Moment of inertia calculator
This moment of inertia calculator covers the most common solid bodies: enter the mass and dimensions to get I in kg·m² with the formula, and optionally for a shifted axis using the parallel axis theorem.
Example: Body and axis of rotation: Solid cylinder or disc (axis along the length) · Mass 2 · Mass unit: kg · Length unit: cm · Radius r 10 → Result: I = 0.01 kg·m². Source: Wikipedia – Massenträgheitsmoment (formulas for simple bodies, parallel axis theorem). Updated: .
How it is calculated
The moment of inertia describes how strongly a body resists a change in its rotation. A solid cylinder of 2 kg and 10 cm radius has I = ½·m·r² = 0.01 kg·m².
The key formulas
- Solid cylinder, disc: I = ½·m·r²
- Hollow cylinder: I = ½·m·(R² + r²)
- Sphere: I = ⅖·m·r², thin hollow sphere: I = ⅔·m·r²
- Rod about the middle: I = m·L²/12, about the end: I = m·L²/3
- Box about the axis through the centre: I = m·(a² + b²)/12
- Thin ring: I = m·r²
Shifted axis: the parallel axis theorem
If the axis lies parallel to the axis through the centre of mass at a distance d, add m·d²: I′ = I + m·d². A 1 m, 2 kg rod has 0.167 kg·m² about its middle and 0.667 kg·m² about its end, because d = 0.5 m adds 2 · 0.25 = 0.5 kg·m².
The further out the mass sits, the larger the moment of inertia: for equal mass and radius the ring with k = 1 has twice as much as the disc with k = ½.
Frequently asked questions
How do you calculate the moment of inertia of a sphere?
With I = ⅖·m·r² for a solid sphere about an axis through its centre, and I = ⅔·m·r² for a thin hollow sphere.
What is the parallel axis theorem?
I′ = I + m·d²: the moment of inertia about an axis parallel to the centre-of-mass axis at distance d equals the moment about the centre-of-mass axis plus m·d².
What unit is the moment of inertia in?
kg·m². The calculator also shows g·cm², a unit common in data sheets (1 kg·m² = 10⁷ g·cm²).
Sources and legal basis
- Wikipedia – Massenträgheitsmoment (formulas for simple bodies, parallel axis theorem)
- Wikipedia – Satz von Steiner
As of: · Editorial policy
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