Short Answer:

The moment of inertia (M.I.) of a rectangle about its centroidal axes measures how the rectangle’s area is distributed relative to those axes and how resistant the section is to bending or rotation. For a rectangle of width and height , the centroid lies at the geometric centre. The standard centroidal moments of inertia are

where is about the horizontal centroidal axis (through the mid-height) and about the vertical centroidal axis (through mid-width).

These formulas are used directly in beam bending, deflection and strength calculations. Units are (e.g., mm or m). If you need the M.I. about a parallel axis, use the parallel-axis theorem .

Detailed Explanation :

Moment of inertia of a rectangle

What is being found
The problem asks for the moment of inertia of a rectangular area about its centroidal axes. In common engineering notation:

= width (horizontal dimension),
= height (vertical dimension),
centroid is at the geometric centre if we place origin at the centre.
We want (about the horizontal axis through centroid) and (about the vertical axis through centroid).

Why it matters
The second moment of area (moment of inertia) quantifies how area is spread from an axis. In beam bending, a larger about the neutral axis means smaller bending deflection and lower bending stress for the same moment.

Derivation

Choose coordinates and element
Place the rectangle symmetrically with its centroid at the origin. Let the vertical coordinate be and horizontal coordinate . The rectangle extends from to and to . Consider a thin horizontal strip (for ) of thickness at coordinate . The strip length = , so elemental area .

Moment of inertia about centroidal x-axis ()
Definition for a plane area:

Substitute and integrate over :

Compute the integral:

Thus

Moment of inertia about centroidal y-axis ()
Similarly take a vertical strip of width at . Elemental area . Then

Compute:

Summary of results

Centroidal moment of inertia about horizontal axis (through centroid): .
Centroidal moment of inertia about vertical axis (through centroid): .

Units and interpretation
Units are length (e.g., mm, m). Note that depends strongly on height (cube of ), so increasing depth (height) is very effective at increasing bending stiffness about the horizontal axis — a key reason deep beams are stiffer.

Example numeric calculation
Take a rectangular cross-section , :

So the section is much stiffer about the horizontal centroidal axis than the vertical one.

Parallel axis note (when needed)
If you need the M.I. about an axis parallel to the centroidal axis at distance , use the parallel axis theorem:

where and is the centroidal M.I. above.

Common uses
These centroidal formulas are used in beam bending (bending stress ), deflection calculations, and in determining natural frequencies or buckling behaviour where stiffness matters.

Conclusion

The moment of inertia of a rectangle about its centroidal axes is given by simple closed formulas: and . These are derived by integrating the square of the distance of elemental areas from the axis. The results show the strong dependence on the dimension perpendicular to the axis (cube), explaining why depth increases bending stiffness dramatically. For off-centred axes apply the parallel-axis theorem.