Polar Moment of Inertia Calculator

Calculate polar moment of inertia (J), shear stress, and angle of twist for solid circles, hollow shafts, rectangles, and custom polygons.

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Published - March 09, 2026
Updated - June 21, 2026

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The Polar Moment of Inertia Calculator helps engineers and students analyze torsional properties of cross-sections. Enter your shaft geometry — solid circle, hollow tube, rectangle, or custom polygon — alongside an applied torque, shaft length, and material to instantly compute J, cross-sectional area, radius of gyration, polar section modulus, shear stress, and angle of twist. Results update live, with a visual diagram and optional side-by-side comparison of multiple cross-sections.

Polar Moment of Inertia (J) Calculator + Torsion Check

Polar Moment of Inertia Calculator - Compute torsional resistance (J) for solid/hollow circles, rectangles, ellipses. Shear stress, angle of twist, and more.

Brand accent: Outdoor-friendly contrast • Large tap targets

Quick presets:

1) Inputs (Geometry, Units, Material)

Shape type

Microcopy: For non-circular shapes, torsion behavior can differ from “circular-shaft” assumptions; this tool shows accuracy notes below.

Length unit (inputs)

J output unit

Stress unit

Tip: Keep all dimensions in one unit to avoid mistakes. The calculator converts internally to SI for reliability.

Geometry

Input mode

Common mistake: mixing radius and diameter. Choose one mode.

Value

Must be > 0.

Torsion (optional, for τ and θ)

Applied torque (T)

Microcopy: Don’t confuse N·mm vs N·m (1000× factor).

Shaft length (L)

Tip: Use actual torque span length (between constraints).

Material (Shear modulus G)

G affects twist θ, not geometric J.

CTA: After you compute, hit Copy report to paste into a lab report, email, or design log.

Accuracy notes (build trust)

2) Results (J, A, k, Zₚ + torsion outputs)

Units are shown per row. Empty torsion fields mean you didn’t supply torque/length/material.
Output	Value	Unit	Notes
Area (A)	—	—	Cross-sectional area
Polar moment (J)	—	—	Computed about centroidal axis
Radius of gyration (k = √(J/A))	—	—	Useful for comparing efficiency
rₘₐₓ (outer fiber distance)	—	—	Used for Zₚ and stress estimate
Polar section modulus (Zₚ = J / rₘₐₓ)	—	—	Directly ties to shear stress
Max shear stress estimate (τ ≈ T·rₘₐₓ / J)	—	—	Exact for circular shafts
Angle of twist (θ = T·L / (G·J))	—	—	Small-angle assumption
Torsional stiffness (kₜ = G·J / L)	—	—	Higher = stiffer shaft

3) Visuals (SVG section + chart)

Cross-section diagram (updates with inputs) SVG: ready

Shear stress shape (normalized) Chart

For circular shafts, \(\tau(r)\) is linear in radius. For non-circular sections, distribution differs—use this chart as a teaching/intuition aid.

Shape Library (quick reference)

Formula transparency • copy-friendly

These are the most common cross-sections engineers use in torsion checks. The same formulas are used by the calculator outputs.

Solid Circle (Shaft)

Exact (circular torsion)

\[ \begin{aligned} J &= \frac{\pi}{2}r^4 \\ &= \frac{\pi}{32}D^4 \end{aligned} \]

Microcopy: Doubling diameter increases \(J\) by \(2^4 = 16\times\).

Hollow Circle (Tube / Annulus)

Exact (circular torsion)

\[ \begin{aligned} J &= \frac{\pi}{2}\left(r_o^4 - r_i^4\right) \\ &= \frac{\pi}{32}\left(D_o^4 - D_i^4\right) \end{aligned} \]

Common mistake: ensure \(D_i < D_o\) (or \(r_i < r_o\)).

Rectangle

J = Iₓ + Iᵧ (polar about centroid)

\[ \begin{aligned} I_x &= \frac{b h^3}{12}, \quad I_y = \frac{h b^3}{12} \\ J &= I_x + I_y \end{aligned} \]

Accuracy note: torsional constant for rectangles can differ from \(I_x+I_y\) due to warping.

