Polar Moment of Inertia Calculator
Calculate J for solid and hollow circular sections
Analyze polar moment of inertia for torsional stress and deformation in shafts.
Select shape
Quick Guide: Enter any one quantity below, and the calculator will determine the others automatically.
Parameters & Result
Introduction / Overview
The polar moment of inertia (usually written as J) is a geometry-based number that tells you how strongly a circular cross-section resists twisting. In practice, you’ll see it whenever you’re looking at shafts, axles, drill bits, couplers, or any round part that transmits torque.
Polar moment vs. area moment (a quick distinction)
- Polar moment (J) is used for torsion (twisting) of circular sections.
- Area moment of inertia (I) is used for bending (deflection/normal stress).
This calculator focuses on the two most common circular cases: a solid circular section and a hollow circular section(often called a tube). It supports both radius and diameter inputs and can also solve the reverse direction: if you know a target J, it can estimate a matching radius/diameter.
Once you have J, you’re one step away from estimating torsional shear stress and angle of twist (you’ll need torque, length, and material shear modulus).
Why do we need the polar moment of inertia?
If a round shaft is carrying torque, two questions usually come up fast: “How high is the shear stress?” and “How much will it twist?”For circular sections, both questions connect directly to J.
Two core torsion relationships (circular shafts)
τ = (T · ρ) / J
Shear stress at radius ρ under torque T
φ = (T · L) / (J · G)
Angle of twist φ over length L (G is shear modulus)
✅ Intuition you can trust: for the same torque, a larger J means lower stress and less twist.
Under the hood, J comes from the cross-section geometry: it is essentially “area elements weighted by squared radius.” That’s why pushing material farther from the center (bigger diameter or a tube shape) has a disproportionate impact.
Formula: solid circle polar moment of inertia
For a solid circular section (a “full” shaft), the polar moment of inertia is:
Solid circle
J = (π/2) · R⁴
If you prefer diameter: J = (π/32) · D⁴ (since R = D/2).
Variable meanings
- J: polar moment of inertia (units of length⁴, e.g., mm⁴, in⁴)
- R: outer radius
- D: diameter (D = 2R)
Formula: hollow cylinder polar moment of inertia
For a hollow circular section (tube), we subtract the “missing” core from the outer circle:
Hollow circle
J = (π/2) · (R⁴ − Rᵢ⁴)
In diameters: J = (π/32) · (D⁴ − d⁴)
🔍 Common design insight: a tube often gives a high J for its weight, because the material sits farther from the center.
Extra variables
- Rᵢ: inner radius
- d: inner diameter (d = 2Rᵢ)
How do I calculate the polar moment of inertia?
You can use the calculator in two ways: (1) enter dimensions and get J, or (2) enter J and let it solve for a required radius/diameter.
Select the shape
Choose Solid circular section or Hollow circular section.
Enter what you know
For a solid circle, enter R or D (or enter J to solve backward). For a hollow circle, enter any two of R, Rᵢ, and J.
Read the result (example calculations)
Example A (solid): D = 50 mm
J = (π/32) · D⁴ = (π/32) · (50⁴) ≈ 613,592 mm⁴
Example B (hollow): D = 60 mm, d = 40 mm
J = (π/32) · (D⁴ − d⁴) ≈ 1,021,018 mm⁴
How to interpret the number
- If J doubles, torsional stress and twist (for the same loading) are roughly cut in half.
- Because of the fourth-power term, small diameter changes can produce big jumps in J.
Real-world examples / use cases
Here are a few common scenarios where J is the first geometry number you’ll want in a torsion check. (These examples are meant to build intuition; final designs should always consider material properties, safety factors, and standards.)
Automotive driveshaft sizing
Background: you’re comparing two diameters for the same material. If diameter increases by 10%, J increases by about 1.1⁴ ≈ 1.46×.
Tooling & drill bits
Background: twist affects accuracy. Input: a solid shaft with D = 12 mm. Result: compute J to estimate how much torsional compliance you’ll have.
Bike axles & spindles
Background: weight matters. Try a hollow section and see how much J you keep while removing the core.
Power transmission shafts
Background: constant torque over long spans. Use J together with shaft length and material shear modulus to estimate twist.
3D-printed couplers
Background: polymers are flexible. A small diameter change can dramatically change torsional stiffness because J scales with the fourth power.
When you’re doing quick comparisons, compute the ratio instead of absolute numbers. For solid circles: J ∝ D⁴. For tubes: J ∝ (D⁴ − d⁴).
