{
  "id": "moment-of-inertia",
  "version": "570257cdfe64",
  "status": "published",
  "name": "Moment of Inertia Calculator",
  "question": "Find the moment of inertia",
  "summary": "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.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/moment-of-inertia-calculator",
  "markdown": "https://www.acalculator.org/physics/moment-of-inertia-calculator.md",
  "kind": "function",
  "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."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "shape": {
        "title": "Shape and axis",
        "description": "The object’s shape and the axis it turns about.",
        "type": "string",
        "enum": [
          "point",
          "hoop",
          "disk",
          "tube",
          "rod-center",
          "rod-end",
          "sphere",
          "shell",
          "hoop-diameter",
          "cylinder-diameter",
          "slab"
        ]
      },
      "m": {
        "title": "Mass (M)",
        "description": "The mass of the object.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "exclusiveMinimum": 0,
        "maximum": 1000000000000
      },
      "r": {
        "title": "Radius (R)",
        "description": "The radius of the object; for a hollow cylinder, the outer radius.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000
      },
      "ri": {
        "title": "Inner radius (R₁)",
        "description": "The inner radius of the hollow cylinder.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000
      },
      "l": {
        "title": "Length (L)",
        "description": "The length of the rod or cylinder.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000
      },
      "a": {
        "title": "Side a",
        "description": "One side of the rectangular plate.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000
      },
      "b": {
        "title": "Side b",
        "description": "The other side of the rectangular plate.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000
      },
      "d": {
        "title": "Axis offset (d)",
        "description": "Optional: move the axis this far, parallel to the shape’s own axis through its center of mass.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 1000000
      }
    }
  },
  "outputs": {
    "inertia": {
      "label": "Moment of inertia (I)",
      "description": "The moment of inertia about the chosen axis, in kilogram square meters.",
      "format": "number"
    },
    "lbft2": {
      "label": "In lb·ft²",
      "description": "The same moment of inertia in pound square feet.",
      "format": "number"
    },
    "icm": {
      "label": "About the shape’s own axis",
      "description": "The moment of inertia before any axis offset, in kilogram square meters.",
      "format": "number"
    },
    "k": {
      "label": "Radius of gyration (k)",
      "description": "The distance at which the whole mass would give the same moment of inertia: √(I ÷ M).",
      "format": "quantity",
      "unit": "m"
    },
    "formula": {
      "label": "Formula used",
      "description": "The moment of inertia formula for the shape.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "shape": "disk",
      "m": 500,
      "r": 2,
      "ri": 0.5,
      "l": 1,
      "a": 1,
      "b": 0.5
    },
    "outputs": {
      "inertia": 1000,
      "lbft2": 23730.360404231935,
      "icm": 1000,
      "k": 1.4142135623730951,
      "formula": "I = ½·M·R²"
    },
    "text": "The moment of inertia is 1,000 kg·m² (I = ½·M·R²)."
  },
  "examples": [
    {
      "given": {
        "shape": "disk",
        "m": 500,
        "r": 2
      },
      "expect": {
        "inertia": 1000
      },
      "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"
    },
    {
      "given": {
        "shape": "rod-end",
        "m": 2,
        "l": 0.5
      },
      "expect": {
        "inertia": 0.16666666666666666
      },
      "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"
    },
    {
      "given": {
        "shape": "sphere",
        "m": 1,
        "r": 0.2,
        "d": 0.7
      },
      "expect": {
        "icm": 0.016000000000000004,
        "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"
    },
    {
      "given": {
        "shape": "tube",
        "m": 3,
        "r": 0.1,
        "ri": 0.08
      },
      "expect": {
        "inertia": 0.0246
      },
      "source": "hand calculation in content.mdx: ½ × 3 × (0.1² + 0.08²) = 0.0246 kg·m²; 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"
    },
    {
      "given": {
        "shape": "slab",
        "m": 6,
        "a": 1,
        "b": 0.5
      },
      "expect": {
        "inertia": 0.625,
        "k": 0.3227486121839514
      },
      "source": "hand calculation in content.mdx: 6 × (1 + 0.25) ÷ 12 = 0.625 kg·m²; k = √(0.625 ÷ 6); 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"
    }
  ],
  "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)"
  ],
  "related": [
    "kinetic-energy",
    "net-force",
    "acceleration",
    "density"
  ],
  "changelog": []
}