###### Abstract  
The purpose of this supplementary method study was to demonstrate, within the hemimaxillary, the three-dimensional skeletal and dental displacements of upper permanent first molars in patients undergoing rapid maxillary expansion (RME) with anchorage on deciduous teeth or paramedian miniscrews. Five post-treatment cases were analyzed on frontal and axial views before (T0) and after expansion (T1), adopting a new hemimaxillary reference system. Three-dimensional landmarks were selected to evaluate molar changes; buccal tipping and rotation were calculated, in addition to interpolar angle, molar angle of rotation, and orthopedic expansion angles. The origins (dental and/or orthopedic) of molar displacements were investigated by alternate interior angle demonstration according to the parallel lines theorem. Through inverse geometric functions of right triangles, it was possible to determine theoretical-experimental forms to calculate angles from periapical radiographs taken at T1. These findings have significant clinical implications, enabling the assessment of treatment outcomes while adhering to radioprotection principles. Reproducible formulas enable the calculation of expansion angles without CBCT. For RME to allow clinical improvement, molar rotation and interpolar angles at T1 should be less than the difference between the respective angles at T0 and the orthopedic expansion angle. This method provides a reproducible, radiation-efficient method to assess maxillary changes, enhancing treatment precision in orthodontics. m x  
with the base of the pyramid at the alveolar ridge level and the apex near the fronto-nasal suture [8]. This results in a buccal inclination of the alveolar halves carrying the teeth with them. After expansion has occurred, however, the inclination of the teeth is not equal to the alveolus one because buccal tipping of the posterior teeth also occurs. It has been reported in the literature that the average expansion in the mid-palatal suture ranges from 0.84 to 2.88 mm, accounting for 12-52.5% of the screw expansion [9], with the remaining part of the expansion being determined by precisely these dento-alveolar effects produced by strong expansion forces [10]. Because of the periodontal ligament hyalinization of the anchor teeth, the force produced by the device should ideally be transferred directly to the sutures, to increase orthopedic effects and decrease alveolar-dental effects [11]. Dental effects are particularly evident at the permanent upper first molar level and include buccal tipping, generally between 0\({}^{\circ}\) and 24\({}^{\circ}\)[12], and a disto-rotation. Understanding the dental effects generated by RME is essential as they represent an indispensable element in terms of dental problems that are not only transverse, correcting the cross-bite, but also sagittal, correcting the molar key. In fact, the disto-rotation of upper molars contributes to the correction of class II molars, improving those situations in which the transverse arch defect is sometimes associated with the abnormal rotation of these dental elements. Moreover, a rotation of the upper first molars represent an element of high clinical relevance for a correct alignment of the lower arch, as it can result in a mesial eruption of the upper first and second premolars and the subsequent appearance of precontacts with the lower arch that would prevent the physiological distal translation of the antagonists [13]. The accurate evaluation of dental and skeletal changes following RME is crucial for optimizing orthodontic treatment outcomes. Traditional methods often rely on two-dimensional imaging, which can fail to capture the complex nature of maxillary displacements. Early research in fact aimed to prove the effectiveness of maxillary expansion on skeletal and dento-alveolar structures using dental casts and two-dimensional cephalography in latero-lateral and posterior-anterior projection [14; 15; 16; 17; 18].