Centroid by moment method is based on the principal of taking moments about any arbitrary point. Centroid by moment method states that “When a number of coplanar parallel forces acts in a certain plane, then the algebraic sum of their moments about any point in the same plane is equal to the moment of their resultant forces about the same point.’’
Moment method is most effective when the plane is irregular or when it is not possible to divide large covered area into rectangles, triangles, circles, etc…

Method

Consider an irregular surface whose centroid is required to determine with respect to two reference axes i.e. x-axis and y-axis which are mutually perpendicular to each other.

Step I

Divide the whole area of a plane into small elementary strips respectively, whose areas are a₁, a₂, a₃ … a_n

Step II

Find the centroidal distance of each separate elementary strip from mutually perpendicular reference x-axis & y-axis.
Centroidal distance from vertical reference axis i.e. y-axis is named as x₁, x₂, x₃, x₄, x₅ .. x_n. Similarly, centroidal distances of each elementary strip from horizontal reference axis i.e. x-axis is named as y₁,
y₂, y₃ …. y_n.

Step III

Sum up all the moments in both direction i.e. x-axis & y-axis of each separate elementary strip.
[sum ax = a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}……a_{n}x_{n}]
[sum ay = a_{1}y_{1}+a_{2}y_{2}+a_{3}y_{3}……a_{n}y_{n}]

Similarly, sum up all the areas separately.
[A = a_{1}+a_{2}+a_{3}……a_{n}]

Step IV

Finally, to get the centroidal distance, Divide total moment along reference x-axis and y-axis with the summation of areas separately as shown below.
[large bar{x} = frac{sum ax}{A}]
Similarly,
[large bar{y} = frac{sum ay}{A}]