{
  "id": "cylinder-volume",
  "version": "4ede7d3a7e52",
  "status": "published",
  "name": "Cylinder Volume Calculator",
  "question": "What is the cylinder volume?",
  "summary": "Finds the volume of a cylinder from its radius and height (V = πr²h), or the radius or height from the volume.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/cylinder-volume-calculator",
  "markdown": "https://www.acalculator.org/math/cylinder-volume-calculator.md",
  "kind": "formulas",
  "method": "V = πr²h: the area of the circular base, πr², times the height h.",
  "assumptions": [
    "The cylinder is a right circular cylinder: two equal circles joined by a straight side. An oblique cylinder with the same base and vertical height has the same volume.",
    "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 distance between the two circular ends.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0
      },
      "v": {
        "title": "Volume",
        "description": "The space inside the cylinder.",
        "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 distance between the two circular ends.",
      "format": "quantity"
    },
    "v": {
      "label": "Volume",
      "description": "The space inside the cylinder.",
      "format": "quantity"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "r": 0.0762,
      "h": 0.127
    },
    "outputs": {
      "v": 0.00231666659443376
    },
    "text": "A cylinder with radius 3 in and height 5 in has a volume of 141.4 in³."
  },
  "examples": [
    {
      "given": {
        "r": 3,
        "h": 5
      },
      "expect": {
        "v": 141.3716694115407
      },
      "source": "OpenStax Prealgebra 2e, Example 9.53 (141.3 with π ≈ 3.14); hand calculation in content.mdx: π × 3² × 5 = 45π = 141.3717"
    },
    {
      "given": {
        "r": 4,
        "h": 13
      },
      "expect": {
        "v": 653.451271946677
      },
      "source": "OpenStax Prealgebra 2e, Example 9.54 (653.12 with π ≈ 3.14); hand calculation in content.mdx: π × 4² × 13 = 208π = 653.4513"
    },
    {
      "given": {
        "v": 1000,
        "h": 10
      },
      "expect": {
        "r": 5.641895835477563
      },
      "source": "hand calculation in content.mdx: r = √(1000 ÷ (π × 10)) = 5.6419"
    },
    {
      "given": {
        "v": 500,
        "r": 5
      },
      "expect": {
        "h": 6.366197723675813
      },
      "source": "hand calculation in content.mdx: h = 500 ÷ (π × 5²) = 6.3662"
    }
  ],
  "sources": [
    "OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (V = πr²h; Examples 9.53 and 9.54). 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.",
    "NIST Handbook 44, Appendix C, General Tables of Units of Measurement: 1 US gallon = 231 in³. https://www.nist.gov/document/2026-nist-handbook-44-appendix-c"
  ],
  "related": [
    "cone-volume",
    "surface-area",
    "circle"
  ],
  "changelog": []
}