How to calculate a rectangle’s second moment of area
This geometric quantity measures how the area is distributed away from a chosen axis. For bending about the horizontal centroidal axis, depth matters strongly because it is cubed.
What the formula is saying
Use the axis through the centre and parallel to width b. Measure depth h perpendicular to that axis. The rectangle formula is I = bh³/12; swapping the bending axis swaps the roles of b and h.
Read the symbols in plain language
- b
- Widthm
- h
- Depthm
Sort out the units first
With b and h in metres, I is in m⁴, not m². 1 mm⁴ = 10⁻¹² m⁴. The example uses b = 0.3 m and h = 0.6 m.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
- b · Width
- 0.3 m
- h · Depth
- 0.6 m
Cube the depth
h³ means h × h × h, not h × 3.
(0.6)^3 = 0.216 m³Multiply by the width
This combines width with the cubed depth.
(0.3) × (0.216) = 0.0648 m⁴Divide by the rectangle factor
The factor 12 belongs to this shape and centroidal axis.
(0.0648) ÷ 12 = 0.0054 m⁴
Does this worked answer make sense?
Keep b fixed and double h: I becomes eight times larger. This is a useful check on the power of three.
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.
- b · Width
- 0.2 m
- h · Depth
- 0.3 m
Find: a rectangle’s second moment of area
A hint, not the answer
Use the axis through the centre and parallel to width b. Measure depth h perpendicular to that axis. The rectangle formula is I = bh³/12; swapping the bending axis swaps the roles of b and h.
With b and h in metres, I is in m⁴, not m². 1 mm⁴ = 10⁻¹² m⁴. The example uses b = 0.3 m and h = 0.6 m.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Cube the depth
h³ means h × h × h, not h × 3.
(0.3)^3 = 0.027 m³Multiply by the width
This combines width with the cubed depth.
(0.2) × (0.027) = 0.0054 m⁴Divide by the rectangle factor
The factor 12 belongs to this shape and centroidal axis.
(0.0054) ÷ 12 = 0.00045 m⁴
Avoid the common trap
Depth and width are not interchangeable for a fixed bending axis. This is an area property, not mass moment of inertia.
When this method applies — and when it does not
Solid rectangle about its centroidal axis parallel to b. Holes, composite sections and offset axes need a different calculation.
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.
One idea understood. Keep going.
This optional checkmark is saved only in this browser. It is your own progress note, not a certificate.
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.