The Center of Mass of an object (of mass m) is the single point that moves in the same way as a point mass (of mass m) would move when subjected to the same external forces that act on the object. That is, if the resultant force acting on an object (or system of objects) of mass m is F, the acceleration of the center of mass of the object (or system) is given by acm = F/m

If the object is considered to be composed of tiny masses m1, m2, m3, and so on, at coordinates (x1,y1,z1), (x2,y2,z2), and so on, then the coordinates of the center of mass are given by

where Σ = sigma, the sequence of

The x-coordinate
Xcm = Σ(xi * mi) / Σ(mi)

The y-coordinate
Ycm = Σ(yi * mi) / Σ(mi)

The z-coordinate
Zcm = Σ(zi * mi) / Σ(mi)

where the sums extend over all masses composing the object. In a uniform gravitational field, the center of mass and the center of gravity coincide.

For the integration method of center of mass, check out Center of Mass: Part Two.