Polar moment of inertia
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, .  Linking these two components of rigidity, one can calculate the angle of twist of a beam, , using: