The Gap-Subtraction Method: An Elementary Pursuit of the Circle without Calculus

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-21
DOI
https://doi.org/10.5281/zenodo.22870228
Primary Topic
Mathematics and Applications
Type
preprint
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preprint

The Gap-Subtraction Method: An Elementary Pursuit of the Circle without Calculus

Parikshit Parikshit Bishnoi
Zenodo (CERN European Organization for Nuclear Research)
Mathematics and Applications
preprint

The Gap-Subtraction Method: An Elementary Pursuit of the Circle without Calculus

Parikshit Parikshit Bishnoi
preprint en

Abstract

This paper introduces an original geometric framework named the Gap-Subtraction Method to calculate the area of a circle and derive the value of π. Bypassing university-level calculus, Taylor series expansions, and complex trigonometry, this methodology relies strictly on 10th-grade geometry and algebraic first principles. We explore how regular polygons converge toward a circular limit by systematically analyzing the shrinking geometric "gaps" between a bounding shape and an inscribed unit circle. A total of 6 distinct formulas are classified under this single unified framework, demonstrating that deep mathematical convergence can be unlocked using foundational geometric axioms.

Zenodo (CERN European Organization for Nuclear Research)
Mathematics and Applications
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The Gap-Subtraction Method: An Elementary Pursuit of the Circle without Calculus — Parikshit Parikshit Bishnoi · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS