A Cone Balancing a Cylinder

I am wanting to balance a cone with a cylinder, in the way that Archimedes did in `The Method' with other objects. The plan is to find the volume of the cone using the volume of the cylinder. However, I seem to be having trouble finding a relationship between the sections of the cone and the sections of the cylinder I am balancing it with.

I know that the centre of gravity of the cone is 1/3 of the way along it, and the centre of gravity of the cylinder is half way along it. Having thought about this problem for a while I was convinced that if we take some proportion then the sections to the left and to the right of the respective centres of gravity by this proportion would balance (so the sections at and of the cylinder would balance with the sections and of the cone).

I could not get this to work, and in retrospect my guess was clearly wrong (just looking at the extremities gives a contradiction!)

Does anyone have any ideas?

Note that as I know what the volume of a cone is, I was attempting to use a cylinder of radius . This should, I believe, work.