Second Moment of Area Calculator
Compute the second moment of area (I) for rectangular, solid circular, and hollow circular cross-sections.
🔲 What is the Second Moment of Area?
The second moment of area (also called the area moment of inertia) is a geometric property of a cross-section that measures how its material is distributed relative to a reference axis, typically the centroidal bending axis. It appears directly in every beam bending and deflection formula: a section with a larger second moment of area resists bending more effectively for the same material and load.
Structural and mechanical engineers use this calculator constantly. Steel beam selection tables list I for every standard section so engineers can pick the lightest section that meets a deflection or stress requirement. Machine shaft design uses I to size shafts against bending under gear and pulley loads. Furniture and product designers use it to decide how thick a shelf or bracket must be to avoid sagging.
A common misconception is that a heavier or larger cross-section is automatically stiffer. In fact, the same amount of material arranged farther from the centroidal axis (a hollow tube instead of a solid bar, or an I-beam instead of a solid rectangle) produces a dramatically larger second moment of area, because I depends on the square of the distance from the axis, integrated over the whole area. This is exactly why structural sections are shaped the way they are.
This calculator computes I for three of the most common cross-sections used in structural and mechanical design: a solid rectangle, a solid circle, and a hollow circular tube, returning results in both mm^4 and cm^4 with full working shown.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Rectangular Timber Beam Section
A 200 mm × 400 mm rectangular timber section
Example 2 — Solid Round Steel Bar
A 150 mm diameter solid steel bar
Example 3 — Hollow Steel Tube (Pipe Section)
A steel tube with 200 mm outer diameter and 180 mm inner diameter
❓ Frequently Asked Questions
🔗 Related Calculators
What is the second moment of area?
The second moment of area (also called the area moment of inertia) measures how a cross-section's material is distributed relative to a reference axis, usually the centroidal axis. It appears directly in beam bending and deflection formulas: a larger I means a stiffer, more bending-resistant section.
What is the formula for the second moment of area of a rectangle?
For a rectangle of width b and height h, about the centroidal axis parallel to the width (the usual bending axis), I = bh^3/12. Height matters far more than width because it is cubed, doubling h increases I by 8x while doubling b only doubles I.
What is the formula for a solid circular section?
For a solid circle of diameter d, I = pi*d^4/64, taken about any diameter through the centroid (a circle is symmetric, so I is the same about every centroidal axis).
What is the formula for a hollow circular tube?
For a tube with outer diameter D and inner diameter d, I = pi*(D^4 - d^4)/64. This subtracts the hollow inner circle's contribution from the solid outer circle's contribution.
Why does the second moment of area use the fourth power of length?
I is computed by integrating the squared distance from the axis (y^2) over the cross-sectional area (dA), and since area itself scales with length squared, the result scales with length to the fourth power overall.
What units does this calculator use?
All dimensions are entered in millimeters (mm), and the result is displayed in both mm^4 (the standard unit for smaller structural and mechanical sections) and cm^4 (divide by 10,000), whichever is more convenient for your reference tables or design code.
Why must the outer diameter be greater than the inner diameter for a tube?
The hollow circle (tube) formula subtracts the inner circle's I from the outer circle's I. If the outer diameter were not larger than the inner diameter, there would be no wall material left, so the calculator validates D greater than d before computing.
Is second moment of area the same as moment of inertia used in dynamics?
No, despite the common name overlap. The second moment of area (used here, for bending stiffness) is a purely geometric property of a 2D cross-section (length^4). Mass moment of inertia (used in dynamics and rotation) depends on mass distribution (mass times length^2). They share mathematical form but different physical meaning.
How is the second moment of area used after I calculate it?
I feeds directly into beam deflection formulas (delta = PL^3/48EI for a point load) and into the section modulus (Z = I/c) used for bending stress calculations. Use the Beam Deflection Calculator and Section Modulus Calculator on this site to take I further.
Which cross-section is stiffest for the same material area?
For a given amount of material, a hollow tube or an I-beam shape is far stiffer in bending than a solid circular or rectangular bar, because moving material away from the centroidal axis increases I much faster than it increases area, this is exactly why structural steel sections are shaped like I-beams and hollow tubes rather than solid bars.