Random question about cones and cylinders volume

A cone's volume with height ##x## and radius ##y## is ##1/3## of the volume of a cylinder with height ##x## and radius ##y##.I was trying to visualize it in my head and struggled a bit.Take a rectangle triangle with height ##x## and the other side of lenght ##y## which isn't the hypothenuse , then you get the cone if you rotates the triangle such that the two corners forming height ##x## rotates on themselves at the center , and you get the the rest of the cylinder if you rotates (from the same center) the triangle that would form a rectangle with the initial rectangle triangle.This means a rotation of a rectangle triangle where only one corner is rotating on itself at the center covers 2 times the volume of a rectangle triangle rotating such that two corners rotate on themselves at the center.



(Obviously I assume the triangle is filling the volume anywhere it goes during the rotation and I assume the rotation is complete)



So at the center of the rotation , in the first exemple constructing the cone the side ##x## is only "filling" the volume at the very center while it's filling all the curved area of the cylinder in the second exemple.I'm not exactly sure what I'm trying to accomplish here but I guess putting more rigor on this phenomenon in my mind would be a good start.



Any help?



thanks