# Find the moment of inertia Finds the moment of inertia of a point mass, hoop, disk, hollow cylinder, rod, sphere, shell or plate from its mass and size, about its own axis or an axis moved by the parallel-axis theorem. - Page: https://www.acalculator.org/physics/moment-of-inertia-calculator - JSON spec: https://www.acalculator.org/physics/moment-of-inertia-calculator.json - Version: 570257cdfe64 ## Default answer Example with the default inputs (Shape and axis Solid cylinder or disk, about its axis, Mass (M) 500 kg, Radius (R) 2 m): The moment of inertia is 1,000 kg·m² (I = ½·M·R²). ## Inputs | Key | Label | Description | | --- | --- | --- | | shape | Shape and axis | The object’s shape and the axis it turns about. | | m | Mass (M) | The mass of the object. | | r | Radius (R) | The radius of the object; for a hollow cylinder, the outer radius. | | ri | Inner radius (R₁) | The inner radius of the hollow cylinder. | | l | Length (L) | The length of the rod or cylinder. | | a | Side a | One side of the rectangular plate. | | b | Side b | The other side of the rectangular plate. | | d | Axis offset (d) | Optional: move the axis this far, parallel to the shape’s own axis through its center of mass. | ## Outputs | Key | Label | Description | | --- | --- | --- | | inertia | Moment of inertia (I) | The moment of inertia about the chosen axis, in kilogram square meters. | | lbft2 | In lb·ft² | The same moment of inertia in pound square feet. | | icm | About the shape’s own axis | The moment of inertia before any axis offset, in kilogram square meters. | | k | Radius of gyration (k) | The distance at which the whole mass would give the same moment of inertia: √(I ÷ M). | | formula | Formula used | The moment of inertia formula for the shape. | ## Method I from the shape’s formula (M·R², ½·M·R², ½·M·(R₁² + R₂²), M·L² ÷ 12, M·L² ÷ 3, ⅖·M·R², ⅔·M·R², ¼·M·R² + M·L² ÷ 12, M·(a² + b²) ÷ 12); with an offset axis, I = I_cm + M·d²; k = √(I ÷ M). ## Assumptions - Each object is rigid with its mass spread evenly (uniform density); a hoop, rod or shell is thin. - The offset axis is parallel to the shape’s own axis, which passes through the center of mass (not for a point mass or a rod about its end). - The model works in kilograms and meters; 1 lb·ft² = 0.0421401 kg·m². ## Worked examples 1. shape = disk, m = 500, r = 2 gives inertia = 1,000. Source: OpenStax, University Physics Volume 1, §10.5 Calculating Moments of Inertia (rod ML²/12 and ML²/3, disk mR²/2, parallel-axis theorem I = I_cm + md²; Example 10.11: 1,025 kg·m²). https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia, retrieved 2026-10-02: ½ × 500 × 2² = 1,000 kg·m². 2. shape = rod-end, m = 2, l = 0.5 gives inertia = 0.166667. Source: OpenStax, University Physics Volume 1, §10.5 Calculating Moments of Inertia (rod ML²/12 and ML²/3, disk mR²/2, parallel-axis theorem I = I_cm + md²; Example 10.11: 1,025 kg·m²). https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia, retrieved 2026-10-02, Example 10.12: (1/3)(2.0 kg)(0.50 m)² = 0.167 kg·m². 3. shape = sphere, m = 1, r = 0.2, d = 0.7 gives icm = 0.016, inertia = 0.506. Source: OpenStax, University Physics Volume 1, §10.5 Calculating Moments of Inertia (rod ML²/12 and ML²/3, disk mR²/2, parallel-axis theorem I = I_cm + md²; Example 10.11: 1,025 kg·m²). https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia, retrieved 2026-10-02, Example 10.12: (2/5)(1.0)(0.2)² + (1.0)(0.7)² = 0.016 + 0.490; Wikipedia, List of moments of inertia (hoop mr², disk mr²/2, rod mL²/12 and mL²/3, sphere 2mr²/5, shell 2mr²/3, tube m(r₁² + r₂²)/2, plate m(h² + w²)/12, cylinder about a diameter m(3r² + h²)/12). https://en.wikipedia.org/wiki/List_of_moments_of_inertia, retrieved 2026-10-02. 