Here is the construction of the solid. Take an ellipse, make a copy of it, and put it on top of the original ellipse. Now turn the top ellipse by $90^\circ$ (quarter turn). Glue the two boundaries. I would like the height, volume, and the equation describing the ridge/boundary of this solid in terms of the major and minor axes.
Edit: Instead of two ellipses if you took a rectangle with sides $a$ and $b$ with $ b < a$ and followed the same procedure, you end up with box base a $b \times b$ square and of height $a -b.$ I wish I could take pictures of both solids I made and post it here. Hopefully this helps.