is-the-volume-of-a-sphere-frac43-pi-r3). The difference is this question doesn't presume we know the volume of a cylinder.2012-02-07
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    Thanks, pepsi, for adding the information about the heights.2012-02-07
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    "This question has not received enough attention." pepsi, have *you* given any attention to the links supplied by Donezo and anon? Maybe instead of protesting that the question has not received enough attention, you could let us know what you find unsatisfactory about those links.2012-06-27
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    @GerryMyerson 'This question has not received enough attention' are not my words--it was the option in the bounty form that fit best. I'm not protesting anything, merely trying to provide incentive for someone to answer the question. Neither of the links in the comments directly answered the question (which is probably why they were posted as comments).2012-06-27
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    To answer your question, the link to Archimedes' notes appears to use center of mass to prove equal volume. The original question asks if/how it's possible to do it *just* by slicing pieces from the cylinder and combining them to form the others, or vice versa. The reference to Cavalieri's principle seems promising, but it still remains to be shown how the principle can be applied.2012-06-27
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    [This](http://en.wikipedia.org/wiki/Cavalieri%27s_principle#Spheres)?2012-06-27

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I do not know how Archimedes did but I believe it was similiar way what I showed below. enter image description here

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Maybe this will explain things.