Radius of Gyration Calculator
The radius of gyration (k) is a single length that summarizes how the mass of a rotating body is distributed relative to its axis of rotation. It satisfies the relation I = m k², or equivalently k = sqrt(I / m). A larger radius of gyration means the mass is spread farther from the axis, resulting in greater rotational inertia for the same total mass. This concept is used in structural engineering (column buckling resistance), vehicle dynamics (handling), and robotics (arm inertia). Enter the moment of inertia (kg m²) and total mass (kg) to calculate the radius of gyration.
Radius of gyration formula
k = √(I / m)
Where I is the moment of inertia (kg m²) and m is the total mass (kg). The radius of gyration k has units of meters (m).
Radius of gyration for common shapes
- Solid disk (axis through center): k = R / sqrt(2) = 0.7071 R
- Thin ring (axis through center): k = R
- Solid sphere: k = R sqrt(2/5) = 0.6325 R
- Thin rod about center: k = L / sqrt(12) = 0.2887 L
- Thin rod about end: k = L / sqrt(3) = 0.5774 L
Radius of gyration: frequently asked questions
What is the radius of gyration?
The radius of gyration (k) is the distance from the axis of rotation at which the entire mass of a body could be concentrated to give the same moment of inertia. It is calculated as k = sqrt(I / m), where I is the moment of inertia and m is the total mass.
What are the units of radius of gyration?
The radius of gyration has units of length (meters in SI). It is derived as sqrt(kg m² / kg) = sqrt(m²) = m.
How is radius of gyration used in structural engineering?
In structural engineering the radius of gyration of a cross-section (r = sqrt(I / A), where A is area) determines a column's resistance to buckling. Slender columns with a small radius of gyration buckle more easily under compressive loads.
What is the radius of gyration for a solid disk?
For a solid disk of mass m and radius R rotating about its central axis, I = (1/2)mR², so k = sqrt((1/2)mR²/m) = R / sqrt(2) = 0.7071 R.
Is the radius of gyration the same as the center of mass?
No. The center of mass is the point where mass can be considered concentrated for translational motion. The radius of gyration describes the effective distance for rotational inertia and depends on how the mass is distributed around the axis, not its center.
Official sources
- OpenStax University Physics Volume 1: Moment of Inertia and Rotational Kinetic Energy.
- NIST Reference on Constants, Units, and Uncertainty: NIST Physics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.