{
  "aliases": [
    "volume-of-a-sphere"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $R \u003e 0$, and let $B = \\{(x, y, z) \\in \\mathbb{R}^3 : x^2 + y^2 + z^2 \\le R^2\\}$.\nThen\n$$\\operatorname{vol}(B) = \\frac{4}{3} \\pi R^3.$$\n\n## Remarks\n\nIn [[spherical coordinates|05WKRC7HWT]], $\\operatorname{vol}(B) = \\int_0^{2\\pi} \\int_0^\\pi \\int_0^R \\rho^2 \\sin \\varphi \\, d\\rho \\, d\\varphi \\, d\\theta = 2\\pi \\cdot 2 \\cdot R^3 / 3$.\nBy slices, the cross-section at height $z$ is a disc of area $\\pi (R^2 - z^2)$, and $\\int_{-R}^{R} \\pi (R^2 - z^2) \\, dz = \\tfrac{4}{3} \\pi R^3$. This is [[Cavalieri's principle|0GTZ2F3PQY]].\nThis is the case $n = 3$ of the [[volume of a ball in R^n|00YH28R457]], since $\\Gamma(5/2) = \\tfrac{3}{4} \\sqrt{\\pi}$.\nArchimedes found this volume, by comparison with a cone and a cylinder.\n\n## Explanations\n\n### L1 human\n\nThe ball of radius $R$ fills two thirds of the cylinder of radius $R$ and height $2R$ around it, since that cylinder has volume $2 \\pi R^3$.\nAny of the standard methods of multiple integration gives the same value.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "volume-of-a-sphere"
    ],
    "kind": "statement",
    "msc": [
      "26B15"
    ],
    "name": "Volume of a ball in R^3",
    "role": "proposition",
    "sources": [
      {
        "locator": "Volume",
        "url": "https://en.wikipedia.org/wiki/Ball_(mathematics)"
      },
      {
        "locator": "Spherical coordinates",
        "url": "https://en.wikipedia.org/wiki/Multiple_integral"
      },
      {
        "locator": "History",
        "url": "https://en.wikipedia.org/wiki/Cavalieri%27s_principle"
      }
    ],
    "tag": "0WMDV48VSN",
    "uses": [
      "05WKRC7HWT",
      "00YH28R457",
      "0GTZ2F3PQY"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0WMDV48VSN/",
  "links": [
    {
      "field": "uses",
      "target": "05WKRC7HWT"
    },
    {
      "field": "uses",
      "target": "00YH28R457"
    },
    {
      "field": "uses",
      "target": "0GTZ2F3PQY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0WMDV48VSN.md",
  "msc": [
    "26B15"
  ],
  "name": "Volume of a ball in R^3",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0WM/0WMDV48VSN.md",
  "status": null,
  "tag": "0WMDV48VSN",
  "uri": "m2://0WMDV48VSN",
  "url": "https://math2.tamnd.com/wiki/volume-of-a-ball-in-r-3/",
  "writing": {
    "tag": "0WMDV48VSN",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}