Radius of Gyration Calculator

Compute radius of gyration from area moment of inertia and cross-sectional area, or rearrange the formula to solve for inertia or area for slenderness and stability checks.

Configuration

Choose which quantity you want to solve for and how the main result should be reported.

Section Properties

Provide any two of the three section properties and the calculator will solve for the third. All computations are performed in consistent SI units internally.

Results Summary

The main result is shown below, with quick stats including consistent-unit values and the optional slenderness ratio \(L/k\).

Structural & Mechanical Design Guide

Radius of Gyration Calculator

Learn what the radius of gyration \(k\) really tells you about a cross-section, how it connects to moment of inertia, area, and slenderness ratio, and how to use the Radius of Gyration Calculator to make safer, code-aware design decisions.

8–10 min read Updated 2025

Quick Start

The Radius of Gyration Calculator is built around the core definition \[ k = \sqrt{\frac{I}{A}} \] where \(I\) is second moment of area about a chosen axis and \(A\) is the cross-sectional area. Follow these steps to get a reliable value and slenderness check.

  1. 1 Choose the axis you care about. Select strong axis, weak axis, or polar radius \(k_0\). Buckling checks usually care about the weak axis radius of gyration.
  2. 2 Select how you will define the section. Use standard shapes (e.g., rectangular, circular, I-section) or enter tabulated values \(I\) and \(A\) directly if you have them.
  3. 3 Keep your units consistent. If \(I\) is in \(\text{mm}^4\), make sure \(A\) is in \(\text{mm}^2\). The calculator handles the conversions, but mixing inches and millimetres without telling it will break the result.
  4. 4 Enter the unbraced length if you want slenderness ratio. The calculator can compute \[ \lambda = \frac{K L}{k} \] for your chosen effective length factor \(K\) and member length \(L\).
  5. 5 Review the outputs. The main result is the radius of gyration \(k\) in your selected units, plus any quick stats such as slenderness ratio and the equivalent radius in alternate units.
  6. 6 Use the steps & equations. Scroll to the calculation steps to see exactly how \(I\), \(A\), and \(k\) were combined, and compare with the worked examples below.
  7. 7 Sanity-check against typical ranges. For a given cross-section depth, \(k\) should feel like a fraction of that size. If a 300 mm deep member shows \(k = 5\) mm, something is wrong with the inputs.

Tip: Start with the weak axis radius of gyration for buckling checks, then compare with the strong axis for a feel of how “floppy” the section is in different directions.

Common mistake: Using the gross area but an effective moment of inertia (cracked section) in the same calculation. \(I\) and \(A\) must refer to the same section definition.

Choosing Your Method

There are several practical ways to get the radius of gyration. The calculator supports each of these approaches, depending on what information you have.

Method A — From Section Properties \(I\) and \(A\)

Fastest when you already have tabulated section properties from a handbook or FEM output.

  • Works for any arbitrary shape, including built-up or composite sections.
  • Matches what design codes typically use for column slenderness checks.
  • Easy to re-use for both axes: just switch \(I_x\) vs \(I_y\).
  • Requires reliable values for both \(I\) and \(A\).
  • Can hide mistakes if the wrong axis or units are selected.
Core definition: \(k = \sqrt{\dfrac{I}{A}}\)

Method B — From Geometry (Standard Shapes)

Ideal for simple shapes or quick hand checks where \(I\) is not already tabulated.

  • Great for rectangles, circles, and thin-walled tubes.
  • Helps build intuition about how shape changes affect stiffness.
  • Calculator can carry out the algebra and unit conversions for you.
  • Becomes tedious for complex or asymmetric sections.
  • Easier to mis-identify the correct axis or dimension.
Example: rectangle about the minor axis: \[ I_y = \frac{b h^3}{12}, \quad A = b h, \quad k_y = \sqrt{\frac{I_y}{A}} \]

Method C — From Design Tables / Software

Best when your shape is a standard rolled section and you have a code-approved table or model.

  • Directly uses manufacturer or code section properties (e.g., \(r_x, r_y\) in steel tables).
  • Reduces manual calculation errors.
  • Can be automated for many sections at once.
  • Less transparent — you see the result, not the formula.
  • Still need to check axis, units, and whether values are gross or effective.
Many steel tables list \(r_x, r_y\) directly, where \(r \equiv k\).

What Moves the Number

The radius of gyration is a compact way to express how spread-out the area is from the centroid. Bigger \(k\) means area is farther from the centroid and the section resists buckling better for a given area. These are the main “levers” that change \(k\).

Moment of inertia \(I\)

For a fixed area, increasing \(I\) (moving material away from the centroid) increases \(k\). Deep I-sections and hollow tubes typically give a much larger radius of gyration than solid bars.

