The parallel-axis theorem is the relationship between the moment of inertia with respect to a centroidal axis and the moment of inertia with respect to any parallel axis. While the explanation sounds complicated, the application is very simple. Let's first try to derive the expressions by considering an arbitrary shape below:
y axis is parallel to yc axis and x axis is parallel to xc axis. yc and xc marks on the yc and xc axes are the coordinates of the centroid of the whole object. b is the distance from y axis to yc axis and a is the distance from x axis to xc axis.
| Since we know that the moment of inertia is given by |
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Also we know that when an axis acts through the centroid, its first moment of area will be zero, then ycdA = 0. Additionally, we know that yc2dA = Ixc. So then,
Ix = Ixc + a2A
Similarly:
