How to optimize the outer pipe diameter?

Suppose that I have a big pipe and I want to put n small pipes in it, say 10 pipes 1 inch radius each. What is the smallest radius for the biggest pipe to contain all of them? Constraint minimization, they must not be inside one another. If we write a matlab function, how will I write the function to minimize the radius for the outer pipe?

Answer 1

[Walter Roberson]

https://en.wikipedia.org/wiki/Circle_packing_in_a_circle

3.813... approximately is the required radius.

The shape is not obvious, even when you look at a diagram of it. You will need to find the 1969 proof (or a translation of it.)