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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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    $\begingroup$ Volumes of cones and cylinders depenc on more than just the radius, so I'm having trouble making sense out of your first sentence. $\endgroup$ Feb 7, 2012 at 4:47
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    $\begingroup$ cut-the-knot.org/pythagoras/Archimedes.shtml $\endgroup$
    – JavaMan
    Feb 7, 2012 at 4:50
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    $\begingroup$ @Gerry: I believe Archimedes worked under the assumption that $h = 2r$. $\endgroup$
    – JavaMan
    Feb 7, 2012 at 4:54
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    $\begingroup$ Very related. The difference is this question doesn't presume we know the volume of a cylinder. $\endgroup$
    – anon
    Feb 7, 2012 at 5:01
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    $\begingroup$ This? $\endgroup$ Jun 27, 2012 at 17:55

2 Answers 2

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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.

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