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Polar moment of inertia calculator
Find the polar moment of inertia J of a solid circle, a hollow circle (tube), a rectangle or a hollow rectangle, in mm, cm, m or inches. You also get the second moments of area and the polar section modulus, with the working.
Units: choose mm, cm, m or inches for the dimensions. The answer is in the same unit to the fourth power, and in mm⁴. This is a learning and checking tool: check your own working and your course’s conventions.
Result
Working
- Polar second moment of areaJ = π (D4 − d4) / 32 = π × (804 − 604) / 32 = 2.75 × 106 mm⁴.
The second moment of area about a diameter is I = J / 2 = 1.37 × 106 mm⁴. - Polar section modulus and unitsThe largest distance from the centroid is rmax = 40.0 mm, so Zp = J / rmax = 68700 mm³.
In millimetres, J = 2.75 × 106 mm⁴ = 2.75 × 10−6 m⁴.
Practise polar moment questions
Press the button for a question with random numbers. Work it out on paper first, then type your answer.
How the polar moment of inertia calculator works
The calculator uses the standard formulas for the torsion of circular shafts, and shows every step. The steps are:
- Choose the section and the length unit, then enter the dimensions.
- For a circle, J = π D⁴ / 32. For a hollow circle, subtract the hole. For a rectangle, add the two second moments of area with the perpendicular axis theorem: J = Ix + Iy.
- Divide by the largest distance from the centroid to get the polar section modulus.
- Read the answer in your own unit to the fourth power and in mm⁴.
Worked example
A solid circular bar is 2 inches in diameter. Find its polar moment of inertia J in in⁴ and in mm⁴, and its polar section modulus.
- Polar second moment of areaJ = π D4 / 32 = π × 24 / 32 = 1.57 in⁴.
For a circle the second moment of area about a diameter is I = J / 2 = π D4 / 64 = 0.785 in⁴. - Polar section modulus and unitsThe largest distance from the centroid is rmax = 1.00 in, so Zp = J / rmax = 1.57 in³.
In millimetres, J = 6.54 × 105 mm⁴ = 6.54 × 10−7 m⁴.
Polar moment of inertia formulas
The last column is the calculator’s own answer for the example sizes, so you can check it against the formulas.
| Section | Polar moment of inertia J | Example | J from the calculator (mm⁴) |
|---|---|---|---|
| Solid circle, diameter D | π D⁴ / 32 | D = 50 mm | 6.14 × 10⁵ |
| Hollow circle, outer D and inner d | π (D⁴ − d⁴) / 32 | D = 80 mm, d = 60 mm | 2.75 × 10⁶ |
| Rectangle, width b and depth h | b h (b² + h²) / 12 | b = 60 mm, h = 120 mm | 1.08 × 10⁷ |
| Hollow rectangle, outer B × H and inner b × h | [B H (B² + H²) − b h (b² + h²)] / 12 | B = 100, H = 150, b = 80, h = 130 mm | 2.04 × 10⁷ |
Which formulas does this use?
Torsional shear stress in a solid shaft, Twist of a solid circular shaft, Strain energy stored in a twisted shaft, Shaft power from torque and speed, Shear modulus from Young’s modulus and Poisson’s ratio, Second moment of area, rectangle, Hollow rectangular section second moment of area.
Related calculators
Shaft torsion calculator, Angle of twist calculator, Shaft diameter calculator, the Mohr’s circle calculator for the stresses on a shaft surface element, and the beam calculator for bending.
Common questions
What is the polar moment of inertia?
It is the second moment of area of a cross-section about an axis perpendicular to the section, through the centroid. For a solid circle it is J = π D⁴ / 32. It measures how well a circular shaft resists twisting, and appears in τ = T r / J and θ = T L / (G J).
What is the polar moment of inertia of a hollow shaft?
J = π (D⁴ − d⁴) / 32, where D is the outer diameter and d is the inner diameter. It is the value for the solid outer circle minus the value for the hole.
How is the polar moment related to the second moment of area?
By the perpendicular axis theorem, J = Ix + Iy for any plane section. A circle has Ix = Iy, so J = 2 I, which gives I = π D⁴ / 64 about a diameter.
Is it the same as the mass moment of inertia?
No. The polar moment of inertia here is an area property, in mm⁴ or in⁴, and depends only on the shape of the section. The mass moment of inertia, in kg·m², includes the mass and tells you the resistance to angular acceleration.
Can I use J for a rectangular shaft?
You can calculate J for a rectangle, but it is not the torsion constant. Non-circular sections warp when twisted, so τ = T r / J and θ = T L / (G J) do not apply to them.
Is the calculator free, and does it keep my numbers?
It is free to use, with no sign-up, no adverts and no personal tracking. The numbers you type stay in your browser.