Quick answer
The second moment of area of a rectangle is bh³ ÷ 12 about its horizontal centroidal axis. A 50 × 100 mm rectangle has I = 4,166,667 mm⁴, I_y = 1,041,667 mm⁴ and an area of 5,000 mm².
Choose a section and enter its dimensions in one length unit. The calculator returns the second moment of area, the second axis value and the area. Use this moment of inertia calculator to check your own figures.
How to use the Moment of Inertia Calculator
- Choose section: Rectangle (a = width, b = height), Circle (a = diameter), Hollow circle (a = outer, b = inner diameter), Hollow rectangle (a, b = outer width, height; c, d = inner width, height) and Triangle (a = base, b = height).
- Enter dimension a (above 0).
- Optionally, enter dimension b (above 0).
- Optionally, enter dimension c (above 0).
- Optionally, enter dimension d (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Section | Choose one: Rectangle (a = width, b = height); Circle (a = diameter); Hollow circle (a = outer, b = inner diameter); Hollow rectangle (a, b = outer width, height; c, d = inner width, height); Triangle (a = base, b = height) | Starts at Rectangle (a = width, b = height) |
| Dimension a | A number | > 0 |
| Dimension b (optional) | A number | > 0 |
| Dimension c (optional) | A number | > 0 |
| Dimension d (optional) | A number | > 0 |
Formula and method
Formula
rectangle bh³/12; circle πd⁴/64; hollow circle π(D⁴ − d⁴)/64; hollow rectangle (BH³ − bh³)/12; triangle bh³/36 about the centroidal axis
IEEE-754 double precision; shown to the decimals indicated.
Results are about the centroidal axes in length⁴ for the unit you use; Iy for the triangle is that of an isosceles triangle.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Section = Rectangle (a = width, b = height); Dimension a = 50; Dimension b = 100.
| Result | Value |
|---|---|
| Moment of inertia (Ix) | 4,166,666.6667 |
| Iy (about the vertical axis) | 1,041,666.6667 |
| Area | 5,000 |
Example 2
Inputs: Section = Circle (a = diameter); Dimension a = 100.
| Result | Value |
|---|---|
| Moment of inertia (Ix) | 4,908,738.5212 |
| Iy (about the vertical axis) | 4,908,738.5212 |
| Area | 7,853.9816 |
Example 3
Inputs: Section = Hollow circle (a = outer, b = inner diameter); Dimension a = 100; Dimension b = 80.
| Result | Value |
|---|---|
| Moment of inertia (Ix) | 2,898,119.2229 |
| Iy (about the vertical axis) | 2,898,119.2229 |
| Area | 2,827.4334 |
The first line is the headline result (moment of inertia (Ix)). Below it the moment of Inertia Calculator lists iy (about the vertical axis) and area, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Why depth matters
Doubling the depth multiplies I by eight (h³), which is why deep beams are stiff. A 50 × 100 beam stood on its deep side has four times the stiffness of one laid flat: 4.17 million against 1.04 million mm⁴.
Other sections
Circle π d⁴ ÷ 64, hollow circle π (D⁴ − d⁴) ÷ 64, hollow rectangle (BH³ − bh³) ÷ 12 and triangle bh³ ÷ 36 about its centroid. Use one length unit throughout, and note that results are in that unit to the fourth power.
How the result changes with dimension a
Holding every other input at the first worked example, this table changes only dimension a. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Dimension a | Moment of inertia (Ix) | Iy (about the vertical axis) | Area |
|---|---|---|---|
| 25 | 2,083,333.3333 | 130,208.3333 | 2,500 |
| 50 | 4,166,666.6667 | 1,041,666.6667 | 5,000 |
| 100 | 8,333,333.3333 | 8,333,333.3333 | 10,000 |
| 200 | 16,666,666.6667 | 66,666,666.6667 | 20,000 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Dimension b: left empty shows "Enter this dimension." beside that field and no result.
Choosing between the Moment of Inertia Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Beam Deflection Calculator](/engineering/beam-deflection-calculator/) | beam and load, load and span | Maximum deflection |
| [Beam Bending Stress Calculator](/engineering/beam-bending-stress-calculator/) | bending moment, distance from neutral axis and second moment of area | Bending stress |
| [Column Buckling Calculator](/engineering/column-buckling-calculator/) | modulus of elasticity, smallest second moment of area and column length | Critical load |
| [Factor of Safety Calculator](/engineering/factor-of-safety-calculator/) | strength or capacity and applied load or stress | Factor of safety |
| [Strain Calculator](/engineering/strain-calculator/) | change in length and original length | Strain |
Background: structural and wiring estimates
Structural and electrical results are only as good as their assumptions. Bending stress is the bending moment divided by the section modulus, and an allowable stress is the yield strength divided by a safety factor that depends on the governing design code and the type of load. For a single-phase or DC run the voltage drop uses the round-trip length: V = 2 × L × R per unit length × I.
Recommended limits, such as keeping branch-circuit voltage drop around 3%, differ between sources and code editions. These pages are estimates: follow the applicable code and a qualified engineer or electrician for real work.
Assumptions and limitations
- Use one length unit throughout
Use results as engineering estimates and check them against the governing standards, safety factors and a qualified engineer's judgement before relying on them.
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for dimension b to confirm this.
Frequently asked questions
Is this the area or the mass moment of inertia?
The area (second) moment, used for beams.
Why is the I-beam efficient?
It puts material far from the axis.
What is the radius of gyration?
The square root of I divided by area.
What does the Moment of Inertia Calculator calculate?
Area moment of inertia of common cross-sections. It returns Moment of inertia (Ix), Iy (about the vertical axis) and Area.
What formula does the Moment of Inertia Calculator use?
rectangle bh³/12; circle πd⁴/64; hollow circle π(D⁴ − d⁴)/64; hollow rectangle (BH³ − bh³)/12; triangle bh³/36 about the centroidal axis. Results are about the centroidal axes in length⁴ for the unit you use; Iy for the triangle is that of an isosceles triangle.
What do I need to enter in the moment of Inertia Calculator?
You enter Section and Dimension a. Optional: Dimension b, Dimension c and Dimension d.
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Use one length unit throughout.
What does the “Section” option change?
It sets how the moment of Inertia is worked out. The choices are Rectangle (a = width, b = height), Circle (a = diameter), Hollow circle (a = outer, b = inner diameter), Hollow rectangle (a, b = outer width, height; c, d = inner width, height) and Triangle (a = base, b = height).
What is an example of a moment of Inertia result?
With section Rectangle (a = width, b = height), dimension a 50 and dimension b 100, the moment of inertia (Ix) is 4,166,666.6667.
Where does the method come from?
The method follows: Standard section properties (e.g. Gere & Goodno, Mechanics of Materials).
Can I rely on this calculator for a decision?
Use results as engineering estimates and check them against the governing standards, safety factors and a qualified engineer's judgement before relying on them. For this calculator in particular: Use one length unit throughout.
Related calculators
- Strain Calculator — Engineering strain from change in length.
- Stress Calculator — Normal or shear stress from force and area.
Sources
- Standard section properties (e.g. Gere & Goodno, Mechanics of Materials)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.