Common scenarios / when to use
Especially useful when…
- You have a shaft diameter from a datasheet and need a quick torsion stiffness estimate.
- You’re deciding between a solid rod and a tube for weight and stiffness.
- You’re reverse-engineering: “What diameter gives me this target J?”
⚠️ Not a good fit: non-circular cross-sections (rectangles, I-beams, channels). Those need a torsion constant, not this J.
If your problem is bending (not torsion), you’ll usually want the second moment of area about a bending axis (often written as I), not J.
Tips & best practices
- •Be consistent with radius vs. diameter. The formula changes only by a constant, but mixing inputs is a common mistake.
- •For hollow sections, confirm Rᵢ < R. If inner radius is larger, the geometry isn’t physically possible.
- •Expect big sensitivity. Because of the fourth power, rounding a diameter too aggressively can noticeably shift J.
- •Unit conversions are length⁴. Converting mm to m is ×1000, but converting mm⁴ to m⁴ is ×10¹².
If you’re deciding between designs, try “reverse mode”: enter a target J and see what diameter a solid shaft would need. Then switch to hollow and see whether you can match that J with less material.
Calculation method / formula explanation
The polar moment of inertia for a plane area is defined (geometrically) as:
Definition
J = ∫ ρ² dA
Where ρ is the distance from the center to a tiny area element dA.
For circles, this integral has a clean closed-form result, which is why engineers love circular shafts in torsion — the math stays tidy. For many non-circular shapes, torsion causes warping, and the simple circular relationships no longer hold.
Related concepts / background info
Perpendicular axis theorem (why you may see different notation)
In many textbooks, the polar moment about the out-of-plane axis equals the sum of second moments about the in-plane axes: J = I_x + I_y. That statement is about the geometry of a plane area.
⚠️ Easy mix-up: this polar moment of area is not the same as a mass moment of inertia used in dynamics. They share similar names, but the units and meaning are different.
Frequently asked questions (FAQs)
What are the units of polar moment of inertia?
The units are length⁴ (for example, mm⁴, cm⁴, m⁴, in⁴, or ft⁴). It’s the same dimensional type as the area moment of inertia used in bending.
What’s the difference between area moment of inertia and polar moment of inertia?
Area moment (I) is mainly used in bending problems (deflection and normal stress).Polar moment (J) is used in torsion problems (twist and torsional shear stress) for circular sections.
What is the polar moment of inertia of a circle of diameter 5 cm?
Using J = (π/32) · D⁴ with D = 5 cm:
J = (π/32) · (5⁴) ≈ 61.36 cm⁴
How do I calculate the polar moment of inertia of a hollow cylinder?
Use J = (π/2) · (R⁴ − Rᵢ⁴) (or J = (π/32) · (D⁴ − d⁴)). The calculator supports either radius or diameter inputs.
Why can’t I use this J for a rectangle or I-beam in torsion?
The neat torsion equations shown earlier assume the cross-section stays plane and does not warp. That assumption is true for circles (and close relatives), but not for most non-circular shapes. For those, you typically use a torsion constant from tables or more advanced analysis.
How do I calculate the “polar moment” of an ellipse?
For torsion of an elliptical section, engineers often use a torsion constant (sometimes written as K or Jt) rather than the circular polar moment. A common closed form for an ellipse is:
K = π a³ b³ / (a² + b²)
Here, a and b are the semi-axes.
Why does my result change so much when I tweak the diameter?
Because J scales with the fourth power: a small diameter change gets amplified. As a quick estimate, +10% diameter gives about +46% in J.
Can the inner radius be zero?
Yes. If Rᵢ = 0, the hollow formula collapses to the solid circle case.
Limitations / disclaimers & sources
- •This calculator applies to circular cross-sections (solid or hollow). Non-circular torsion requires different methods.
- •Results depend on ideal geometry. Real parts may have keyways, splines, fillets, holes, or manufacturing tolerances.
- •We compute J only. Stress/twist checks require torque, length, and material properties (e.g., shear modulus).
- •This content is for educational use and should not replace professional engineering judgment for safety-critical designs.
External references / further reading
- •Wikipedia: Polar moment of inertia
- •Standard mechanics of materials textbooks (e.g., “Mechanics of Materials” by Hibbeler) for torsion of circular shafts.
- •“Roark’s Formulas for Stress and Strain” for section properties and torsion constants across many shapes.
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