Moment of Inertia or second moment of area is a geometrical property of a section of structural member which is required to calculate its resistance to bending and buckling. Mathematically, the moment of inertia of a section can be defined as

Moment of Inertia about x-x axis

Moment of Inertia about y-y axis

Moment of Inertia of some standard areas can be found below.

**1. Rectangular section**;

(a) I_{xx} = (bd^{3})/12

(b) I_{yy} = (db^{3})/12

where b= width of the section, and d= depth of section.

The axes x-x and y-y are passing through the
centroid and x-x axis is parallel to the width of section and y-y
parallel to the depth.

**2. Circular Section**

I_{xx}
= I_{yy} = (πD^{4})/64,

where D is the diameter of the section and x-x and y-y axes are passing through the centroid.

**Moment of inertia of hollow sections** can also
be determined by subtracting the moment of inertia of the removed
area from the moment of inertia of original area.

We can use **parallel axes theorem **to find the
moment of inertia about an axis parallel to x-x or y-y,

For example if p-q is an axis parallel to x-x
and it is at a distance of 'h' from x-x axis. Then the moment of
inertia I_{pq} about p-q axis can be determined
as given below;

I_{pq} = I_{xx
}+ Ah^{2}

where A= the area of the section

Parallel axes theorem is also used to determine the **moment of inertia of built-up sections**.

**Polar Moment of inertia** is required in
case of torsion of structural member. Polar Moment of Inertia
is defined as the Moment of inertia bout an axis perpendicular to
the plane of the section and can be calculated by applying
**perpendicular axes theorem **which says that

I_{xx}
+ I_{yy} = I_{zz} , where zz-axis is perpendicular to both xx-axis and yy-axis.

Use our **
Moment
of Inertia calculator** to determine centroid, moment of inertia,
section modulus and radius of gyration for
different sections including angle, circle, rectangle, Channel, I or H-section, T-section, pentagon, hexagon.
Moment of inertia is required to determine bending stress, shear stress and
deflection of beam.

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