Arch perimeter prediction using Ramanujan’s equation for ellipse: A mathematical approach in borderline cases

Objectives: The objective is to evaluate the accuracy of Ramanujan’s equation for the perimeter of an ellipse in predicting pre- and post-treatment maxillary arch perimeter in borderline orthodontic cases. Material and Methods: This retrospective cohort study analyzed 70 maxillary dental study models (35 pre-treatment and 35 post-treatment) from non-extraction orthodontic cases with borderline crowding or generalized spacing. The semi-major and semi-minor axes were measured using a digital Vernier caliper, and the arch perimeter was recorded manually using a brass wire. Ramanujan’s equation for the perimeter of an ellipse was used to calculate theoretical arch perimeter values, which were then compared with measured values. Measurement reliability was evaluated using Cronbach’s alpha. Validity was assessed through paired t-tests and intra-class correlation coefficients (ICC). The study was conducted following the strengthening of the reporting of observational studies in epidemiology guidelines. Results: The mean difference between calculated and manual arch perimeter values was below 1.2% for both pre- and post-treatment models. ICC values were 0.985 (pre-treatment) and 0.977 (post-treatment), indicating excellent reliability. Paired t -tests showed no significant difference between manual and calculated values ( P > 0.05). A strong linear correlation was confirmed through scatter plot analysis. Conclusion: Ramanujan’s equation demonstrated high reliability and predictive accuracy for estimating maxillary arch perimeter in borderline orthodontic cases. It may serve as a valuable mathematical adjunct in clinical decision-making, particularly in evaluating space availability for non-extraction treatment planning.