{
  "id": "cone-volume",
  "version": "f31dfd323894",
  "status": "published",
  "name": "Cone Volume Calculator",
  "question": "What is the cone volume?",
  "summary": "Finds the volume of a cone from its base radius and height (V = πr²h ÷ 3), or the radius or height from the volume.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/cone-volume-calculator",
  "markdown": "https://www.acalculator.org/math/cone-volume-calculator.md",
  "kind": "formulas",
  "method": "V = πr²h ÷ 3: one third of the base area πr² times the height h.",
  "assumptions": [
    "The cone has a circular base. The height is the vertical distance from the base to the tip, so the formula holds for a slanted (oblique) cone too.",
    "π is the full-precision value 3.141592653589793….",
    "The radius and height are in the length units shown next to them, and the volume is in the volume unit shown next to it.",
    "Each value must be greater than 0."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "r": {
        "title": "Radius",
        "description": "The radius of the circular base: half its diameter.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0
      },
      "h": {
        "title": "Height",
        "description": "The vertical height from the base to the tip, not the slanted side.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0
      },
      "v": {
        "title": "Volume",
        "description": "The space inside the cone.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "volume",
        "exclusiveMinimum": 0
      }
    }
  },
  "solve": {
    "fillAny": 2,
    "variables": [
      "r",
      "h",
      "v"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "r": {
      "label": "Radius",
      "description": "The radius of the circular base: half its diameter.",
      "format": "quantity"
    },
    "h": {
      "label": "Height",
      "description": "The vertical height from the base to the tip, not the slanted side.",
      "format": "quantity"
    },
    "v": {
      "label": "Volume",
      "description": "The space inside the cone.",
      "format": "quantity"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "r": 0.0508,
      "h": 0.1524
    },
    "outputs": {
      "v": 0.00041185183901044616
    },
    "text": "A cone with radius 2 in and height 6 in has a volume of 25.13 in³."
  },
  "examples": [
    {
      "given": {
        "r": 2,
        "h": 6
      },
      "expect": {
        "v": 25.132741228718345
      },
      "source": "OpenStax Prealgebra 2e, Example 9.55 (25.12 with π ≈ 3.14); hand calculation in content.mdx: π × 2² × 6 ÷ 3 = 8π = 25.1327"
    },
    {
      "given": {
        "r": 2.5,
        "h": 8
      },
      "expect": {
        "v": 52.35987755982989
      },
      "source": "OpenStax Prealgebra 2e, Example 9.56, diameter 5 (52.33 with π ≈ 3.14); hand calculation in content.mdx: π × 2.5² × 8 ÷ 3 = 52.3599"
    },
    {
      "given": {
        "v": 100,
        "h": 6
      },
      "expect": {
        "r": 3.989422804014327
      },
      "source": "hand calculation in content.mdx: r = √(3 × 100 ÷ (π × 6)) = √(50 ÷ π) = 3.9894"
    },
    {
      "given": {
        "v": 50,
        "r": 2
      },
      "expect": {
        "h": 11.93662073189215
      },
      "source": "hand calculation in content.mdx: h = 3 × 50 ÷ (π × 2²) = 11.9366"
    }
  ],
  "sources": [
    "OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (V = πr²h ÷ 3; Examples 9.55 and 9.56). https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area",
    "NIST Digital Library of Mathematical Functions, §3.12 Mathematical Constants, equation 3.12.1 (π). https://dlmf.nist.gov/3.12",
    "NIST Special Publication 811 (2008), Appendix B: 1 in = 0.0254 m exactly, for unit conversions."
  ],
  "related": [
    "cylinder-volume",
    "surface-area",
    "area-of-a-circle"
  ],
  "changelog": []
}