Area \(A\)

For a fixed \(I\), increasing area decreases \(k\) because mass is concentrated closer to the centroid. Wide, compact sections can have high capacity but modest \(k\) values.

Axis of interest

Every section has different radii of gyration about different axes: \(k_x\) (strong axis), \(k_y\) (weak axis), and polar \(k_0\) for torsion. Buckling usually cares about the smallest \(k\).

Slenderness ratio \(\lambda\)

The effective slenderness \(\lambda = K L / k\) does not change \(k\) itself, but it tells you how “column-like” the member behaves. Longer unbraced length or smaller \(k\) both increase \(\lambda\).

Composite action

When steel, concrete, or timber act together, the combined \(I\) and \(A\) may change significantly. Using the wrong composite properties can give misleading \(k\) values.

Unit system

Radius of gyration carries length units. Switching from mm to m or in to ft must be handled consistently. The calculator normalizes internally, but your inputs need to match their declared units.

Worked Examples

These examples mirror what the Radius of Gyration Calculator does under the hood. Use them to cross-check results or to explain the concept to students and colleagues.

Example 1 — Rectangular Column in mm Units

  • Section: rectangular column, width \(b = 200\ \text{mm}\), depth \(h = 300\ \text{mm}\)
  • Axis: minor axis \(y\) (buckling about the weak axis)
  • Length: \(L = 3.0\ \text{m} = 3000\ \text{mm}\)
  • Effective length factor: \(K = 1.0\) (pinned–pinned as a starting assumption)
1
Compute area \(A\). \[ A = b h = 200 \times 300 = 60{,}000\ \text{mm}^2 \]
2
Compute second moment of area about the minor axis. \[ I_y = \frac{b h^3}{12} = \frac{200 \times 300^3}{12} = 4.50 \times 10^8\ \text{mm}^4 \]
3
Calculate radius of gyration \(k_y\). \[ k_y = \sqrt{\frac{I_y}{A}} = \sqrt{\frac{4.50 \times 10^8}{60{,}000}} \approx 86.6\ \text{mm} \]
4
Compute slenderness ratio. \[ \lambda_y = \frac{K L}{k_y} = \frac{3000}{86.6} \approx 35 \] The calculator reports both \(k_y\) and \(\lambda_y\) in the quick stats so you can compare to code limits.

In the Radius of Gyration Calculator, you can choose a “Rectangular Section” mode, enter \(b\), \(h\), \(L\), and \(K\), and it will perform exactly these steps with unit checks and rounding.

Example 2 — Solid Circular Bar Using Standard Formulas

  • Section: solid circular bar, diameter \(d = 150\ \text{mm}\)
  • Axis: any centroidal axis (section is axisymmetric)
  • Length: \(L = 2.4\ \text{m} = 2400\ \text{mm}\)
  • Effective length factor: \(K = 0.7\) (e.g., fixed–pinned condition)
1
Use standard formulas for a solid circle. \[ I = \frac{\pi d^4}{64}, \quad A = \frac{\pi d^2}{4} \]
2
Simplify the ratio \(I/A\). \[ \frac{I}{A} = \frac{\pi d^4 / 64}{\pi d^2 / 4} = \frac{d^2}{16} \Rightarrow k = \sqrt{\frac{I}{A}} = \frac{d}{4} \]
3
Insert the diameter. \[ k = \frac{150}{4} = 37.5\ \text{mm} \] The calculator will confirm this and also give the value in metres if requested: \(k = 0.0375\ \text{m}\).
4
Check slenderness. \[ \lambda = \frac{K L}{k} = \frac{0.7 \times 2400}{37.5} \approx 44.8 \] You can compare this against any code limits for axial compression members.

In the calculator’s “Circular Section” mode, you only enter \(d\), \(L\), and \(K\). It uses the same algebra and keeps track of unit conversions, so you can quickly test alternative diameters or lengths.

Common Layouts & Variations

Different cross-section layouts can have similar area but very different radii of gyration. The table below shows typical behaviours and where a higher or lower \(k\) matters in practice.

Section TypeRadius of Gyration BehaviourTypical UsesDesign Notes
Rolled I-section (strong axis)Large \(k_x\) because flanges put material far from the neutral axis.Beams and columns with major-axis bending / buckling.Great buckling resistance about strong axis; check weak-axis \(k_y\) separately.
Rolled I-section (weak axis)Much smaller \(k_y\) since web depth dominates.Columns in frames where lateral stability is critical.Often governs column slenderness; bracing and effective length are important.
Rectangular hollow section (RHS)High \(k\) in both directions relative to area, especially for thin walls.Columns, masts, frames, and architectural members.Efficient for buckling; local buckling limits may control wall slenderness.
Solid round barEqual \(k\) about any axis through centroid; moderate values for a given area.Tension/compression members, pins, and small columns.Simple to detail; may be heavier than hollow sections for the same buckling performance.
Built-up lattice or angle columnEffective \(k\) depends on spacing between chords and lacing details.Towers, trusses, industrial structures.Requires careful calculation of combined \(I\) and \(A\); check both main and local buckling modes.
Composite steel–concrete columnEffective \(k\) increases as concrete contributes to stiffness.Multi-storey buildings with composite frames.Use code-specified transformed section properties; cracked vs uncracked assumptions matter.
  • Confirm that the radius of gyration corresponds to the axis you are checking.
  • Use section properties from the same code or manufacturer used for capacity design.
  • Check for local buckling limits as well as overall slenderness.
  • For compound sections, verify the connection stiffness justifies treating them as fully composite.
  • Re-run the calculator if you change unbraced length or effective length factor \(K\).
  • Compare new designs to similar past projects to see if \(k\) and slenderness “feel right.”

