How to calculate radius of gyration of an area
This radius is a geometric summary of how far an area is spread from an axis. It is useful when forming slenderness ratios, but is not necessarily a physical radius you can measure on the shape.
What the formula is saying
The definition I = Ai² gives i = √(I/A). Divide first, then take the square root. Both I and A must describe the same section.
Read the symbols in plain language
- I
- Second moment of aream⁴
- A
- Aream²
Sort out the units first
m⁴ divided by m² gives m²; its square root gives m. Use I about the axis relevant to your question.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
- I · Second moment of area
- 0.0054 m⁴
- A · Area
- 0.18 m²
Find the squared radius
Dividing the area moment by area leaves a squared length.
(0.0054) ÷ (0.18) = 0.03 m²Take the square root
Undo the square to return to an ordinary length.
√((0.03)) ≈ 0.1732050808 m
Does this worked answer make sense?
For the example rectangle, h/√12 = 0.6/√12 ≈ 0.173205 m is an independent check. Keep extra digits until the final answer.
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Use these new values. Work it out first, then check your answer.
- I · Second moment of area
- 0.00045 m⁴
- A · Area
- 0.06 m²
Find: radius of gyration of an area
A hint, not the answer
The definition I = Ai² gives i = √(I/A). Divide first, then take the square root. Both I and A must describe the same section.
m⁴ divided by m² gives m²; its square root gives m. Use I about the axis relevant to your question.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find the squared radius
Dividing the area moment by area leaves a squared length.
(0.00045) ÷ (0.06) = 0.0075 m²Take the square root
Undo the square to return to an ordinary length.
√((0.0075)) ≈ 0.08660254038 m
Avoid the common trap
Stopping after I/A leaves i², not i. Do not use the radius of a circle unless the problem actually asks for it.
When this method applies — and when it does not
Area radius of gyration, not mass radius of gyration. This calculation alone does not determine column capacity or an effective length.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Lesson updated: · Worked examples checked against the implemented formula; not an independent engineering certification.