Deriving the formula for volume of a cone - middle school 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!" 
 A: 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.
