Parallel-Axis Theorema voor een Oppervlakte

What is the moment of inertia for the area of a thin plate about an arbitrary axis?

Consider a differential element at a certain distance from the centroidal axis parallel to the given axis. Its differential moment of inertia can be calculated.

Integrating the expression over the entire area yields the plate's moment of inertia about the planar axis.

Here, the first term represents the moment of inertia about the centroid, while the third term equals the total area. The second term is zero as this axis passes through the centroid.

The resulting expression gives the moment of inertia about the planar axis. Similarly, the moment of inertia about the other planar axis is obtained.

Summing these moments of inertia gives the polar moment of inertia.

So, the moment of inertia about any axis equals the moment of inertia about a centroidal axis parallel to it plus the product of the area and the square of the distance between the axes. The obtained expression is called the parallel axis theorem for an area.