Cross-sections of a cone inscribed in a cylinder

Question

The Problem:

Consider a right circular cone inscribed within a right circular cylinder. In any cross-section of the cylinder which passes through its axis, 50% of the area of the resultant rectangle lies within the triangular cross-section of the cone, as in this image:

I have seen and been convinced by many proofs for the fact that such a cone occupies precisely 1/3 the volume of the cylinder.

However, this conflicts with my (deeply amateur) intuition around this cross-section. Clearly, all the infinitely-many cross sections that could be taken revolving around the axis would be identical, with 50% of their area within the cone, this leads to the feeling that the cone occupies half the volume of the cylinder.

The Question:

Why is this a bad approach to this problem, and how is my intuition leading me astray?

I don't need to be convinced that the cone occupies 1/3 of the volume of the cylinder, I just want to understand where and why my approach falls down.

Some caveats/thoughts:

I am (surprise surprise) not a mathematician, although I am an enthusiast! I'm not great with equations and formulae, so a literary answer would be appreciated (if at all possible).

I am most interested in set and number theories, and these have doubtless guided my intuition on this problem.

I understand calculus' approach to this problem using disc integration.

I sense this misunderstanding relates to the idea that my suggested cross-sections are somehow "overlapping" closer to the axis, but they are two-dimensional, so how could that be? I can see that every cross section would contain all the points along the cylinder's axis, but that accounts for almost no elements of the set of points contained within each cross-section of the cone.

Is this a mistake of trying to intuit three-dimensional volume using only two-dimensional cross sections? Is this just infinity stuff being unhelpful again? Help me, Stack Exchange!

Answer 1

Imagine first dividing the cylinder into many thin circular slices, and then cutting up each circular slice as shown in this diagram:

The result is that the cylinder is divided into many tiny pieces of different sizes with some straight sides and some slightly rounded sides.

On every plane through the central axis, the cone's cross-section is half of the cylinder's cross-section. This is essentially the same as the observation that approximately half the tiny pieces will be contained in the cone. (The approximation gets better and better as the pieces get smaller and smaller.)

However, the planes through the central axis can't "see" that the outer tiny pieces are bigger than the inner tiny pieces. (Every tiny piece has an identical cross-section.) The cone grabs many of the inner pieces, and few of the outer pieces. So even though it contains half of the tiny pieces, those pieces only add up to a third of the volume.

Answer 2

For a concrete example, let's suppose the cylinder has height $1$ and radius $1.$

When you are taking your cross sections along the axis and using them to estimate the relative volumes of the cone and cylinder, you are assuming that all parts of the cross-section are equally representative of the volumes of the cone and cylinder.

This is not accurate. The area of the cross-section within distance $0.01$ of the axis (a strip of width $0.02$ down the middle of the rectangle) is equal to the area of the cross-section within distance $0.01$ of the sides parallel to the axis (a pair of strips of width $0.01$ running down the edges of the rectangle). But the volume of the cone within $0.01$ of the axis is much smaller than the volume of the cone within $0.01$ of the outer curved surface of the cylinder.

A more representative "cross-section" would be a wedge with a (small but non-zero) thickness at the outer curved surface of the cylinder, tapering down to a sharp (zero thickness) edge at the axis. This wedge can be approximated by a polyhedron in which the portion occupied by the cone is a tetrahedron. The tetrahedron can be shown to have $1/3$ the volume of the wedge.