Radius of Gyration Calculator
Calculate the radius of gyration (r) for structural sections. Essential for column buckling analysis and slenderness ratio calculations.
Input Method
Minimum Radius of Gyration
3.1623 in
Example Slenderness (12 ft column)
Formula
r = √(I/A)
The radius of gyration represents the distance from the centroid at which the entire area could be concentrated to give the same moment of inertia.
What is Radius of Gyration?
The radius of gyration (r) is a geometric property of a cross-section that represents the distance from the neutral axis at which the entire cross-sectional area could be concentrated to produce the same moment of inertia. It is a fundamental parameter in structural engineering, particularly for column buckling analysis and slenderness ratio calculations.
The radius of gyration is calculated as the square root of the moment of inertia divided by the cross-sectional area: r = √(I/A). A larger radius of gyration indicates that the area is distributed farther from the neutral axis, making the member more resistant to buckling.
This calculator computes the radius of gyration for rectangular, solid circular, and hollow circular cross-sections. It also accepts direct input of moment of inertia and area for custom or complex sections. The results include radii about both axes and an example slenderness calculation for a 12-foot column.
The Radius of Gyration Formula
The fundamental formula for radius of gyration is derived from the definition of moment of inertia:
Radius of Gyration
Where:
- r= Radius of gyration (inches)
- I= Moment of inertia about the axis (in⁴)
- A= Cross-sectional area (in²)
Radius of Gyration for Common Shapes
The radius of gyration varies significantly depending on the cross-sectional shape and the axis of consideration:
| Shape | rx (Strong Axis) | ry (Weak Axis) |
|---|---|---|
| Rectangle (b × h) | h / √12 | b / √12 |
| Solid Circle (dia D) | D / 4 | D / 4 |
| Hollow Circle (Do, Di) | √(Do² + Di²) / 4 | √(Do² + Di²) / 4 |
For rectangles, the strong axis (rx) uses the height while the weak axis (ry) uses the width. The minimum radius of gyration governs column buckling design because buckling occurs about the weakest axis.
Connection to Slenderness Ratio
The radius of gyration is directly used in the slenderness ratio calculation: λ = L/r, where L is the unsupported length of the member. The slenderness ratio determines whether a column is classified as short (stocky) or slender:
- Short columns (λ < 22): Strength is governed by material capacity, not buckling.
- Intermediate columns (22 < λ < 100): Inelastic buckling may govern — reduce capacity using empirical formulas.
- Slender columns (λ > 100): Elastic buckling governs — use Euler's formula for capacity.
A larger radius of gyration produces a lower slenderness ratio for the same length, resulting in higher column capacity. This is why engineers prefer wide-flange sections and hollow tubes over solid bars for columns — they distribute area farther from the neutral axis.
How to Use This Calculator
The calculator offers two input methods:
- From I and A: Enter the moment of inertia and cross-sectional area directly. This is useful for complex or built-up sections where you have the properties from a table or analysis.
- From Dimensions: Select a shape (rectangle, solid circle, or hollow circle) and enter the dimensions. The calculator computes I, A, and r automatically.
Results include rx, ry, the minimum radius of gyration, section area, moments of inertia about both axes, and an example slenderness calculation for a 12-foot column.
Real-World Applications
The radius of gyration is essential for column design in steel and concrete structures. Steel column selection tables (AISC Manual) list r values for every section, allowing engineers to quickly determine the slenderness ratio and look up the compressive strength.
Engineers use the minimum radius of gyration to determine the buckling resistance of compression members, design bracing systems, and verify that slender elements meet code requirements. It is also used in the design of masonry walls, precast concrete columns, and timber posts.
Worked Examples
Rectangle Radius of Gyration
Problem:
Calculate the radius of gyration for a 4-inch × 8-inch rectangular section about both axes.
Solution Steps:
- 1Area = 4 × 8 = 32 in²
- 2Ix = 4 × 8³ / 12 = 170.67 in⁴
- 3Iy = 8 × 4³ / 12 = 42.67 in⁴
- 4rx = √(170.67 / 32) = 2.309 in (= 8 / √12)
- 5ry = √(42.67 / 32) = 1.155 in (= 4 / √12)
Result:
rx = 2.309 in, ry = 1.155 in, rmin = 1.155 in
Hollow Circle Radius of Gyration
Problem:
Calculate the radius of gyration for a hollow circular section with 8-inch OD and 6-inch ID.
Solution Steps:
- 1Area = π/4 × (8² - 6²) = 21.99 in²
- 2Ix = π/64 × (8⁴ - 6⁴) = 137.44 in⁴
- 3r = √(137.44 / 21.99) = 2.50 in
- 4Verify: r = √(8² + 6²) / 4 = √100 / 4 = 10/4 = 2.50 in
Result:
r = 2.50 inches (equal for both axes due to symmetry)
Slenderness Check
Problem:
A 12-foot column has rmin = 1.5 inches. Is it short, intermediate, or slender?
Solution Steps:
- 1Convert length: L = 12 × 12 = 144 inches
- 2Slenderness ratio: λ = L / rmin = 144 / 1.5 = 96
- 3Compare: λ = 96 is in the range 22–100, so it is an intermediate column
- 4The column may be subject to inelastic buckling — check AISC Table 4-22 for φFc.
Result:
Slenderness ratio = 96 (intermediate column, inelastic buckling may govern)
Tips & Best Practices
- ✓Always use the minimum radius of gyration for column buckling calculations — it governs the critical buckling mode.
- ✓Hollow circular sections (pipes) are excellent columns because they distribute area far from the centroid, giving a high r for their weight.
- ✓For W-shapes, r_y is typically much smaller than r_x — weak-axis buckling usually governs unless braced.
- ✓The radius of gyration is independent of the material — it depends only on the cross-sectional geometry.
- ✓When comparing sections, a higher r/A ratio indicates a more efficient use of material for buckling resistance.
- ✓Use the slenderness ratio to quickly screen whether a column needs detailed buckling analysis.
- ✓For concrete columns, the radius of gyration of the gross section is approximately 0.3 times the dimension (r ≈ 0.3h).
Frequently Asked Questions
Sources & References
Last updated: 2026-06-06
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
Formula Source: Standard Mathematical References
by Various