Specs, Logistics & Sanity Checks

While the radius of gyration is a geometric quantity, it drives real-world decisions about section selection, bracing, and construction details. Use the following points to keep calculations grounded in how members are actually specified and built.

Specification & Code Checks

  • Identify which design standard you are using (steel, concrete, timber, aluminium, etc.).
  • Check how the code defines slenderness limits and which radius (\(k_x, k_y\), or \(k_0\)) is relevant.
  • Make sure section properties come from a trusted, up-to-date source.
  • Use the calculator’s slenderness output as input to design formulas, not as a standalone pass/fail.

Practical Modelling Choices

  • Match the effective length factor \(K\) to your frame analysis assumptions (pinned, fixed, sway, non-sway).
  • Use the same unbraced length in both the calculator and your global analysis model.
  • For partially restrained connections, be conservative with \(K\) until you have more detailed evidence.
  • Consider the effect of lateral bracing, sheathing, or slabs that may reduce the effective length.

Sanity Checks Before Finalizing

  • Compare \(k\) values for alternative sections with the same area — does the “stiffer-looking” shape really give a larger \(k\)?
  • Run a quick hand calculation using \(k = \sqrt{I/A}\) for at least one member to verify calculator settings.
  • Check that units in drawings, tables, and the calculator match (e.g., mm vs in, m vs ft).
  • Document the axis, length, and \(K\) assumptions used with each radius of gyration in design notes.

Once you are comfortable with the inputs, the Radius of Gyration Calculator becomes a quick “dial” to test different section shapes, lengths, and boundary conditions before committing to a final design.

Frequently Asked Questions

What is the radius of gyration in structural engineering?
The radius of gyration \(k\) is a geometric property that combines moment of inertia \(I\) and area \(A\) into a single length term: \(k = \sqrt{I/A}\). It describes how far the cross-section’s area is distributed from the centroid and is widely used in column slenderness and buckling checks.
How do I calculate radius of gyration from moment of inertia and area?
Use the definition \(k = \sqrt{I/A}\), where \(I\) is the second moment of area about the axis you care about (e.g., \(I_x\) or \(I_y\)) and \(A\) is the total cross-sectional area. Keep units consistent, such as \(I\) in \(\text{mm}^4\) and \(A\) in \(\text{mm}^2\), which gives \(k\) in millimetres.
How does radius of gyration relate to slenderness ratio?
Slenderness ratio is typically defined as \(\lambda = K L / k\), where \(L\) is the member length, \(K\) is an effective length factor that captures end conditions, and \(k\) is the radius of gyration about the relevant axis. Larger \(k\) reduces \(\lambda\) and generally improves buckling resistance for the same length and area.
Which axis radius of gyration should I use for buckling checks?
For axial compression, you normally check buckling about both principal axes and use the governing case. That means you compute slenderness with the smaller radius of gyration (the weak axis) and the matching effective length. The calculator lets you switch between axes so you can compare both quickly.
Does the radius of gyration depend on units?
Yes. Radius of gyration has length units, so if you input \(I\) and \(A\) in millimetres, you will get \(k\) in millimetres; if you use inches, you will get \(k\) in inches. The numeric value changes with the unit system but represents the same physical geometry. The calculator handles conversions as long as you pick the correct units.
Is radius of gyration the same as moment of inertia?
No. Moment of inertia \(I\) has units of length to the fourth power (e.g., \(\text{mm}^4\)) and measures bending stiffness about an axis, while radius of gyration \(k\) is a length derived from both \(I\) and area \(A\). They are related by \(k = \sqrt{I/A}\), but they are not interchangeable in design formulas.
Can I use the same radius of gyration for bending and buckling?
For elastic column buckling, the same geometric radius of gyration that appears in slenderness calculations is based on the same \(I\) used for bending about that axis. In other words, if you are checking buckling about the strong axis, use the \(k\) derived from the strong-axis moment of inertia. For inelastic or cracked sections, make sure that the \(I\) you choose matches the assumptions in the design code.
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