I realize there are several solutions for calculus-based proofs, but how would you explain the formula for the volume of a cone to middle school students?

Please do not give the un-rigorous "imagine filling up a cylinder with N cones... you got it, that's 3 cones!"

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    $\begingroup$ You might want to check this link out: math.stackexchange.com/questions/623/… I'm not sure if the first answer is rigorous enough for you though. $\endgroup$ – cheeseinthetrap Apr 10 '15 at 3:26
  • $\begingroup$ Middle school is not the place for mathematical rigor. Also, you don't have to imagine anything; you can actually make the experiment with a conical paper cup and a bucket of water. $\endgroup$ – Wildcard Jan 5 '17 at 5:09
  • $\begingroup$ A rigorous proof would require a somewhat rigorous definition of volume. $\endgroup$ – Henricus V. Jan 5 '17 at 5:15
  • $\begingroup$ Why the downvote? $\endgroup$ – ina Jan 5 '17 at 5:56
  • $\begingroup$ @ina, because of the conflict within the question that you appear to be asking for a rigorous explanation for middle school students. I'll suggest an edit. $\endgroup$ – Wildcard Jan 5 '17 at 23:19

It would help if you can explain Archimedes' method to them. The basic principle is, if you have two three-dimensional objects ($A$ and $B$) and a reference plane, and for every plane parallel to the reference plane, the cross-section of $A$ in that plane has area exactly $k$ times the cross-section of $B$ in the same plane, then $A$ has $k$ times the volume of $B$.

You can partition a cube into three square pyramids, so each pyramid's volume is $1/3$ the product of the base area times the height. If you start with a cube of the same height as your right circular cone, Archimedes' method shows that the ratio of the volume of the cone to the volume of one of the square pyramids is the ratio of the circular base of the cone to the square base of the pyramid. Follow through on this and you find that the volume of the cone is also $1/3$ the product of the base area times the height.

Update: This answer, mentioned in a comment, says much the same thing, but more thoroughly and with good diagrams. Too bad I didn't find that link in my own search, but there it is anyway.

  • $\begingroup$ Unfortunately, there is no finite way to prove this by slicing/partitioning for pyramids and cones - Hilbert's third problem. en.wikipedia.org/wiki/Hilbert%27s_third_problem $\endgroup$ – ina Apr 10 '15 at 4:02
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    $\begingroup$ That's a different problem entirely. The methods of Archimedes or Cavalieri are not dissections, and they are not finite partitions. They might also not be considered rigorous proofs nowadays, but you asked for a way to explain the formula to third-graders. I don't think third-graders are ready for mathematical rigor. (The question is whether they are even ready for Archimedes; the method is a precursor to integral calculus.) $\endgroup$ – David K Apr 10 '15 at 4:24

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