The Napkin Ring Problem

Geometrical proof

This problem or paradox that I discovered via Vsauce states that the volume of a towel ring depends on the height of the ring but not on the radius of the sphere. See link below. I propose with this picture a geometrical proof, that does not use calculations involving the radius R of the sphere. Using Cavalieri's principle, one only needs to show that the cross section has an area that does not depend on R but solely on h and y. This section is a ring with area pi.(r_o²-r_c²). Here I show using the intersecting chords theorem that this number is equal to a product in terms of h y and h only.