Polar Moment of Inertia Calculator finds the polar moment J of a solid shaft, pipe, rectangle, ellipse or triangle. It also gives shear stress, angle of twist and torsion constant.
Enter the shaft diameter. Torque, length and material are under the section below.
Torque, Material and Units
Step-by-Step Working
- Known values: d = 50 mm
- Raise the diameter to the fourth power: 50⁴ = 6,250,000
- Multiply by π and divide by 32: J = 613,592.3 mm⁴
- Peak shear stress: T × r ÷ J = 500 N·m × 0.025 m ÷ 6.136 × 10⁻⁷ m⁴ = 20.37 MPa
- Angle of twist: T × L ÷ (G × J) = 0.0102 rad = 0.5836°
- Result: J = 613,592.3 mm⁴ for the solid round shaft
Polar Moment at Other Sizes
| Size | Polar Moment | Compared |
|---|---|---|
| 50% of every dimension | 38,349.5 mm⁴ | 0.0625 times |
| 75% of every dimension | 194,144.4 mm⁴ | 0.3164 times |
| Your size | 613,592.3 mm⁴ | Your result |
| 125% of every dimension | 1,498,028.1 mm⁴ | 2.44 times |
| 150% of every dimension | 3,106,311.1 mm⁴ | 5.06 times |
| 200% of every dimension | 9,817,477 mm⁴ | 16 times |
Cross-Section Diagram
Polar Moment of Inertia Calculator Shapes
The polar moment of inertia, written J, measures how well a cross-section resists twisting. It adds up every bit of area times the square of its distance from the center. Material far from the center counts the most, which is why J rises with the fourth power of size.
The Polar Moment of Inertia Calculator covers five cross-sections. They are a solid round shaft, a hollow shaft or pipe, a rectangle or square, an ellipse and an equilateral triangle. The default is a 50 mm solid shaft, which has a polar moment of 613,592.3 mm⁴.
Each result comes with the area, the two bending moments Ix and Iy, the polar section modulus and the radius of gyration. With a torque, a length and a material entered, round shafts also get the peak shear stress, the angle of twist and the torsional stiffness.
Polar Moment of Inertia Formulas
For any shape, the polar moment about the center is the sum of the two area moments of inertia through that point. This is the perpendicular axis theorem, and it is the basic way to calculate the polar moment of inertia when Ix and Iy are already known.
$$J = I_x + I_y$$
The polar moment of inertia of a solid shaft uses the diameter d alone. A rod, a solid cylinder and any solid circular section all share this formula, since only the round cross-section matters.
$$J = \frac{\pi d^4}{32}$$
A pipe or tube has the bore removed. With outer diameter D and inner diameter d, the hollow part is subtracted from the solid one. This is the formula for the polar moment of inertia of a tube of any wall thickness.
$$J = \frac{\pi (D^4 – d^4)}{32}$$
The table lists the formula the Polar Moment of Inertia Calculator applies to each shape, with the answer for its opening dimensions. Every one is the sum of Ix and Iy for that shape, taken about its center.
| Cross-section | Polar moment formula | Example | J |
|---|---|---|---|
| Solid round shaft | π d4 ÷ 32 | d = 50 mm | 613,592.3 mm⁴ |
| Hollow shaft or pipe | π (D4 − d4) ÷ 32 | 60 mm and 40 mm | 1,021,017.6 mm⁴ |
| Rectangle | b h (b2 + h2) ÷ 12 | 40 mm by 90 mm | 2,910,000 mm⁴ |
| Square | s4 ÷ 6 | 50 mm side | 1,041,666.7 mm⁴ |
| Ellipse, half-axes a and b | π a b (a2 + b2) ÷ 4 | 80 mm by 30 mm overall | 860,011 mm⁴ |
| Equilateral triangle | √3 s4 ÷ 48 | 70 mm side | 866,386.2 mm⁴ |
In a spreadsheet the solid shaft formula is =PI()*A1^4/32 with the diameter in cell A1. For a pipe it is =PI()*(A1^4-B1^4)/32, with the outer diameter in A1 and the inner in B1. The step-by-step panel of the Polar Moment of Inertia Calculator shows the same working for every entry.
Solid Shaft Polar Moment by Diameter
Because the diameter is raised to the fourth power, small changes in size move J a long way. Doubling the diameter multiplies the polar moment by 16. The polar section modulus, J divided by the outer radius, rises by 8.
| Diameter | Polar moment J | Polar section modulus |
|---|---|---|
| 10 mm | 981.7 mm⁴ | 196.3 mm³ |
| 20 mm | 15,708 mm⁴ | 1,570.8 mm³ |
| 25 mm | 38,349.5 mm⁴ | 3,068 mm³ |
| 30 mm | 79,521.6 mm⁴ | 5,301.4 mm³ |
| 40 mm | 251,327.4 mm⁴ | 12,566.4 mm³ |
| 50 mm | 613,592.3 mm⁴ | 24,543.7 mm³ |
| 75 mm | 3,106,311.1 mm⁴ | 82,835 mm³ |
| 100 mm | 9,817,477 mm⁴ | 196,349.5 mm³ |
The polar moment of inertia is measured in units of length to the fourth power. The 50 mm shaft’s 613,592.3 mm⁴ is the same as 61.36 cm⁴, 1.47 in⁴ or 6.136 × 10⁻⁷ m⁴. The fourth card of the Polar Moment of Inertia Calculator lists these conversions.
