Okay, I understand most people don't have a lathe. I also understand that if someone does have a lathe, there is an 86.4% chance they are better at it than I am. I'm in the other 13.6%.
But I do have a lathe hanging around the shop, and I wanted to put a slight taper on the legs to give this chunky table slender legs a la 1950s furniture. As for wood selection, this table is built out of leftovers excepting the top. The legs are turned out of a stick of Lyptus I had in my supplies. Later you will see the legs blackened to match our dark bookcases.
I cut the three legs apart on the table saw (be sure to support the taper to keep the leg from falling when free from the other piece). I set my miter sled to about a 2 1/2 deg. and trimmed the top of each leg. This makes the legs splay out a little bit; an aesthetic decision mostly. An example of how I wanted the legs to look like is here
The gist of this leg design is to have the legs protrude through the tire. Each leg is attached to a plate. These plates are screwed to a stack of plywood (see next step) that has been cut to the profile of the tire interior, transferring the load around the whole tire, not just one sidewall. It is necessary to break this leg assembly up into two parts: leg + plate and plywood stack. If you join them up prior to getting them in place, it just won't fit and you will be fighting the tire.
It didn't occur to me how pliable tire sidewalls were when they weren't inflated. My first go at this didn't have the stacked plywood leg mounts and the table felt like a jelly fish. After correcting that, it is solid.
I drilled a pilot hole into the top of each leg so I wouldn't split it during attachment. I don't have a chuck on my lathe, so I clamped them on my drill press.
Next you will want to make a cardboard profile pattern of the "en-tire" interior (just terrible).
For this example, cardboard is better than corrugated. If you don't know the difference, cardboard is on the back of a pad of paper, corrugated is used in bigger boxes. I used corrugated and about halfway through wondered why I didn't use plain cardboard. The corrugations made it more difficult to cut and I find it harder to trace around...
Transfer the pattern onto the end of some 1x4 of hardwood stock (I think the scrap I used was Alder) that is approximately three times longer than a top plate. Use the belt sander (very noisy, dusty option) or a hand plane (the quite, no-dust option) to trim the board to match the profile. Contrary to the way I marked the board in the photo, you should leave a flat spot on the bottom of the plate where the leg will attach so they will have a stable joint.
After shaping the board, I cut it into the three support plates, then pre-drilled a pilot hole. I used a Forstner bit to recess the lag bolt head into the plate so it wouldn't interfere with the stack support. I attached the legs to their support plates using a 4" lag bolt (probably overkill, but it is what I had on hand).