Area moment of inertia is the name commonly used to describe the second moment of area. For example, consider an arbitrary area A shown at right.
Its moment of inertia about the designated x-axis is given asArea moment of inertia is the name commonly used to describe the second moment of area. For example, consider an arbitrary area A shown at right.
Its moment of inertia about the designated x-axis is given as
 (1) |
where y is the perpendicular distance measured from the x axis to the centroid of the differential area dA. Similarly the moment of inertia about the y-axis is given as
 (2) |
Application: The area moments of inertia are used extensively in mechanics of materials for calculations of flexural stress and deflection of beams as well as buckling stresses in columns. Notice that if we consider the mass of the object instead of its area in Eqs. (1) and (2), then we obtain what are known as mass moments of inertia, which are commonly used in dynamic problems involving accelerating bodies.
Polar Moment of Inertia: 
The polar moment of inertia of area A shown below about point O is defined as
 (3) |
Since x and y form the rectangular components of distance r, we have the equality
Thus, the polar moment of inertia is simply the sum of Ix and Iy defined by Eqs. (1) and (2).
Application: The polar moment of inertia is used in calculations of shear stress and twist angle of shafts under torsion.
Product of Inertia: 
The area product of inertia is defined as
 (4) |
The x and y terms inside the integral denote the centroidal position of the differential area measured from the y and x axes, respectively. Similar to moments of inertia discussed previously, the value of product of inertia depends on the position and orientation of selected axes. It is possible for the product of inertia to have a positive, negative, or even a zero value.
If, for example, either x or y represents an axis of symmetry, then the product of inertia Ixy would be zero. To see why this is the case, take a look at the figure to the right. Consider the small area A1 to the right of y axis at the distance of x1. Then consider a similar area to the left of this axis of symmetry at the distance of -x1. Since both areas are at the same vertical position from the x-axis, they have the same value of y. The contribution from the left area is -x1yA1 and that from the right is x1yA1 which add up to zero. Since every point on one side of the axis of symmetry has an equal counterpart on the other side, the total value of the integral would be zero.
However, if we were to consider the product of inertia with respect to the x' and y' axes, then Ix'y' would not be zero. We will have more discussion about the product of inertia in the section on principal axes.
Application: Product of inertia is used in structural mechanics for analyzing unsymmetric bending of beam sections
Parallel-Axis Theorem: 
In many instances the moment of inertia of an area about an axis that is parallel to its centroidal axis is sought. The equations for moments of inertia about such parallel axes are given as
 (5) |
 (6) |
where Ixc and Iyc are the moments of inertia of the area about the centroidal axes xc and yc, respectively. dy is the perpendicular distance between the centroidal axis xc and the parallel axis denoted by x', while dx is the perpendicular distance between yc and y'. The x and y axes in this case serve as reference axes for finding the centroidal location of the area.
The parallel-axis theorem also applies to the polar moment of inertia
(7) |
where Jo is the polar moment of inertia about point o, Jc is the polar moment of inertia about the centroid, and d2 = dx2 + dy2.
Similarly, the product of inertia with respect to x'y' axes can be found using the parallel-axis theorem as
 (8) |
The parallel-axis theorem is used in calculating the moments of inertia of a composite shape, one made up of a collection of elementary shapes. In that case, the moment of inertia of each elementary shape about its own centroidal axes are obtained first, then the corresponding parallel axis terms are added to determine the moments of inertia of the composite area. This procedure is demonstrated in the following example.
Radius of Gyration: The radius of gyration is a parameter that relates the moment of inertia to the area of a section. The radii of gyration corresponding to Ix, Iy, and J are expressed as
 (9) |
We can think of the radii of gyration rx and ry as the x and y coordinates of an imaginary point at which the entire area is lumped. As such, a radius of gyration does not have any physical significance. It is simply a way of relating the moment of inertia about a particular axis to the area.
Application: In buckling analysis of columns a quantity of importance is the so-called slenderness ratio denoted by L/r. This quantity relates the length (L) of the column to its cross-sectional radius of gyration (r).
Principal Axes and Principal Moments of Inertia: 
The principal axes are those for which the product of inertia is zero. For any arbitrary shape there exists a set of axes which result in zero product of inertia. The orientation of principal axes with respect to the centroidal coordinates x and y can be obtained using
 (10) |
where Ix, Iy, and Ixy represent the moments of inertia about the x-axis, moment of inertia about the y-axis, and the product of inertia with respect to x and y axes, respectively. The angle qp is measured positive counter clockwise from the centroidal x-axis.
Based on the above definition, if either x or y is an axis of symmetry, then they both are considered as principal axes as Ixy and hence qp would be zero in that case.
The moments of inertia about the principal axes are expressed as
 (11) |
 (12) |
Key Observations:
A key factor to remember is that the sum of moments of inertia about any two perpendicular axes in the plane of the area is constant. This implies the following:
Another important fact to remember is that between Ixp and Iyp one represents the minimum while the other represents the maximum moment of inertia for the shape considered. This fact is crucial in design of beams for minimum deflection. If the load on a beam is applied perpendicular to the cross-sectional axis with the largest moment of inertia, then the resulting deflection is the minimum for that shape and size beam.