The centroid of a shape is the weighted average of the centroids of its component parts.

A_1.C_1 + A_2.C_2 = A.C

Where A_1 is the area of the unshaded region, C_1 is the centroid height of the unshaded region etc.

We require C_2, so making that the subject:

C_2 = (A.C - A_1.C_1)/A_2

We can calculate the area and centroid height (A and C) of the overall figure from first principles, as it is simply an equilateral triangle, and we can find A_2 by simply taking A_1 from A.

C_2 = (rt(3)/4 x rt(3)/6 - rt(3)/6 x rt(3)/10) / (rt(3)/12)

C_2 = 3rt(3)/10 =~ 0.5196…