Radius of Gyration Calculator
Calculate radius of gyration from section properties, common shape dimensions, or column slenderness inputs.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Use direct section properties, common section shapes, or column slenderness inputs.
Choose the unknown or section-property output. Required inputs update automatically.
Section radii mode calculates rx, ry, rmin, and polar radius from real x/y section properties.
In direct formula modes, this selector updates the equation labels and tells you which matching I and r values to enter.
End-condition presets fill in a typical idealized K value. Verify final design against the governing standard.
Enter the known values
Only the values needed for the selected calculation are shown.
Visual Check
The diagram updates by solve mode so labels stay separated and readable on desktop and mobile.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Check—
Show solution steps See equations, unit conversions, substitutions, checks, and assumptions
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Radius of gyration is calculated using the standard relationship between area moment of inertia and cross-sectional area.
- Assumptions will appear after a valid calculation.
Engineering Calculator Guide
How to Use the Radius of Gyration Calculator
The radius of gyration calculator above finds r from moment of inertia and area, solves for I or A, calculates rx, ry, rmin, and ro from common shapes, and checks column slenderness using KL/r. Use this guide to choose the correct inputs, units, and axis.
Article Contents
Radius of Gyration Calculator Quick Answer
Radius of gyration is calculated from the relationship between area moment of inertia and cross-sectional area. It describes how far a section’s area is effectively distributed from a selected centroidal axis. For column checks, the smallest radius, usually called rmin, is commonly used in the slenderness ratio.
Best For
Calculating radius of gyration, section radii, and basic column slenderness checks.
Main Result
r, rx, ry, rmin, ro, or KL/r.
Key Input
Use the moment of inertia and radius that match the same axis.
Quick Answer
If you already know moment of inertia and area, use r = √(I/A). If you are checking a column, calculate both rx and ry, then use the smaller value as rmin unless your design method specifies otherwise.
Important Axis Warning
Do not mix axes. Use Ix with rx, Iy with ry, and J with ro. Changing the axis selector in a direct formula mode changes the equation labels and interpretation; the number changes only when the entered matching values change.
Calculator Inputs and Outputs
The calculator supports direct formula solving, shape-based section properties, and slenderness ratio checks. The exact inputs shown above depend on the selected solve mode.
| Type | Value | What It Means | Common Units |
|---|---|---|---|
| Input | \(I_x\), \(I_y\) | Area moment of inertia about centroidal x and y axes. | in⁴, ft⁴, mm⁴, cm⁴, m⁴ |
| Input | \(A\) | Total cross-sectional area. | in², ft², mm², cm², m² |
| Input | \(r\), \(r_x\), \(r_y\) | Radius of gyration about the selected axis. | in, ft, mm, cm, m |
| Input | \(K\), \(L\) | Effective length factor and unsupported column length. | dimensionless, length |
| Output | \(r_x\), \(r_y\), \(r_{min}\) | Axis-specific radii and the controlling least radius. | length |
| Output | \(r_o\) | Polar radius of gyration based on \(I_x + I_y\). | length |
| Output | \(\lambda = KL/r\) | Slenderness ratio for preliminary column checks. | dimensionless |
Radius of Gyration Formula
The basic radius of gyration formula is:
For a cross-section with x and y centroidal axes:
The polar radius of gyration uses the polar second moment:
The calculator can also rearrange the formula:
For a slenderness ratio check:
Variable Definitions
| Variable | Name | Meaning |
|---|---|---|
| \(r\) | Radius of gyration | Effective distance from an axis where area could be concentrated to produce the same moment of inertia. |
| \(r_x\) | Radius of gyration about x-axis | Calculated using \(I_x\) and area \(A\). |
| \(r_y\) | Radius of gyration about y-axis | Calculated using \(I_y\) and area \(A\). |
| \(r_{min}\) | Least radius of gyration | The smaller of \(r_x\) and \(r_y\), often controlling for column buckling. |
| \(r_o\) | Polar radius of gyration | Calculated using \(J = I_x + I_y\). |
| \(I\) | Area moment of inertia | Second moment of area about a selected axis. |
| \(I_x\) | Moment of inertia about x-axis | Used with \(r_x\). |
| \(I_y\) | Moment of inertia about y-axis | Used with \(r_y\). |
| \(J\) | Polar second moment of area | For centroidal perpendicular axes, \(J = I_x + I_y\). |
| \(A\) | Cross-sectional area | Total area of the section. |
| \(K\) | Effective length factor | Accounts for idealized column end restraint. |
| \(L\) | Unsupported length | Unbraced member length before applying \(K\). |
| \(\lambda\) | Slenderness ratio | Dimensionless ratio \(KL/r\). |
How to Use the Radius of Gyration Calculator
- Choose what to solve for. Select radius of gyration, section radii, moment of inertia, area, or slenderness ratio.
