How to prove that a torus has the same volume as a cylinder (with the height equal to the torus' perimeter)

Solution 1:

(i) Slice the torus into a million near-disks.
(ii) Rotate every second disk through $180^\circ$.
(iii) Stick them all together again.
You get a near-cylinder whose height is nearly $2\pi b$. Now let a million tend to infinity.

Solution 2:

To use Cavalieri's theorem, lay the torus and cylinder on a table and slice them with planes parallel to the table. Then it suffices to show that the torus slices (annuli) have the same area as the cylinder slices (rectangles).

At height $h$ (measured from the centre of the torus or cylinder), the annulus has inner and outer radii $$R_\pm = |b| \pm \sqrt{r^2-h^2},$$ so its area is $$A_1 = \pi R_+^2 - \pi R_-^2 = \pi(R_+ + R_-)(R_+ - R_-) = \pi \cdot 2|b| \cdot 2\sqrt{r^2-h^2}.$$ The rectangle has width $2\sqrt{r^2-h^2}$ and length $2\pi |b|$, so its area is $$A_2 = 2\sqrt{r^2-h^2} \cdot 2\pi|b|.$$ The areas are equal, so we're done!

I prefer Pappus's centroid theorem though...