Radius of Gyration Calculator

Compute radius of gyration for common structural shapes. Essential for column buckling analysis, stability checks, and mechanical design.

Rect 6x12 (in)
Circle Ø10 (in)
Pipe 10x8 (in)
I 12x6 (in)
Square 5 (in)
Calculating...

What is Radius of Gyration?

The radius of gyration (symbol: k or r) is a geometric property of a cross-section that describes how the area is distributed around a centroidal axis. It is defined as the root mean square distance of the area’s points from the axis:

k = √( I / A )

where I = second moment of area (area moment of inertia) about the given axis, and A = cross-sectional area.

Physical Interpretation

  • If you imagine concentrating the entire cross-sectional area into a thin ring at a distance k from the axis, that ring would have the same moment of inertia as the actual shape.
  • A larger radius of gyration indicates that the material is spread farther from the axis, giving greater resistance to bending and buckling.
  • For the same area, a hollow section has a larger k than a solid section, which is why tubes are efficient for columns.

Why It Matters in Structural Engineering

  • Column Buckling: The Euler buckling load for a slender column is \( P_{cr} = \frac{\pi^2 E I}{(KL)^2} = \frac{\pi^2 E A k^2}{(KL)^2} \). Thus, k directly influences the critical load.
  • Slenderness Ratio: Defined as \( \lambda = \frac{L_{eff}}{k} \), where Leff is the effective length. Design codes (AISC, Eurocode) use λ to classify columns as short, intermediate, or slender, and to determine allowable stresses.
  • Vibrations: Natural frequencies of beams and frames depend on k. Higher k leads to higher stiffness and frequency.
  • Section Optimization: Engineers often choose shapes that maximize k for a given area to achieve lightweight, stable structures.

About Different Axes

The radius of gyration can be computed about any axis (e.g., x‑x, y‑y). For unsymmetrical sections, the minimum k governs buckling. In this calculator, we provide the radius of gyration about the horizontal centroidal axis (strong axis for most shapes). For I‑beams, the value about the weak axis would be different, but this tool focuses on the strong axis.

Common Formulas (for principal axes)

Shape Radius of Gyration
Rectangle (width b, height h) \( k_x = h / \sqrt{12} \) (about horizontal), \( k_y = b / \sqrt{12} \)
Circle (diameter D) \( k = D / 4 \)
Hollow circle (outer D, inner d) \( k = \frac{1}{4}\sqrt{D^2 + d^2} \)
Square (side a) \( k = a / \sqrt{12} \)
I‑section (symmetric) \( k = \sqrt{I / A} \) (calculated from flange/web dimensions)

Example

Consider a rectangle 6 in wide by 12 in tall. \(A = 72\ \text{in}^2\), \(I = \frac{6 \times 12^3}{12} = 864\ \text{in}^4\), so \(k = \sqrt{864 / 72} = \sqrt{12} = 3.464\ \text{in}\). This matches \(h/\sqrt{12} = 12/3.464 = 3.464\ \text{in}\).

Frequently Asked Questions

You can choose inches, millimeters, or centimeters from the dropdown. All input values are interpreted in the selected unit. Results will be displayed in the same unit (length), square unit (area), and fourth power unit (moment of inertia). No conversion is needed – just pick your preferred system.

The I‑beam is modeled as three rectangles: top flange, bottom flange, and web. The moment of inertia is computed about the horizontal centroidal axis using the parallel axis theorem. The result is accurate for symmetric I‑sections (equal flanges). Note that real rolled sections may have fillets, but this approximation is sufficient for most educational and preliminary design purposes.

Buckling resistance depends on the slenderness ratio (L/k). A larger k means the section is more resistant to buckling. Design codes specify maximum slenderness ratios to prevent failure. The radius of gyration appears directly in Euler's formula and all subsequent column design equations.