Custom Polygon

Exact polygon moments (vertex sums)

\[ \begin{aligned} J &= I_x + I_y \\ I_x, I_y &\text{ from vertex-sum polygon formulas (Green's theorem).} \end{aligned} \]

Microcopy: polygon must be simple (not self-intersecting) and vertices ordered consistently.

4) Formulas used (MathJax)

\[ \begin{aligned} J &= \frac{\pi}{2} r^4 \\ &= \frac{\pi}{32} D^4 \\ A &= \pi r^2 \\ Z_p &= \frac{J}{r} \\ k &= \sqrt{\frac{J}{A}} \end{aligned} \]

\[ \begin{aligned} \tau_{\max} &\approx \frac{T \, r_{\max}}{J} \\ \theta &= \frac{T L}{G J} \\ k_t &= \frac{GJ}{L} \end{aligned} \]

Note: The torsion equations above are standard for circular shafts. For non-circular sections, warping can change the effective torsional constant.

5) Comparison list (batch-like quick checks)

No items yet. Compute a case and click Add to comparison list.

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Engineering Reference Guide

Polar Moment of Inertia (J)
Complete User Guide & Formula Reference

Step-by-step instructions, all formulas explained, unit conversion reference, input validation, accuracy notes, and expert tips for torsion shaft design.

4 Cross-Section Shapes τ = Tr / J θ = TL / GJ SI & Imperial Units Non-Circular Warnings

Section 01

What Is Polar Moment of Inertia? A Plain-English Explanation

The polar moment of inertia J (also called the polar second moment of area) is a geometric property of a cross-section that describes its resistance to twisting (torsion). A larger J means the shaft resists more torque before it deforms or fails.

Think of it this way: when you apply a torque to a round shaft, every particle of material at radius r from the center contributes to resisting that twist — and the farther away from the center, the more leverage it has. J is the mathematical sum of all those contributions: J = ∫ r² dA integrated over the entire cross-section.

J is used in two fundamental engineering equations that appear throughout shaft design, drive shaft calculations, fastener torque analysis, and structural engineering:

🔩

Why J Matters in Practice J appears directly in the shear stress formula (τ = Tr/J) and twist angle formula (θ = TL/GJ). Underestimating J leads to over-conservative (heavy, costly) designs; overestimating it leads to unsafe failures. Getting J right is the starting point for all torsion analysis.

Do not confuse J with the mass moment of inertia (used in dynamics/rotation problems) — J is a purely geometric (area) property.

Section 02

Step-by-Step Calculator Walkthrough

Follow these steps in order. The calculator computes in real-time as you type — no need to click "Compute" unless you want to force a refresh.

Choose Your Cross-Section Shape

Use the Shape dropdown at the top of the Inputs panel. Four shapes are supported: Solid Circle, Hollow Circle / Tube, Rectangle, and Arbitrary Polygon. Each shape reveals its own geometry input fields below.

Set Your Length Unit

Pick the unit for all geometry inputs: mm, cm, m, or in (inches). All conversions happen internally — you never need to manually convert. Choose the unit your drawing dimensions are already in to avoid transcription errors.

Enter the Geometry Dimensions

Fill in the geometry fields that appear for your selected shape. All values must be positive numbers. For a hollow circle you can enter outer diameter + inner diameter, outer radius + inner radius, or outer diameter + wall thickness — pick the mode that matches your drawing.

(Optional) Enter Torsion Inputs for Stress & Twist

If you also want shear stress τ, twist angle θ, and torsional stiffness k_t, fill in the three torsion fields: applied torque T (with its unit), shaft length L, and material shear modulus G. For standard materials pick from the preset list (Steel = 79 GPa, Aluminium = 26 GPa, Brass = 39 GPa); for others choose Custom G and type your value.

Choose Output Units

Use the J output unit dropdown (mm⁴, cm⁴, m⁴, in⁴) and Stress unit dropdown (MPa, Pa, psi) to get results in your preferred system. These can be changed any time — results update instantly.

Read the Results Panel

The Results panel shows eight output values: A, J, k, r_max, Z_p, τ, θ, and k_t. The diagram panel shows a labeled sketch of your cross-section. See Section 5 for what each output means.

Export, Compare, or Print

Copy Report puts a full text summary on your clipboard. Add to Comparison List lets you store multiple cases side by side. Copy CSV exports comparisons for pasting into Excel. Print / Save PDF uses your browser's print function.