Polar Moment of Inertia of a Pipe
To calculate the polar moment of inertia of a pipe, raise both diameters to the fourth power, subtract, multiply by π and divide by 32. A tube with a 60 mm outer and 40 mm inner diameter has J = 1,021,017.6 mm⁴. A 2 in pipe with a 1.5 in bore has 1.07 in⁴.
The hollow section is efficient, and the Polar Moment of Inertia Calculator reports how much. That 60 mm tube keeps 80.2% of the polar moment of a solid 60 mm shaft while using 55.6% of the material. The metal near the center of a solid shaft adds little to J, so removing it costs little stiffness and saves weight.
Shear Stress and Angle of Twist
J turns a torque into a stress and a twist. For a round shaft, the peak shear stress is the torque T times the outer radius r, divided by J. The angle of twist is the torque times the length L, divided by the shear modulus G times J. Cornell University’s torsion notes derive both.
$$\tau = \frac{T r}{J}$$
$$\theta = \frac{T L}{G J}$$
These are the figures on the third card of the Polar Moment of Inertia Calculator. A torque of 500 N·m on the 50 mm steel shaft gives a peak shear stress of 20.37 MPa. Over a 1 m length it twists 0.5836°, and its torsional stiffness is 49,087.4 N·m per radian. The same torque on the 60 mm by 40 mm tube gives 14.69 MPa and 0.3507°.
The material only affects the twist, not the stress. The presets use the shear moduli in OpenStax College Physics, which are 80 GPa for steel, 40 for cast iron, 35 for brass and 25 for aluminum.
| Material | Shear modulus | Twist of the 50 mm shaft |
|---|---|---|
| Steel | 80 GPa | 0.5836° |
| Cast iron | 40 GPa | 1.17° |
| Brass | 35 GPa | 1.33° |
| Aluminum | 25 GPa | 1.87° |
Rectangle and Other Non-Round Shapes
The polar moment of inertia of a rectangle is b h (b² + h²) ÷ 12. A 40 mm by 90 mm bar has Ix = 2,430,000 mm⁴ and Iy = 480,000 mm⁴, so J = 2,910,000 mm⁴. A 50 mm square has 1,041,666.7 mm⁴.
There is a catch with these shapes. A section that is not round warps when it is twisted, so its torsional stiffness is lower than J suggests. The twist then depends on a separate torsion constant, which the Polar Moment of Inertia Calculator reports in place of a stress.
| Cross-section | Polar moment J | Torsion constant | Share of J |
|---|---|---|---|
| Round shaft, 50 mm | 613,592.3 mm⁴ | 613,592.3 mm⁴ | 100% |
| Square, 50 mm | 1,041,666.7 mm⁴ | 880,208.3 mm⁴ | 84.5% |
| Equilateral triangle, 70 mm | 866,386.2 mm⁴ | 519,831.7 mm⁴ | 60% |
| Rectangle, 40 mm by 90 mm | 2,910,000 mm⁴ | 1,384,148 mm⁴ | 47.6% |
| Ellipse, 80 mm by 30 mm | 860,011 mm⁴ | 371,826.9 mm⁴ | 43.2% |
The flatter the shape, the larger the gap. Using J for the twist of the 40 by 90 rectangle would make the bar look about twice as stiff as it is. J remains the correct section property for these shapes in other work, such as bolt groups and combined loading.
An irregular shape has no single formula. It is split into simple parts, and each part’s own moment is moved to the common center with the parallel axis theorem, which adds the part’s area times the square of its offset. The tool handles the five standard shapes only.
Moment of Inertia vs Polar Moment
What is the difference between moment of inertia and polar moment of inertia?
The area moment of inertia, Ix or Iy, is taken about an axis lying in the cross-section and governs bending. The polar moment is taken about the axis running along the shaft, through the center of the section, and governs twisting. The first card of the Polar Moment of Inertia Calculator lists Ix and Iy beside the area for this reason.
Both are properties of a shape alone and share the unit of length to the fourth power. Another name for this kind of moment of inertia is the second moment of area, and J is the polar second moment of area. To calculate a second moment of area for a rectangle about its center, use b h³ ÷ 12.
Is this the same as the moment of inertia of a spinning cylinder?
That is a different quantity. The mass moment of inertia depends on mass and is measured in kg·m². It sets how hard an object is to spin up. A solid cylinder has ½ m r² about its axis, so a 10 kg cylinder of 25 mm radius has 0.003125 kg·m². It is the mass version that decides how fast a cylinder rolls down a slope in the Inclined Plane Calculator.
Is there a trick to remember the formulas?
Learn one and derive the rest. A circle has an area moment of π d⁴ ÷ 64 about any diameter. The polar moment is twice that, π d⁴ ÷ 32, because Ix and Iy are equal. A hollow shaft is the outer circle minus the inner one, and a rectangle is b h³ ÷ 12 plus h b³ ÷ 12.