A generalized expression for filling congruent circles in a circle

TL;DR

This paper derives a generalized mathematical expression for filling a larger circle with congruent smaller circles, considering various arrangements and inscribed configurations, useful for practical packing problems.

Contribution

It introduces a new generalized formula for inscribing multiple congruent circles within a larger circle, extending previous methods.

Findings

Derived a generalized expression for the largest inscribed circle in a circle segment.

Extended the formula to multiple touching circles filling the larger circle.

Provided a method to estimate the total number of such circles for filling purposes.

Abstract

The paper reports a generalized expression for filling the congruent circles (of radius r) in a circle (of radius R). First, a generalized expression for the biggest circle (r) inscribed in the nth part of the bigger circle (R) was developed. Further, it was extended as n such circles (r) touching each other and the bigger circle (R). To fill the bigger circle (R), the exercise was further repeated by considering the bigger circle radius as R-2r, R-4r and so on. In the process, a generalized expression was deduced for the total no. of such circles (r) which could be inscribed in this way of filling the bigger circle (R). The approach does not claim the closest packing always though it could be helpful for practical purposes.