Section 03

Cross-Section Shape Guide: Which Shape to Choose

Fig 1 — Four supported cross-section shapes. Orange dot = centroid (C). Orange dashed line = outer radius rₒ. Blue dashed line = inner radius rᵢ or height h. Grey dots = polygon vertices.

Solid Circle (Round Shaft)

The most common shaft cross-section. Enter either the diameter D (most engineering drawings) or the radius r. The calculator uses D/2 = r internally. This is the only shape for which the torsion formulas (τ, θ) are exact.

Hollow Circle (Tube, Pipe, Hollow Shaft)

Three input modes let you match whatever your drawing shows:

Outer & Inner Diameters (Do, Di) — most common for pipe specs
Outer & Inner Radii (ro, ri) — common in analysis/theory contexts
Outer Diameter + Wall Thickness (Do, t) — Di is calculated as Do − 2t

⚠️

Hollow Circle Validation The inner value must be strictly less than the outer value. If inner ≥ outer, the calculator shows a geometry error. For the thickness mode, make sure t < Do/2 (i.e., the wall doesn't exceed the full radius).

Rectangle

Enter width b (horizontal) and height h (vertical). The calculator computes J = I_x + I_y about the centroid — this is the polar second moment of area, not the torsional constant. For rectangular sections under actual torsion, see the accuracy note in Section 8.

Arbitrary Polygon

Paste vertex coordinates one per line as x, y pairs using your chosen length unit. The calculator applies Green's theorem to find A, centroid, I_x, I_y, then J = I_x + I_y. At least 3 non-collinear points are required. Vertices can be listed clockwise or counter-clockwise — the sign is normalized automatically.

💡

Polygon Input Example (L-Section) To define an L-section with flanges 50 mm wide, paste:

Section 04

All Formulas Used — Full Derivation & Explanation

Every formula is derived from first principles. Inputs are converted to SI (meters, N·m, Pa) internally before calculation, then converted back to your chosen output units.

Formula F1 — Solid Circle: Polar Moment J Exact · Circular shafts only

J = (π / 2) × r⁴ = π × D⁴ / 32

JPolar moment of inertia [length⁴] e.g. mm⁴, in⁴

rRadius of solid shaft [length]

D = 2rDiameter [length]

πPi ≈ 3.14159265…

Formula F2 — Hollow Circle / Tube: Polar Moment J Exact · Circular tubes only

J = (π / 2) × (rₒ⁴ − rᵢ⁴) = π × (Dₒ⁴ − Dᵢ⁴) / 32

rₒOuter radius [length]

rᵢInner radius [length] (= 0 for solid)

Dₒ, DᵢOuter and inner diameters [length]

Formula F3 — Rectangle: Polar Second Moment of Area Polar 2nd moment (J = Ix + Iy) · Not torsional constant

Ix = b·h³/12 Iy = h·b³/12 J = Ix + Iy = (b·h/12)·(h² + b²)

bWidth of rectangle [length]

hHeight of rectangle [length]

IxSecond moment about centroidal x-axis [length⁴]

IySecond moment about centroidal y-axis [length⁴]

Formula F4 — Arbitrary Polygon: Area, Centroid & J via Shoelace / Green's Theorem Exact for planar polygon · Vertices in order

A = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| Cx = (1/6A) Σ (xᵢ+xᵢ₊₁)·(xᵢyᵢ₊₁−xᵢ₊₁yᵢ) Cy = (1/6A) Σ (yᵢ+yᵢ₊₁)·(xᵢyᵢ₊₁−xᵢ₊₁yᵢ) Ix_c = (1/12)|Σ (yᵢ²+yᵢyᵢ₊₁+yᵢ₊₁²)·cross| − A·Cy² J = Ix_c + Iy_c

(xᵢ, yᵢ)Vertex coordinates, index i = 0…n−1 (wraps around)

crossxᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ (signed area contribution)

Cx, CyCentroid coordinates

Ix_c, Iy_c2nd moments about centroidal axes (parallel axis shift applied)

Derived Geometric Quantities

Formula F5 — Radius of Gyration k All shapes

k = √(J / A)

kPolar radius of gyration [length] — the equivalent radius at which all area could be concentrated to give the same J

