Polar Moment of Inertia Calculator
Find the polar moment of inertia of a shaft cross-section and use it to get torsion shear stress and the angle of twist.
⭕ What is the Polar Moment of Inertia?
The polar moment of inertia of a cross-section, also called the second polar moment of area and written J, measures how a shape's area is distributed around a point, usually its centroid. It is defined as the integral of r squared over the area, so material far from the centre counts much more than material near it. In engineering, J is the geometric property that governs how a shaft resists twisting: for a given torque, a larger J means a lower shear stress and a smaller angle of twist.
You meet J whenever a shaft carries torque. Designers size drive shafts, axles, propeller shafts, drill rods and torsion bars with it. Machine designers compare solid and hollow shafts to save weight, and students use it in strength of materials to compute shear stress from the torsion formula and the twist from the angle formula. For a circle the answer has a clean closed form, J = pi D to the fourth over 32, and for a hollow tube you subtract the bore, J = pi (D to the fourth minus d to the fourth) over 32.
The name causes confusion. In physics, moment of inertia usually means the mass moment of inertia, measured in kg m squared, which describes how hard it is to change a body's rotational speed. The polar moment of inertia of an area is a different quantity with units of length to the fourth power, such as mm4 or in4. It contains no mass at all, only geometry. If you need the energy or acceleration of a spinning body, use the rotational kinetic energy calculator instead. For bending rather than twisting, you need the second moment of area.
This calculator handles both jobs. Circular shaft mode gives J for a solid or hollow circle in mm4, in4 and m4, and, if you enter a torque, length and shear modulus, the peak shear stress and twist angle with a plot of stress against radius. Other sections mode gives J, Ix, Iy and area for a rectangle, ellipse or equilateral triangle. Remember that the torsion equations are valid for circular shafts in the elastic range only.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Solid Steel Shaft Under Torque
40 mm solid steel shaft, 800 mm long, 300 N·m, G = 79 GPa
Example 2 — Hollow Tube, Section Property Only
Tube with 40 mm outside and 35 mm inside diameter, no torque entered
Example 3 — Imperial Shaft in Inches and lbf·in
2 in solid steel shaft, 30 in long, 2000 lbf·in, G = 79 GPa
Example 4 — Hollow Aluminium Shaft
50 mm outside, 40 mm inside diameter, 1000 mm long, 500 N·m, G = 26 GPa
Example 5 — Rectangular Section
Rectangle 60 mm wide and 40 mm high
Example 6 — Elliptical Section
Ellipse with full axes 80 mm and 40 mm
Example 7 — Equilateral Triangle
Equilateral triangle with a 100 mm side
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for the polar moment of inertia of a circle?
For a solid circular section J = pi D^4 / 32 = pi r^4 / 2. For a 40 mm diameter shaft this gives 251,327 mm4, which is 0.6038 in4. The polar moment is the sum of the second moments about two perpendicular centroidal axes, Ix + Iy.
How do you calculate the polar moment of inertia of a hollow shaft?
Use J = pi (D^4 - d^4) / 32, where D is the outer and d the inner diameter. A tube with a 40 mm outside and 35 mm inside diameter has J = 104,004 mm4. The bore removes a large share of J even though it removes less area.
What are the units of polar moment of inertia?
The area polar moment has units of length to the fourth power, such as mm4, cm4, in4 or m4. One in4 equals 416,231 mm4. The mass moment of inertia used in rotational dynamics has different units, kg m2.
What is the difference between polar moment of inertia and mass moment of inertia?
The polar moment of inertia of an area, in mm4, measures a cross-section's resistance to twisting. The mass moment of inertia, in kg m2, measures a body's resistance to changes in rotational speed. They share a name and a form, but not a meaning or a unit.
How do you find shear stress in a shaft from torque?
The shear stress at radius r is tau = T r / J, largest at the outer surface where r = D/2. A 40 mm solid steel shaft carrying 300 N m has J = 251,327 mm4, giving a peak stress of 23.87 MPa.
How do you calculate the angle of twist of a shaft?
The twist angle in radians is phi = T L / (G J), where L is the length and G is the shear modulus. For the 40 mm shaft with L = 800 mm, T = 300 N m and G = 79 GPa the twist is 0.01209 rad, which is 0.693 degrees.
What is the shear modulus of steel and aluminium?
Typical values are about 79 GPa for steel and about 26 GPa for aluminium alloys, but they vary by grade and temperature. The presets use these typical values, so use the figure from your material datasheet when accuracy matters.
Does the J formula work for square or rectangular shafts?
The geometric J = Ix + Iy can be computed for any section, but the torsion formulas tau = T r / J and phi = T L / (G J) only apply to circular sections. Non-circular sections warp when twisted and need a different torsion constant.
Why does a hollow shaft transmit torque efficiently?
Shear stress is highest at the outer radius and zero at the centre, so core material does little work. A hollow shaft puts material where it matters, giving a high J for its weight. For the same outer diameter it needs less material than a solid shaft.
How does the polar moment relate to the second moment of area?
The polar moment J about the centroid equals the sum of the second moments of area about two perpendicular axes in the plane, J = Ix + Iy (the perpendicular axis theorem). For a circle Ix = Iy = pi D^4 / 64, so J = pi D^4 / 32.