- Select the axis or radius used. For direct formula modes, choose \(r_x\), \(r_y\), \(r_{min}\), or \(r_o\). This updates the equation labels and tells you which values to enter.
- Enter matching values. Use \(I_x\) with \(r_x\), \(I_y\) with \(r_y\), and \(J\) with \(r_o\).
- Choose units carefully. Moment of inertia uses length⁴, area uses length², and radius of gyration uses length.
- Review the result. Check the quick stats, warnings, visual diagram, and solution steps to verify the calculation path.
Shape Modes
In section radii mode, the calculator can calculate \(A\), \(I_x\), \(I_y\), \(r_x\), \(r_y\), \(r_{min}\), and \(r_o\) from common shapes such as rectangles, hollow rectangles, circles, and pipes.
How to Interpret the Results
Radius of gyration is not the same thing as the physical radius of a shape. It is a section-property distance that describes how efficiently the area is distributed away from an axis.
| Result | Interpretation |
|---|---|
| Larger \(r\) | More area is effectively distributed farther from the selected axis. |
| Smaller \(r\) | The section is weaker about that axis for buckling-related behavior. |
| \(r_{min}\) | The least radius of gyration; commonly used for column slenderness checks. |
| \(r_o\) | Polar radius based on \(I_x + I_y\), not normally used as the controlling column radius. |
| High \(KL/r\) | The member is more slender and more sensitive to buckling. |
Input Checklist Before Calculating
- Use centroidal moments of inertia unless the problem states otherwise.
- Pair \(I_x\) with \(r_x\), \(I_y\) with \(r_y\), and \(J\) with \(r_o\).
- Use cross-sectional area, not surface area.
- Make sure inner dimensions are smaller than outer dimensions for hollow sections.
- Do not mix inches and millimeters unless the unit selectors convert them intentionally.
- Use \(r_{min}\) for most column slenderness checks.
- Remember that \(I\) uses length⁴ units and area uses length² units.
Worked Example: Finding rx, ry, and rmin
Suppose a section has:
- \(I_x = 120 \, \text{in}^4\)
- \(I_y = 45 \, \text{in}^4\)
- \(A = 24 \, \text{in}^2\)
Calculate \(r_x\):
Calculate \(r_y\):
The least radius of gyration is:
Result
The section’s weak-axis radius is \(r_y = 1.369\) in, so \(r_{min}\) is 1.369 in. For a column slenderness check, this is the radius that would typically be used.
Radius of Gyration Diagram
The diagram below shows the idea behind radius of gyration: a cross-section has centroidal x and y axes, and the calculated radius depends on which moment of inertia is paired with the area.
Reference Formulas for Common Shapes
Radius of gyration does not have one universal “good” value. It depends on geometry and axis. These formulas are useful checks for common ideal shapes.
| Shape | Area and Inertia | Radius of Gyration |
|---|---|---|
| Rectangle | \(A = bh\), \(I_x = bh^3/12\), \(I_y = hb^3/12\) | \(r_x = h/\sqrt{12}\), \(r_y = b/\sqrt{12}\) |
| Solid Circle | \(A = \pi D^2/4\), \(I = \pi D^4/64\) | \(r = D/4\) |
| Hollow Circle / Pipe | \(A = \pi(D_o^2-D_i^2)/4\), \(I = \pi(D_o^4-D_i^4)/64\) | \(r = \sqrt{I/A}\) |
| Hollow Rectangle | \(A = B_oH_o-B_iH_i\) | \(r_x = \sqrt{I_x/A}\), \(r_y = \sqrt{I_y/A}\) |
Design Ranges and Practical Meaning
Radius of gyration itself is a geometric property, so it does not have universal pass/fail limits. A larger value generally means the section’s area is distributed farther from the selected axis, while a smaller value often identifies the weak axis.
Column Design Caution
The slenderness ratio \(KL/r\) is only a screening calculation. Final column design depends on material, loads, boundary conditions, unbraced length, code provisions, safety factors, resistance factors, and applicable local requirements.
| Check | How to Think About It |
|---|---|
| Compare \(r_x\) and \(r_y\) | The smaller value usually indicates the weak axis. |
| Use \(r_{min}\) | Commonly used for column slenderness checks. |
| Review \(KL/r\) | Higher values indicate a more slender member and greater buckling sensitivity. |
| Check units | Unexpectedly large or small results usually come from unit mismatch. |
Radius of Gyration Units
Radius of gyration has length units because the formula divides a fourth-power length quantity by an area quantity, then takes the square root.
