The exact perimeter of an ellipse involves the complete elliptic integral of the second kind, which lacks a closed-form expression in elementary functions. As a result, analytical approximations have been widely studied for applications requiring fast and accurate evaluation of elliptical geometries. In this work, we present an extended framework of exponential corrections to Ramanujan’s second formula, building upon our previously published results. The proposed approach introduces a new family of ultra-accurate and compact closed-form approximations, including multi-exponential models with enhanced flexibility. The resulting formulas preserve simplicity while achieving maximum relative errors ranging from 0.57 down to 0.02 ppm over the full eccentricity range. This represents a substantial improvement over classical and modern approximations, while maintaining low computational cost and single-line analytical structure. Due to their robustness, quasi-exact behavior at both circular and highly eccentric limits, and suitability for numerical algorithms and embedded implementations, the proposed approximations are particularly attractive for engineering applications involving elliptical geometries. This work is an extended and significantly expanded version of a previously published article in MDPI.
Ayala-Raggi et al. (Sat,) studied this question.
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