ACross-sectional area [length²]

Formula F6 — Polar Section Modulus Zp All shapes

Zp = J / r_max

ZpPolar section modulus [length³]

r_maxDistance from centroid to outermost fiber [length] (= rₒ for circles; = √((b/2)²+(h/2)²) for rectangles; max vertex distance for polygons)

Torsion Stress & Twist Formulas

Formula F7 — Maximum Shear Stress τ_max Exact for circular · Estimate for others

τ_max = T · r_max / J

τ_maxMaximum shear stress at outer fiber [Pa, MPa, or psi]

TApplied torque [N·m, N·mm, lbf·in, or lbf·ft]

r_maxOuter fiber distance [m internally]

JPolar moment [m⁴ internally]

Formula F8 — Angle of Twist θ Exact for circular · Estimate for others

θ = T · L / (G · J)

θAngle of twist [radians and degrees shown]

LShaft length [mm, cm, m, or in → converted to m]

GShear modulus of material [GPa, MPa, Pa, or psi → converted to Pa]

Formula F9 — Torsional Stiffness k_t Exact for circular · Estimate for others

k_t = G · J / L

k_tTorsional stiffness [N·m/rad] — torque required per radian of twist

Section 05

Understanding Your Results: What Each Output Means

Output	Symbol & Formula	Typical Unit	What It Tells You
Cross-sectional Area	A	mm², cm², m², in²	Material area of the cross-section; needed for stress calculations and mass estimation.
Polar Moment of Inertia	J = ∫ r² dA	mm⁴, cm⁴, m⁴, in⁴	The primary output. Larger J = more twist resistance. Compare sections by J to find the most efficient cross-section for a given area.
Radius of Gyration	k = √(J/A)	mm, cm, m, in	Effective radius for distributing J over A. Useful for comparing sections of different sizes on equal footing.
Outer Fiber Distance	r_max	mm, cm, m, in	Farthest point from the centroid. The shear stress is maximum here under torsion.
Polar Section Modulus	Zp = J / r_max	mm³, cm³, m³, in³	Use Zp to directly find τ_max without looking up J separately: τ = T / Zp.
Max Shear Stress	τ = T · r_max / J	MPa, Pa, psi	Shear stress at the outer fiber. Compare with the material's allowable shear strength (typically 0.5–0.6 × yield strength).
Angle of Twist	θ = TL / (GJ)	radians and °	Total rotation of one end relative to the other. Typical design limits: 0.5°–1° per meter for precision shafts; up to 3°–5° for general engineering.
Torsional Stiffness	k_t = GJ / L	N·m/rad	Torque needed to produce 1 radian of twist. Higher k_t = stiffer shaft. Use for vibration / dynamic analysis (natural frequency ∝ √k_t).

Section 06

Unit Conversion Quick Reference: Length, Torque, Stress & Shear Modulus

All conversions are handled automatically. This table shows the factors used internally — useful if you need to cross-check results manually.

Length Units

Unit	To Meters (×)	Typical Use
mm	0.001	Mechanical engineering, shafts, fasteners
cm	0.01	Structural members, sometimes civil engineering
m	1.0	SI analysis, large structural elements
in	0.0254	US customary, ASME / AISC standards

Torque Units

Unit	To N·m (×)	Notes
N·m	1.0	SI standard
N·mm	0.001	Common in detailed shaft analysis
lbf·in	0.11299	US customary, engine torque specs
lbf·ft	1.35582	US customary, larger torque values

Shear Modulus G — Built-in Material Values

Material	G (GPa)	Typical Applications
Steel	79	Drive shafts, fasteners, structural members
Aluminium	26	Aerospace, lightweight machinery, extrusions
Brass	39	Valves, fittings, precision instruments

ℹ️

Using Custom G For titanium (~41 GPa), cast iron (~40–50 GPa), composites, or plastics, select Custom G and enter the value from your material datasheet. You can choose GPa, MPa, Pa, or psi as input units.

Section 07

Input Validation Rules: What the Calculator Accepts

All values are validated in real-time. If an input is invalid, you will see an inline warning and the affected result will show "—" rather than a wrong number.