Common Unit Trap
Moment of inertia conversion uses the fourth power of length. For example, converting from in⁴ to mm⁴ is not the same as converting inches to millimeters. The length conversion factor must be raised to the fourth power.
Radius of Gyration Compared With Related Terms
| Term | What It Means | How It Relates |
|---|---|---|
| Physical radius | Actual geometric distance on a circular object. | Not the same as radius of gyration, although for a solid circle \(r = D/4\). |
| Area moment of inertia | Measures how area is distributed about an axis. | Radius of gyration is derived from \(I/A\). |
| \(r_x\) and \(r_y\) | Axis-specific radii of gyration. | Used to compare strong and weak section axes. |
| \(r_{min}\) | Smallest radius of gyration. | Often controls column slenderness. |
| \(r_o\) | Polar radius of gyration. | Uses \(J = I_x + I_y\), not usually the controlling radius for column buckling. |
Common Mistakes to Avoid
Do
- Use \(I_x\) with \(r_x\).
- Use \(I_y\) with \(r_y\).
- Use \(J\) with \(r_o\).
- Use \(r_{min}\) for most slenderness checks.
- Convert length⁴ and length² units correctly.
Don’t
- Use surface area instead of cross-sectional area.
- Mix \(I_x\) with \(r_y\).
- Use polar radius for column slenderness.
- Assume axis selection alone changes the result.
- Use non-centroidal inertia unless the problem requires it.
Troubleshooting Radius of Gyration Results
| Problem | Likely Cause | Fix |
|---|---|---|
| Result is blank or not a number | A required input is missing, zero, or negative. | Enter positive nonzero values for required fields. |
| Radius is extremely large | Moment of inertia units may be wrong. | Check whether the input should be in⁴, ft⁴, mm⁴, or m⁴. |
| \(r_x\) and \(r_y\) seem swapped | Width and height orientation may be reversed. | Confirm which direction is x and which is y for the section. |
| Slenderness seems too low | The larger radius may have been used instead of \(r_{min}\). | Use the least radius of gyration for column checks unless otherwise specified. |
| Area result changes unexpectedly | Inconsistent inertia and radius pair. | Use \(I_x/r_x^2\), \(I_y/r_y^2\), or \(J/r_o^2\) consistently. |
Assumptions and Limitations
- This calculator is for educational and preliminary engineering calculations.
- Moments of inertia are assumed to be based on centroidal axes unless otherwise stated.
- Shape formulas assume ideal geometry with no fillets, rounded corners, holes beyond those entered, or manufacturing tolerances.
- The calculator does not apply material strength, load factors, resistance factors, or safety factors.
- The slenderness ratio output does not verify structural capacity or code compliance.
- Final structural design should be checked by a qualified professional using the applicable design standard.
For deeper background on section properties and column behavior, consult the applicable structural mechanics textbook or governing design code used for your project. This page is intended to support calculator use, not replace formal engineering design.
Glossary
- Radius of Gyration
- A length-based section property calculated from \(r = \sqrt{I/A}\).
- Area Moment of Inertia
- A geometric property that measures how area is distributed about an axis.
- Centroidal Axis
- An axis passing through the centroid of a cross-section.
- Weak Axis
- The axis associated with the smaller radius of gyration or lower bending resistance.
- Polar Radius of Gyration
- A radius based on \(J = I_x + I_y\), commonly written as \(r_o\).
- Slenderness Ratio
- A dimensionless column check calculated from \(KL/r\).
Radius of Gyration Calculator FAQ
What does a radius of gyration calculator calculate?
A radius of gyration calculator finds \(r\) from the relationship \(r = \sqrt{I/A}\), where \(I\) is the area moment of inertia and \(A\) is cross-sectional area. More complete versions also calculate \(r_x\), \(r_y\), \(r_{min}\), polar radius \(r_o\), and slenderness ratio \(KL/r\).
What is the formula for radius of gyration?
The radius of gyration formula is \(r = \sqrt{I/A}\). For axis-specific section properties, \(r_x = \sqrt{I_x/A}\) and \(r_y = \sqrt{I_y/A}\).
What is the difference between rx, ry, and rmin?
\(r_x\) is the radius of gyration about the x-axis, \(r_y\) is the radius of gyration about the y-axis, and \(r_{min}\) is the smaller of \(r_x\) and \(r_y\). The least radius \(r_{min}\) is commonly used for column slenderness checks.
What units should I use for I, A, and r?
Moment of inertia uses length to the fourth power, such as in⁴ or mm⁴. Area uses length squared, such as in² or mm². Radius of gyration uses length units, such as inches or millimeters.
Can this calculator be used for final column design?
No. The calculator can compute radius of gyration and \(KL/r\) slenderness ratio, but it does not check material strength, load factors, resistance factors, buckling capacity, or code compliance.