The radius of gyration is defined as the distance between an axis and the point of maximum inertia in a rotating system. Alternate names include gyration radius and gyradius. The root mean square distance between a rotating object’s parts relative to an axis or gravitational center is a key element of calculating radius of gyration.

The gyration radius has applications in structural, mechanical and molecular engineering. It is denoted by the lowercase letter k or r and the uppercase letter R. The gyradius calculation is used by structural engineers to estimate beam stiffness and the potential for buckling. From a structural standpoint, a circular pipe has an equal gyradius in every direction, making the cylinder the most sufficient column structure to resist buckling.

Alternately, radius of gyration inertia can be described for a rotating object as the distance from the axis to the heaviest point on the body of the object that does not alter the rotational inertia. For these applications, the radius of gyration (R) formula is represented as the root mean square the second moment of inertia (I) divided by the cross-sectional area (A). Other formulas are used for mechanical and molecular applications.

For mechanical applications, the mass of an object is used to calculate the radius of gyration (r) instead of the cross-sectional area (A) as used in the previous formula. The mechanical engineering formula can be calculated using mass moment of inertia (I) and total mass (m). Therefore, the radius of gyration cylinder formula is equal to the root mean square of mass moment of inertia (I) divided by total mass (m).

Molecular applications are rooted in the study of polymer physics where the gyradius polymer represents the size of a protein for a specific molecule. The formula for determining the generation radius in a molecular engineering problem is facilitated by considering the mean distance between two monomers. It follows that the gyration radius in this sense is equivalent to the root mean square of that distance. Provided the nature of polymer chains, the gyration radius in a molecular application is understood to be a mean of all polymer molecules for a given sample over time. In other words, the gyration radius protein is an average gyradius.

Theoretical polymer physicists can use X-ray scattering technology and other light scattering techniques to compare models to reality. Static light scattering and small-angle neutron scattering are also used to verify the accuracy and preciseness of theoretical models used in polymer physics and molecular engineering. These analyses are used to study the mechanical properties of polymers and the kinetic reactions that can involve changes in molecular structures.