Moment of Inertia
Moment of inertia of a body about any axis is defined as the second moment of all elementary areas about that axis.
Theorem of Parallel Axis
It states that the Moment of Inertia (MI) of a plane section about any axis parallel to the centroidal axis is equal to the MI of the section about the centroidal axis plus the product of the area of the section and the square of the distance between the two axes.
MI about PQ = IPQ = IG + Ah2
Perpendicular Axis Theorem
It states that if IXX and IYY are the moments of inertia of a plane section about two mutually perpendicular axes meeting at O, then the moment of inertia about the third axis Z-Z (IZZ) is equal to the sum of the moments of inertia about the X-X and Y-Y axes.
IP = IZZ = IXX + IYY
Radius of Gyration
Radius of gyration of a body about an axis of rotation is defined as the radial distance of a point from the axis of rotation at which, if the whole mass of the body is assumed to be concentrated, its moment of inertia about the given axis would be the same as with its actual distribution of mass.
Mathematically, the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application.
Consider area A with moment of inertia Ix. Imagine that the area is concentrated in a thin strip parallel to the x-axis with equivalent Ix.
