The polar moment of inertia calculator above computes the polar second moment of area, written J or Ip, for four common cross-sections. It is the geometric property that decides how stiffly a shaft resists twisting and how large the shear stress becomes at a given torque. The tool also reports the strong-axis second moment, the torsion constant appropriate to the shape, the peak shear stress and the angle of twist over a stated length.
Arb Digital builds free tools that are explicit about the boundary of the theory they use, because that boundary is where most wrong answers come from. This page publishes no allowable-stress table and no material selection guidance of any kind. It gives you a stress number in megapascals and a twist in degrees, and it stops there. Whether a shaft, axle or structural member is fit for its duty is a question for a qualified engineer working to the design code that applies, taking fatigue, stress concentration, keyways, welds and load history into account.
What This Polar Moment of Inertia Calculator Does
The polar second moment of area is defined as the integral of r2 over the cross-sectional area, where r is measured from the centroid. Because r2 = x2 + y2, that integral splits neatly into the two rectangular second moments, which is the relationship set out in the open-access engineering statics text at Engineering LibreTexts, section 10.5 on the polar moment of inertia: J = Ix + Iy. That identity holds for every shape, and this calculator uses it for every shape.
What does not hold for every shape is the assumption that J is the right number to put in the torsion equations. Those equations are derived on the assumption that plane cross-sections stay plane when the member twists, which is true for a circle and false for essentially everything else. So the tool reports two separate figures: J, the true polar second moment of area, and the torsion constant, which is what actually goes into the stiffness and stress calculations. For round bars and tubes the two are identical. For rectangles they are not, and the difference is displayed rather than hidden.
How to Use It
- Pick the cross-section first. The meaning of the two dimension boxes changes with it: diameters for round shapes, side lengths for rectangular ones. The hint under each box tells you which is in play.
- Enter the outer dimension before the inner one. For a tube the inner diameter must be smaller than the outer; for a rectangle the tool sorts the two sides itself, so the order does not matter there.
- Set the wall thickness only for the hollow rectangle. It is ignored by the other three shapes, and it must be less than half the shorter side or the walls would meet in the middle.
- Enter the torque and the twisted length. The length is the distance between the point where torque goes in and the point where it is reacted, not the total length of the part.
- Read the torsion constant, not just J. If the two differ, the second figure is the one that governs stiffness, and the page explains why below.
The Formula: How the Polar Second Moment Is Calculated
For a solid round bar of diameter d, J = πd4/32, and each rectangular second moment is half of that, πd4/64. For a tube of outer diameter do and inner diameter di, J = π(do4 − di4)/32. For a solid rectangle of sides a and b, Ix = ab3/12 and Iy = ba3/12, so J = ab(a2 + b2)/12. For the hollow rectangle the second moments are the outer box minus the inner void.
The torsion relations themselves are the ones set out in the Cambridge-derived teaching material hosted as Engineering LibreTexts section 7.6, twisting moments and torsional stiffness, which gives the torque as T = GIp dθ/dL. Integrated along a uniform shaft that is the familiar θ = TL/(GJ), and the shear stress at radius r is τ = Tr/J, largest at the outer surface.
Work the defaults for the round case. A 50 mm solid steel bar has J = π × 0.054/32 = 6.136 × 10−7 m4, which is 613,592 mm4. At 500 N·m the surface shear stress is 500 × 0.025 ÷ 6.136 × 10−7 = 20.4 MPa. Over a metre, with G = 79.3 GPa, the twist is 500 ÷ (79.3 × 109 × 6.136 × 10−7) = 0.01028 radians, or 0.589 degrees.
Why J and the Torsion Constant Are Not the Same Thing
This is the single most common error made with this quantity, and it is worth spelling out. The derivation of θ = TL/(GJ) assumes that when the member twists, every cross-section rotates rigidly in its own plane and stays flat. A circular section does exactly that, because the shape is unchanged by rotation about its own axis. A square section does not: it warps, with points moving out of plane in a saddle pattern.
If the warping is free to happen, the section is more flexible than the plane-sections theory predicts, so using J gives an answer that is too stiff. The correction is a torsion constant, written K, which replaces J in the stiffness equation. For a solid rectangle with long side a and short side b, the standard series approximation is K = ab3[1/3 − 0.21(b/a)(1 − b4/12a4)], which this tool uses.
The gap surprises people. For a square of side a, J = a4/6 ≈ 0.167a4, while K ≈ 0.141a4: about 16 per cent stiffer than reality. For a thin strip of ratio ten to one the gap is far worse, because J is dominated by the long dimension cubed while the torsion constant is governed by the thin dimension cubed. Using J for a flat bar can overstate torsional stiffness several times over. The calculator shows both numbers side by side so the gap is visible rather than assumed away.
The peak shear stress in a rectangle also moves. It is not at the corner, where intuition puts it, but at the midpoint of the longer side; the corners are actually stress-free in pure torsion. The tool uses the classical approximation τmax = T(3a + 1.8b)/(a2b2), which for a square reduces to 4.8T/a3, matching the accepted exact-series value to within a fraction of a per cent.
Solid Versus Hollow: Where the Material Earns Its Keep
Because the integrand is r2, material near the axis contributes almost nothing to torsional resistance while still costing full weight. That is why drive shafts, bicycle frames and aircraft structures are tubes rather than bars. The effect is easy to quantify with this tool: take a 50 mm solid bar and hollow it out to a 30 mm bore. The removed core is 36 per cent of the area, but J falls only from 613,592 to 534,072 mm4, a loss of 13 per cent. You keep 87 per cent of the torsional stiffness for 64 per cent of the weight.
