Centroids of Cross-Sectional Areas        Click here for Moments of Inertia of Areas



Key Concepts:  In solid mechanics properties of X˗ sectional areas including centroid and

moment of inertia are important.   Centroids are frequently at the point of symmetry of

the X˗section such as at the center of a circular, annular, rectangular, or box  X-section.

The centroid of a X˗sectional area is the idealized location where all its area can be thought to be concentrated.  Again, to locate the centroid use the principle of first moments:


The moment of the sum equals the sum of the moments of individual parts.




In a Nutshell:  Two common methods can be used to located the centroid – the method of integration and the method of summation for composite areas where individual areas are joined together.  Note:  If there is a void (i.e. a hole such as a circular area), then the void has a negative contribution.  


For the method of integration:            A ycg   = ʃ  y dA    and    A zcg  =  ʃ  z dA


where   A is the total area, ycg is the y-coordinate of the centroid, zcg is the z-coordinate of the

centroid, and  (y,z) are the coordinates to the element of area, dA



For the method of summation:


Areas can be combined together to form “composite sectional areas”.  The principle of

first moments also applies to each “composite sectional area”.  Use “summation” form.


For example:  (Σ Ai )ycg  =   Σ Ai yi      (Also,  (Σ Ai )zcg  =   Σ Ai zi   )


i.e.                  (area1 + area2) ycg  =   (z1 )area1  +  (y2)area2   

So           ycg  = [ (y1 )area1  +  (y2)area2  ]  / (area1 + area2)  

where                ycg    is  the y-component of the centroid of the composite area


Click here for examples.


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