***Note:*** *Different disciplines use the term* moment of inertia *to refer to different moments. In physics,* moment of inertia *is strictly the second moment of **mass** with respect to distance from an axis, which characterizes an object's angular acceleration due to an applied torque. In engineering (especially mechanical and civil),* moment of inertia *commonly refers to the second moment of the area. When reading* polar moment of inertia *take care to verify that it is referring to "polar second moment of area" and not* moment of inertia*. Polar second moment of area will have units of length to the fourth power (e.g. {\displaystyle m^{4}} or {\displaystyle in^{4}}), while moment of inertia is mass times length squared (e.g. {\displaystyle kg\*m^{2}} or {\displaystyle lb\*in^{2}}).* 

The **polar second moment of area** (also referred to as "polar moment of inertia") is a measure of an object's ability to resist torsion as a function of its shape. It is one aspect of the second moment of area linked through the perpendicular axis theorem where the **planar** second moment of area uses a beam's cross-sectional shape to describe its resistance to deformation (bending) when subjected to a force applied in a plane parallel to its neutral axis, the **polar** second moment of area uses a beam's cross-sectional shape to describe its resistance to deformation (torsion) when a moment (torque) is applied in a plane perpendicular to the beam's neutral axis. While the **planar** second moment of area is most often denoted by the letter, {\displaystyle I}, the **polar** second moment of area is most often denoted by either, {\displaystyle I\_{z}}, or the letter, {\displaystyle J}, in engineering textbooks.

The calculated values for the polar second moment of area are most often used describe a solid or hollow cylindrical shaft's resistance to torsion, as in a vehicle's axle or drive shaft. When applied to non-cylindrical beams or shafts, the calculations for the polar second moment of area becomes erroneous due to warping of the shaft/beam. In these instances, a torsional constant should be used, where a correctional constant is added to the value's calculation. 

A schematic showing how the **polar second moment of area** ("Polar Moment of Inertia") is calculated for an arbitrary shape of area, R, about an axis *o*, where *ρ* is the radial distance to the element *dA*.The polar second moment of area carries the units of length to the fourth power ({\displaystyle L^{4}}); meters to the fourth power ({\displaystyle m^{4}}) in the metric unit system, and inches to the fourth power ({\displaystyle in^{4}}) in the imperial unit system. The mathematical formula for direct calculation is given as a multiple integral over a shape's area, {\displaystyle R}, at a distance {\displaystyle \rho } from an arbitrary axis {\displaystyle O}. 

{\displaystyle J\_{O}=\iint \limits \_{R}\rho ^{2}dA}. 

In the most simple form, the **polar** second moment of area is a summation of the two **planar** second moments of area, {\displaystyle I\_{x}} and {\displaystyle I\_{y}}. Using the Pythagorean theorem, the distance from axis {\displaystyle O}, {\displaystyle \rho }, can be broken into its {\displaystyle x} and {\displaystyle y} components, and the change in area, {\displaystyle dA}, broken into its {\displaystyle x} and {\displaystyle y} components, {\displaystyle dx} and {\displaystyle dy}.

Given the two formulas for the **planar** second moments of area:

{\displaystyle I\_{x}=\iint \limits \_{R}x^{2}dxdy}, and {\displaystyle I\_{y}=\iint \limits \_{R}y^{2}dxdy}

The relation to the **polar** second moment of area can be shown as:

{\displaystyle J\_{O}=\iint \limits \_{R}\rho ^{2}dA}

{\displaystyle J\_{O}=\iint \limits \_{R}(x^{2}+y^{2})dxdy}

{\displaystyle J\_{O}=\iint \limits \_{R}x^{2}dxdy+\iint \limits \_{R}y^{2}dxdy}

{\displaystyle \therefore J=I\_{x}+I\_{y}}

In essence, as the magnitude of the polar second moment of area increases (i.e. large object cross-sectional shape), more torque will be required to cause a torsional deflection of the object. However, it must be noted that this does not have any bearing on the torsional rigidity provided to an object by its constituent materials; the polar second moment of area is simply rigidity provided to an object by its shape alone. Torsional rigidity provided by material characteristics is known as the shear modulus, {\displaystyle G}. Linking these two components of rigidity, one can calculate the angle of twist of a beam, {\displaystyle \theta }, using:

{\displaystyle \theta ={\frac {Tl}{JG}}}

Where {\displaystyle T} is the applied moment (torque) and {\displaystyle l} is the length of the beam. As shown, higher torques and beam lengths lead to higher angular deflections, where higher values for the polar second moment of area, {\displaystyle J}, and material shear modulus, {\displaystyle G}, reduces the potential for angular deflections.

Related pages
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* Moment (physics)
* Second moment of area
* List of second moments of area for standard shapes
* Shear modulus

Other websites
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* Torsion of Shafts - engineeringtoolbox.com
* Elastic Properties and Young Modulus for some Materials - engineeringtoolbox.com
* Material Properties Database - matweb.com