Input Field	Rule	Valid Example	Invalid Example
Diameter / Radius	Must be a finite positive number	50 \| 25.4	0 \| −10 \| abc
Hollow — Inner value	≥ 0 and strictly < outer value	30 (when outer = 60)	60 \| 70 \| −5
Wall Thickness (t)	Positive, and t < outer diameter / 2	5 (when Do = 60)	30 (when Do = 60)
Rectangle b, h	Both must be finite and positive	50 × 30	0 × 30 \| 50 × −5
Polygon vertices	At least 3 non-collinear points; "x,y" one per line	0,0 / 50,0 / 50,30 / 0,30	Only 2 points; all on same line
Torque T	Finite, ≥ 0 (zero allowed; blank = skip torsion)	0 \| 250 \| 1500.5	−100 \| text
Shaft Length L	Finite, ≥ 0 (positive required to compute θ)	1200 \| 2.5	0 (gives — for θ) \| negative
Custom G	Finite, strictly positive	200 \| 79	0 \| blank

Section 08

Accuracy Notes & Engineering Limitations

Knowing where results are exact and where they are estimates lets you use this tool confidently in real engineering decisions.

Solid Circle — J

Exact

Closed-form: J = πr⁴/2. No approximation involved.

Hollow Circle — J

Exact

Exact annular formula. Assumes perfect concentricity.

Rectangle — J = Ix + Iy

Exact (geometric)

Exact polar 2nd moment of area. However, J ≠ torsional constant for non-circular sections.

Rectangle — τ, θ

Estimate only

True torsional constant C < J = Ix+Iy for rectangles (can differ by 50%+). Use Timoshenko's formula or FEA for design.

Polygon — J via Green's

Exact (for planar polygon)

Green's theorem gives exact area, centroid, and 2nd moments for any straight-sided polygon.

Polygon — τ, θ

Rough estimate

Non-circular torsion requires warping constant and torsional constant. J = Ix+Iy underestimates C significantly for irregular shapes.

✅

When Results Are Fully Trustworthy For solid or hollow circular shafts, all eight outputs (A, J, k, r_max, Zp, τ, θ, k_t) are based on exact closed-form solutions from Saint-Venant torsion theory. You can use these directly in design with appropriate safety factors.

⚠️

Non-Circular Torsion Caveat For rectangles and polygons, τ = Tr/J and θ = TL/GJ are approximations because the true torsional constant C (from Saint-Venant's warping theory) differs from J = Ix + Iy. For critical non-circular shaft design, use Roark's Formulas for Stress & Strain (Table 10.1) or FEA software. The calculator clearly labels these rows as estimates.

Section 09

Common Mistakes to Avoid: Expert Microcopy Tips

Entering diameter when "Radius" mode is selected

Match your input to the mode shown: D = 50 mm means select "Diameter" and type 50

This is the most frequent error. A 50 mm diameter shaft entered as a radius gives J ≈ 16× too large. Always check which mode is selected before entering a value.

Entering torque in N·mm but leaving unit as N·m

Change the torque unit dropdown to match your value (e.g., N·mm → select "N·mm")

A torque of 250,000 N·mm = 250 N·m. Using the wrong unit makes shear stress appear 1000× off. Always set the unit dropdown before entering the number.

Comparing J values that were computed in different length units

Use the Comparison List feature — it stores the unit alongside each result for clear comparison

J in mm⁴ is 10⁸× larger than J in cm⁴ and 10¹² × larger than J in m⁴. Never compare raw J numbers without checking their units first.

Using τ and θ results from a Rectangle or Polygon for structural design decisions

Treat τ and θ for non-circular sections as order-of-magnitude estimates; verify with handbook tables or FEA

The calculator shows a yellow warning banner when non-circular torsion inputs are detected. Do not dismiss this warning for safety-critical applications.

Mixing length unit (mm) with length unit for L (also mm) but selecting steel G in GPa without noting that T is in N·mm

Use consistent SI unit pairs: T in N·m, L in m, G in Pa → θ in radians (all internally consistent)

The calculator handles all conversions automatically — enter each value in its own unit dropdown and let the tool convert. Never manually scale before entering.

Listing polygon vertices out of order or including duplicate points

List vertices consecutively around the perimeter (CW or CCW) with no repeated points except at the closure (which is automatic)

Crossing edges (a self-intersecting polygon) give incorrect area and J values. The calculator cannot detect self-intersections; check your vertex order visually using the diagram.

