The equations used to calculate the centroid of an arbitrary shape are based on the* parallel force theory* used to calculate the center of gravity (C.G.) of objects. Consider, for example, the system of particles shown below. Each particle has a finite weight and is located at some position in space. This distributed system can be represented by a single large particle that is equal in weight to that of all the particles combined, and is placed at the center of gravity of the system.

Notice that the system of particles can be viewed as a non-concurrent parallel force system. Based on the Cartesian coordinate system shown, the center of gravity is determined by balancing the total moment created by the weight of each individual particle about an axis to that of the resultant weight. For example, considering the moment about the x axis, we have

Which can be rewritten as

If each particle is viewed as a circle of finite area, then the centroid of the distributed system can be calculated by replacing the weight by area in the above equation.

A similar procedure can be used to derive the equations for the other two coordinates of the centroid, i.e., and .

We can also use the theory of parallel forces to derive the equations for centroids of lines and volumes in addition to areas of arbitrary shape.