FBinNY 

Nice analysis, but most folks don't have access to Autocad, and fortunately don't need it to work a problem like this, just a pencil or some visualization skill. A simple sketch will help.

Start with the premise that the inside of the inflated tire outside the rim is a near perfect circle. (This is true for the same reason bubbles are round)

So you have a circle of X circumference nesting on the rim. If you mark the points where the tire meets the rim, the outside arc has a fixed length. So the total perimeter is that arc plus the rim width. Add to the rim and the new perimeter is increased by the same amount. Fill it with air and you have a new larger circle with a diameter increased by the same amount. The rest is easy.

Note, there's a bit of fudge here because the rim width is a secant on the circle, and shorter of the arc, but that difference is negligible for this purpose