Section 10

Frequently Asked Questions (FAQ)

🔢 What is the difference between Polar Moment of Inertia (J) and the Torsional Constant (C or J_t)? ▼

For circular sections (solid or hollow), J = C exactly — the polar second moment of area equals the torsional constant. For non-circular sections (rectangles, I-beams, L-sections, etc.), C < J = Ix + Iy because of warping. The calculator correctly labels J = Ix + Iy for non-circular shapes and flags the τ and θ outputs as estimates.

⚖️ Why is a hollow tube more efficient than a solid shaft of the same weight? ▼

In J = ∫ r² dA, material near the center (small r) contributes almost nothing to J but still adds weight. Moving that material outward (to larger r) increases J without increasing mass. A tube uses the same material more effectively — this is why hollow drive shafts and structural tubes are preferred in weight-sensitive applications. Use the Comparison List to quantify the difference for your specific case.

📐 My drawing shows a wall thickness — which hollow circle mode should I use? ▼

Select "Outer Diameter + Wall Thickness" mode. Enter the outside diameter as "Outer" and the wall thickness as "Inner/t". The calculator derives the inner diameter as Di = Do − 2t automatically. This is the most common specification on pipe and tube drawings.

🔄 What does the angle of twist (θ) output mean practically? ▼

θ is the total angular rotation of one end of the shaft relative to the other end under torque T. A value of 1° over 1 m length (1°/m) is a common maximum for precision machinery. If your shaft produces θ > 3–5° under design torque, consider increasing the shaft diameter, changing the material (higher G), or shortening the shaft. Both degrees and radians are shown side by side.

🏗️ Can I use this calculator for I-beams, C-channels, or T-sections? ▼

Yes — use the Polygon shape and trace the full outline of the section with x,y vertices. The polygon formulas give exact J = Ix + Iy about the centroid. However, for torsion analysis, J is not equal to the torsional constant for thin open sections (I, C, T) — these sections have very low torsional resistance due to warping. Use the AISC Steel Construction Manual or FEA for open thin-walled torsion analysis.

💾 How do I save or share my calculation results? ▼

Three options: (1) Copy Report — copies a formatted text summary to your clipboard, ready to paste into a document or email. (2) Print / Save PDF — uses your browser's print dialog; choose "Save as PDF" to get a printable document. (3) Comparison CSV — if you've added multiple cases to the comparison list, "Copy CSV" gives a comma-separated table ready for Excel or Google Sheets.

🧮 Why does the radius of gyration (k) matter? ▼

The polar radius of gyration k = √(J/A) normalises J by cross-sectional area, enabling fair comparison of sections with different sizes. A higher k means the material is better distributed away from the centroid — more efficient use of material. When optimising shaft weight for a given J requirement, maximise k.

Polar Moment of Inertia Calculator

Polar Moment of Inertia (J) Calculator + Torsion Check

1) Inputs (Geometry, Units, Material)

Geometry

Torsion (optional, for τ and θ)

Accuracy notes (build trust)

2) Results (J, A, k, Zₚ + torsion outputs)

3) Visuals (SVG section + chart)

Shape Library (quick reference)

Solid Circle (Shaft)

Hollow Circle (Tube / Annulus)

Rectangle

Custom Polygon

4) Formulas used (MathJax)

5) Comparison list (batch-like quick checks)

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What Is Polar Moment of Inertia? A Plain-English Explanation

Step-by-Step Calculator Walkthrough

Cross-Section Shape Guide: Which Shape to Choose

Solid Circle (Round Shaft)

Hollow Circle (Tube, Pipe, Hollow Shaft)

Rectangle

Arbitrary Polygon

All Formulas Used — Full Derivation & Explanation

Derived Geometric Quantities

Torsion Stress & Twist Formulas

Understanding Your Results: What Each Output Means

Unit Conversion Quick Reference: Length, Torque, Stress & Shear Modulus

Length Units

Torque Units

Shear Modulus G — Built-in Material Values

Input Validation Rules: What the Calculator Accepts

Accuracy Notes & Engineering Limitations

Common Mistakes to Avoid: Expert Microcopy Tips

Frequently Asked Questions (FAQ)

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