Finding the volume of a cone by cross sections looked difficult to me at first.  Then I starting thinking about calculus and how you might be able to use some calculus in trying to estimate the volume.  If you partition the cone into 2, 4 or more pieces, you could integrate the pieces.  Which essentially is what integration is as it takes small pieces of area and adds them together.  I would try to make horizontal cross sections through the cone, with a known height and radius.  Integrate each part and add up the pieces to get an estimation.  I’m sure there are other ways, but that is what came to mind first.  

I played with Cross Section Flyer and I tried to come up with a general solution different from what I had in my mind (cone is one third of a cylinder)using portioning the cone into many circles which are close to each other and then use the formula for cylinder to estimate the volume of the cone. I think based on the numbers of circles (or small cylinders inside the cone) the estimation can be more or less accurate. With more cylinders we have less free space between cone and cylinders.

Is there a partition into 2 or 4 that might be more accurate than another?

I’m thinking about the volume of a cone is made up of numerous different 2-D circles.

As the picture shows, I make the slice shape a circle and find its area as r² (r ranges from 0 to the radius (R) of the bottom circle), then the volume of the cone is (0² + 1² + 2² +……..R²) (here we only use integer radius to estimate the cone volume).

I think Benito’s idea is really good. When teaching students the measurement of cone volume, the question—how does a cone form? –should be emphasized. But sorry I don’t quite understand how Benito gets to 5(2r)h. Could you explain more about it?