A machine for balancing vehicle wheels, such wheels including a rim with a pair of axially spaced, ring-shaped or annular, flanges, and a tire seated up on the rim. The machine includes a frame with motor driven devices for gripping the vehicle wheel to be balanced and rotating same about a horizontal axis. A mechanical feeler, or probe, determines the position of the first of the pair of flanges on the rim. A contact-less, or remotely spaced, sensor, such as a laser, ascertains the profile of the tire by taking a series of discrete measurements without touching the tire. The mechanical feeler, or probe, and the contact-less sensor, are connected to a control unit in the machine. The control unit interpolates the values provided by the mechanical feeler and the contact-less sensor, and determines the location of the second flange on the wheel rim. The control unit may include an archive, or database, of the dimensions of known rims, to facilitate the accurate determination of the location of the second flange, on the wheel rim.