Polar Moment of Inertia, J ( I_{p} )
The Polar Moment of Inertia is a geometric property of a cross section. Physically, it is a measure of how difficult it is to turn a crosssection about an axis perpendicular to it (the inherent rotational stiffness of the crosssection). This means:
 The greater the Polar Moment of Inertia, the more torque is require to turn the shaft by a certain angle.
 The greater the Polar Moment of Inertia, the smaller the Shear Stress required to produce a given torque.
 For the same area, the further away the material of a crosssection is away from the axis about which it twists, the greater the Polar Moment of Inertia.
Crosssection 
Polar Moment of Inertia, J 
Crosssection 
Solid Circular Shaft 
J_{solid} = 
pR^{4}
2 


Hollow (Thickwalled) Shaft 
J_{thick} = 
p(R_{o}^{4} R_{i}^{4})
2 


ThinWalled Shaft, t<<R 
J_{thin} = 2pt(R_{ave})^{3}, R_{ave} = 
R_{o} + R_{i}
2 


The units of J are generally expressed in: mm^{4} or in^{4}.