Centroid of Area

Centroid of Area: The centroid of an area of arbitrary shape, such as the one shown below, can be determined using the integral equations

 (1)

where dA= differential area and x y z= moment arms of dA (identifying the coordinates of the centroid of dA, see figure)

Depending on the choice of differential area, we can have a single or a double integral in the numerator and denominator of each centroidal coordinate in Eq. (1) as demonstrated in the following examples.

Example 2 (updated)

If the surface being considered has boundaries that allow it to be represented by a collection of simple shapes, then a modified version of Eq. (1) can be used. For such composite shapes, as shown in the figure below, we find the location of centroid as

 (2)

where xi, yi, zi denote the centroidal coordinates of the ith element of the composite shape.

A composite shape should be subdivided in such a way as to simplify the calculation of centroidal location. In some cases it may be easier to think of the composite shape as a larger surface with a portion of it cut out. In the summation equation the area of the cutout portion would have a negative sign associated with it. The calculation of centroidal coordinates is demonstrated in the following example.

Example 3