# Determination of Second Moment of Area (or Area Moment of Inertia) of Plane Area like Rectangle, Triangle, Circle etc. from Integration

The area moment of inertia of the following sections will be determined by the method of integration:

1. Moment of inertia of a rectangular section,
2. Moment of inertia of a circular section,
3. Moment of inertia of a triangular section,
4. Moment of inertia of a uniform thin rod.

4.8.1 Moment of Inertia of a Rectangular Section

1st Case. Moment of inertia of the rectangular section about X-X axis passing through the C.G. of the section.

Fig. 4.22 shows a rectangular section ABCD having width = b and depth = d. Let X-X is the horizontal axis passing through the C.G. of the rectangular section. We want to determine the moment of inertia of the rectangular section about X-X axis. The moment of inertia of the given section about X-X axis is represent by IXX.

Consider a rectangular elementary strip of thickness dy at a distance y from the X-X axis as shown in Fig. 4.22.

2nd Case. Moment of  inertia of the rectangular section about a line passing through the base.

Fig. 4.23 shows a rectangular section ABCD having width = b and depth = d. We want to find the moment of inertia of the rectangular section about the line CD, which is the base of the rectangular section.

Let X-X is the axis passing through the C.G. of the triangular section and parallel to the base.

The distance between the C.G. of the triangular section and base AB=h/3.

Now, from the theorem of parallel axis, given by equation (4.8), we have