engineering (especially mechanical and civil), moment of inertia commonly refers to the second moment of the area. Definition The second moment of area for an arbitrary shape with respect to an arbitrary axis is defined as where is the infinitesimal area element, and is the perpendicular distance from the axis . For example, when the desired reference axis is the x-axis, the second moment of area (often denoted as ) can be computed in Cartesian coordinates as The second moment of the area is crucial in Euler–Bernoulli theory of slender beams. Product moment of area More generally, the product moment of area is defined as Parallel axis theorem It is sometimes necessary to calculate the second moment of area of a shape with respect to an axis different to the centroidal axis of the shape. However, it is often easier to derive the second moment of area with respect to its centroidal axis, , and use the parallel axis theorem to derive the second moment of area with respect to the axis. The parallel axis theorem states where is the area of the shape, and is the perpendicular distance between the and axes. A similar statement can be made about a axis and the parallel centroidal axis. Or, in general, any centroidal axis and a parallel axis. Perpendicular axis theorem For the simplicity of calculation, it is often desired to define the polar moment of area (with respect to a perpendicular axis) in terms of two area moments of inertia (both with respect to in-plane axes). The simplest case relates to and . This relationship relies on the Pythagorean theorem which relates and to and on the linearity of integration. Composite shapes For more complex areas, it is often easier to divide the area into a series of "simpler" shapes. The second moment of area for the entire shape is the sum of the second moment of areas of all of its parts about a common axis. This can include shapes that are "missing" (i.e. holes, hollow shapes, etc.), in which case the second moment of area of the "missing" areas are subtracted, rather than added. In other words, the second moment of area of "missing" parts are considered negative for the method of composite shapes. Examples See list of second moments of area for other shapes. Rectangle with centroid at the origin Consider a rectangle with base and height