The Closed Box, the Open Section, and the Slit That Ruins Everything
For a rectangular hollow section this calculator does not use J. It uses the Bredt thin-walled closed-section result, K = 4Am2t/s, where Am is the area enclosed by the wall mid-line and s is the mid-line perimeter, and the corresponding shear stress τ = T/(2Amt). A closed thin wall carries a constant shear flow all the way round the loop, like current in a circuit, and the enclosed area converts that flow into resisting torque.
The consequence is dramatic and worth knowing before you cut anything. Slit a closed tube lengthwise and the shear flow can no longer complete its circuit. The section reverts to open-section behaviour, where the torsion constant is roughly the sum of one third of each wall's length times its thickness cubed, and the torsional stiffness collapses by one or two orders of magnitude. A 100 by 50 by 3 mm closed box is thousands of times stiffer in torsion than the same box with a hairline slot down one corner. This is why a cable entry cut into a torsionally loaded member is a structural change, not a detail.
Reading the Shear Stress Number Honestly
The stress this calculator reports is the nominal elastic shear stress in a smooth prismatic member under pure torque. Almost nothing in service is that. Keyways, splines, cross-holes, circlip grooves, shoulder fillets and weld toes all raise the local stress above the nominal figure, sometimes by a factor of two or three, and torsional fatigue cracks start there. A shaft that passes on nominal stress can still fail at a keyway.
Torsion also rarely arrives alone: a shaft usually carries bending at the same time, and the combined state has to be resolved with the Mohr's circle calculator before it means anything. None of that is on this page, and it is why no allowable stress appears anywhere on it. Publishing a table of permissible shear stresses without the load spectrum, the surface condition, the stress concentration factors and the governing design code would be worse than publishing nothing, because it would let a number that means very little look like an answer. The calculator gives geometry and elastic response. A qualified engineer takes it from there.
How This Differs From the Adjacent Section Tools
The boundary in one sentence: this page computes the second moment of area about the axis running along the member, which governs twisting, while the section modulus calculator computes the second moment about axes running across it, which governs bending. Same integral, different axis, entirely different failure mode.
The moment of inertia calculator and the mass moment of inertia calculator answer a different question again: they deal with mass distribution and rotational dynamics, measured in kilogram metres squared, and are used with angular acceleration rather than stress. The parallel axis theorem that links those to an offset axis is set out in OpenStax University Physics Volume 1, section 10.5 on calculating moments of inertia. A section property has units of length to the fourth power; a mass property has units of mass times length squared. If your units do not match that, you have the wrong tool open.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using J for a non-circular section — the plane-sections assumption fails, and the torsion constant is smaller, often much smaller, than the polar second moment.
- Confusing area and mass properties — millimetres to the fourth power and kilogram metres squared are not interchangeable, and neither are the equations that consume them.
- Looking for peak shear stress at the corners of a rectangle — the corners are unstressed in pure torsion, and the maximum sits at the midpoint of the longer side.
- Applying the thin-wall box formula to a heavy-walled section — Bredt's result assumes the shear stress is uniform through the wall, which stops being true as the wall thickens.
- Treating the nominal stress as the design stress — keyways, holes and fillets multiply it locally, and fatigue rather than yield usually sets the limit.
Related Free Tools From Arb Digital
For bending properties of the same cross-sections, use the section modulus calculator and the beam deflection calculator. For rotational dynamics rather than section geometry, the moment of inertia calculator and mass moment of inertia calculator apply. The stress strain calculator covers axial behaviour and elastic moduli, the Mohr's circle calculator combines shear with direct stress, the hoop stress calculator handles pressurised tubes, and the torque calculator works out the applied moment in the first place. Browse the full free online tools hub for everything else.
Frequently Asked Questions
It is a measure of how far a cross-section spreads its area away from its own centre, weighted by the square of that distance. A section with more area far from the centre resists twisting more stiffly, which is why a tube outperforms a solid bar of the same weight.
Only for circular sections. For any other shape the cross-section warps out of plane as it twists, so the true torsional stiffness is lower than the polar second moment predicts. This calculator reports both figures separately.
Length to the fourth power, so millimetres to the fourth or metres to the fourth in SI, and inches to the fourth in imperial. If your answer is in kilogram metres squared you are looking at a mass moment of inertia, which is a different quantity entirely.
Because the contribution of each element of area scales with the square of its distance from the axis, material near the centre adds very little torsional resistance. Removing the middle 60 per cent of a bar's diameter costs about 13 per cent of its polar second moment while removing over a third of its weight.
At the midpoint of the longer side of the section, not at the corners. The corners carry no shear stress in pure torsion at all, which is the opposite of what most people expect from the bending case.
Its torsional stiffness collapses. A closed section carries shear flow round a complete loop and gets its stiffness from the enclosed area; once the loop is broken the section behaves as an open one, and the torsion constant can fall by one or two orders of magnitude.
From the material supplier's data for the specific alloy, temper and temperature, not from a generic figure. The default of 79.3 GPa is a common value quoted for structural steel and is offered only as a starting point you should replace.
No. It is a nominal elastic value for a smooth member under pure torque, and it ignores stress concentration at keyways, holes and fillets, combined bending loads, fatigue and every code requirement. A qualified engineer assesses fitness for purpose against the applicable design standard.
This tool is provided for educational and estimating use only. It is not structural or mechanical design advice, and it publishes no allowable stress, material grade or safety factor of any kind. Load-bearing and power-transmitting components must be designed, checked and signed off by a qualified engineer working to the design code that applies in your jurisdiction.
