This calculator estimates the maximum number of smaller circles of radius r that fits into a larger circle of radius R. It could be number of small pipes inside a large pipe or tube, number of wires in a conduit, number of cut circles from circle-shaped plate, and so on.
One may think that there should be formula for that, but, in fact, there is no formula. This is an optimization problem knows as Circle packing in a circle. It belongs to a class of optimization problems in mathematics which are called packing problems and involve attempting to pack objects together into containers. Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. See Circle packing in a circle.
For this problem, optimal solution needs to be found and proved. Wikipedia article lists first 20 solutions (in other words, it lists smallest possible radius of larger circle which is enough to pack specified number of unit circles (circles with radius of one). Default calculator parameters, by the way, allow to pack 11 circles, which will give you the layout presented below:
Happily, there is a project on the internet, solely devoted to packing problems. It is the site called Packomania. Today, it summarises all found solutions. Site author, Eckard Specht, also participates in search for solutions, and, in fact, most solutions were found by him, and there are solutions for up to 2600 circles in a large circle, with pictures of layouts. For each number of circles the ratio of r/R is given, and this can be used to find answer.
The calculator below evaluates ratio r/R and then looks for closest optimal solution among those 2600. If ratio r/R falls outside of research data, the calculator returns error.
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