Calculating centroid for complex geometric shape -- decomposing the shape

For this type of problem, represent each segment as a point. These points are to be located at the center of mass of their respective segments.

You can compute the individual centroid for each segment by calculating its geometric center (in the case of a uniformly dense material).

Then, apply the following equation for the center of mass $bf vec R$,

$${bf vec R} = dfrac{displaystyle sum_i m_i {bf vec r}_i}{displaystylesum_i m_i}.$$

It should look something like this:

Each pink ball represents a point mass, $m_i$.

Note that the mass itself does not matter, but the proportions do. Easiest way is to define the linear density as $mu equiv dfrac{1 rm mass unit}{1 rm mm}$ such that a length of 1 mm corresponds to a mass of 1 unit.