4. shape = tube, m = 3, r = 0.1, ri = 0.08 gives inertia = 0.0246. Source: Wikipedia, List of moments of inertia (hoop mr², disk mr²/2, rod mL²/12 and mL²/3, sphere 2mr²/5, shell 2mr²/3, tube m(r₁² + r₂²)/2, plate m(h² + w²)/12, cylinder about a diameter m(3r² + h²)/12). https://en.wikipedia.org/wiki/List_of_moments_of_inertia, retrieved 2026-10-02; OpenStax, University Physics Volume 1, §10.4 Moment of Inertia and Rotational Kinetic Energy, Figure 10.20 (moments of inertia of common shapes). https://openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy, retrieved 2026-10-02. 5. shape = slab, m = 6, a = 1, b = 0.5 gives inertia = 0.625, k = 0.322749. Source: Wikipedia, List of moments of inertia (hoop mr², disk mr²/2, rod mL²/12 and mL²/3, sphere 2mr²/5, shell 2mr²/3, tube m(r₁² + r₂²)/2, plate m(h² + w²)/12, cylinder about a diameter m(3r² + h²)/12). https://en.wikipedia.org/wiki/List_of_moments_of_inertia, retrieved 2026-10-02. ## FAQ ### What is the moment of inertia? A measure of how hard it is to change an object’s rotation, the rotational version of mass. It depends on the mass and on how far that mass is from the axis: I = Σ m·r² over every bit of the object. ### What is the moment of inertia of a disk? A solid disk or cylinder about its axis has I = ½·M·R². A 500 kg merry-go-round of radius 2.0 m has I = ½ × 500 × 2² = 1,000 kg·m². ### What is the moment of inertia of a rod? A thin rod of length L has I = M·L² ÷ 12 about its center and M·L² ÷ 3 about one end. A 2.0 kg rod 0.50 m long, turned about one end, has I = 2 × 0.25 ÷ 3 = 0.167 kg·m². ### What is the parallel-axis theorem? To find the moment of inertia about an axis parallel to one through the center of mass, add M·d², where d is the distance between the axes: I = I_cm + M·d². A 1 kg sphere of radius 0.2 m on an axis 0.7 m from its center has I = 0.016 + 0.49 = 0.506 kg·m². ### Why does a hoop have more inertia than a disk of the same mass? All of a hoop’s mass is at the rim, at distance R, so I = M·R². A disk has much of its mass close to the axis, so I = ½·M·R², half as much. ### What is the radius of gyration? The distance k from the axis at which the whole mass, put in one place, would give the same moment of inertia: k = √(I ÷ M). For a solid disk, k = R ÷ √2. ## Sources - OpenStax, University Physics Volume 1, §10.4 Moment of Inertia and Rotational Kinetic Energy, Figure 10.20: moments of inertia of ten common shapes. https://openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy (retrieved 2026-10-02) - OpenStax, University Physics Volume 1, §10.5 Calculating Moments of Inertia: thin rod (1/12)ML² about its center and (1/3)ML² about one end, thin disk (1/2)mR², parallel-axis theorem I = I_cm + md²; Example 10.11 (disk of 500 kg, radius 2.0 m) and Example 10.12 (rod 2.0 kg, 0.50 m; sphere 1.0 kg, radius 0.20 m, 0.70 m from the axis). https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia (retrieved 2026-10-02) - Wikipedia, List of moments of inertia: the formulas of Figure 10.20 written out as text (hoop mr², hoop about a diameter mr²/2, disk mr²/2, solid cylinder about a diameter m(3r² + h²)/12, tube m(r₁² + r₂²)/2, rod mL²/12 and mL²/3, sphere 2mr²/5, shell 2mr²/3, plate m(h² + w²)/12). https://en.wikipedia.org/wiki/List_of_moments_of_inertia (retrieved 2026-10-02)