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Moment of inertia formulas

20 standard bodies and axes, each with a derivation and a calculator.

Updated 28 October 2026Free · no sign-up

Body and axisI
Thin ring, through the centre, perpendicular to its planeMR²
Thin ring, about a diameterMR²/2
Thin ring, about a tangent in its plane3MR²/2
Thin ring, about a tangent perpendicular to its plane2MR²
Uniform disc, through the centre, perpendicular to its planeMR²/2
Uniform disc, about a diameterMR²/4
Uniform disc, about a tangent in its plane5MR²/4
Uniform disc, about a tangent perpendicular to its plane3MR²/2
Thin rod, through the centre, perpendicular to its lengthML²/12
Thin rod, through one end, perpendicular to its lengthML²/3
Solid sphere, about a diameter2MR²/5
Thin hollow sphere (spherical shell), about a diameter2MR²/3
Solid sphere, about a tangent7MR²/5
Thin hollow sphere, about a tangent5MR²/3
Solid cylinder, about its own axisMR²/2
Thin hollow cylinder, about its own axisMR²
Solid cylinder, through the centre, perpendicular to its axisM(R²/4 + L²/12)
Rectangular plate (sides a and b), through the centre, perpendicular to the plateM(a² + b²)/12
Annular disc (radii R₁ and R₂), through the centre, perpendicular to its planeM(R₁² + R₂²)/2
Solid cone, about its own axis3MR²/10

The two theorems

  • Parallel axis theorem: I = Icm + Md², where d is the distance between the new axis and a parallel axis through the centre of mass. Works for any body.
  • Perpendicular axis theorem: Iz = Ix + Iy, for a flat (laminar) body, with x and y in its plane and z perpendicular, all through one point.

Radius of gyration

Write I = Mk². Then k is the distance at which all the mass could sit and give the same moment of inertia. For a disc about its axis, k = R/√2.

Rolling shortcut: for a body rolling without slipping, KE = ½Mv²(1 + k²/R²). A ring (k²/R² = 1) is slowest down a slope and a solid sphere (2/5) is fastest.

Questions students ask

What is the moment of inertia of a disc?

MR²/2 about its central axis, and MR²/4 about a diameter.

What is the moment of inertia of a rod about its end?

ML²/3. About its centre it is ML²/12.

What is the moment of inertia of a solid sphere?

2MR²/5 about a diameter. A thin hollow sphere is 2MR²/3.

What is the SI unit of moment of inertia?

kg m². Its dimensional formula is [ML²].

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