{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "XVNB6HV2Y1"
    }
  ],
  "body": "\n## Example\n\nLet $w \u003e 0$, and let $A$, $B$, $C$ be the vertices of an equilateral triangle of side $w$ in the plane.\nThe *Reuleaux triangle* is the intersection of the three closed discs of radius $w$ centred at $A$, $B$ and $C$.\nIts boundary consists of three arcs of [[circles|060W5CBZP6]] of radius $w$: the arc from $B$ to $C$ centred at $A$, and the two similar arcs.\nEach arc has angle $\\pi/3$.\n\nIts boundary is a [[curve of constant width|B4YD1ZXRVH]] $w$.\nCheck: take a unit vector $u$ and the two supporting lines orthogonal to $u$.\nThe arc centred at $A$ consists of the points $A + w v$ with $v$ in an angle $V_A$ of size $\\pi/3$.\nIts outer normal at $A + w v$ is $v$.\nThe outer normals of the body at the vertex $A$ are the vectors $-v$ with $v \\in V_A$.\nThese six angles of size $\\pi/3$ cover all directions.\nSo for some vertex, say $A$, one of $u$ and $-u$ lies in $V_A$ and the other is an outer normal at $A$.\nThen one supporting line passes through $A$ and the other through a point $A \\pm w u$ of the arc centred at $A$.\nThe distance between the two lines is $w$.\n\nThe perimeter is $3 \\cdot w \\cdot \\pi/3 = \\pi w$.\nThe area is the sum of three circular sectors of angle $\\pi/3$ minus twice the triangle: $3 \\cdot \\frac{\\pi w^2}{6} - 2 \\cdot \\frac{\\sqrt{3}}{4} w^2 = \\frac{\\pi - \\sqrt{3}}{2} w^2$.\n\n## Explanations\n\n### L1 human\n\nDraw an equilateral triangle and replace each side by an arc of a circle centred at the opposite corner.\nThe result is a rounded triangle that has the same width in every direction, just like a coin.\nIt can turn between two parallel boards that stay at a fixed distance, and it is used in some coin shapes and drill designs.\nAmong all shapes of a given constant width it has the smallest area.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "B4YD1ZXRVH"
    ],
    "kind": "example",
    "msc": [
      "52A10",
      "52A38"
    ],
    "name": "Reuleaux triangle",
    "same_as": {
      "wikidata": "Q959203",
      "wikipedia": "Reuleaux triangle"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reuleaux_triangle"
      }
    ],
    "tag": "KMNV11B91V",
    "uses": [
      "B4YD1ZXRVH",
      "060W5CBZP6"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/KMNV11B91V/",
  "links": [
    {
      "field": "instance_of",
      "target": "B4YD1ZXRVH"
    },
    {
      "field": "uses",
      "target": "B4YD1ZXRVH"
    },
    {
      "field": "uses",
      "target": "060W5CBZP6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KMNV11B91V.md",
  "msc": [
    "52A10",
    "52A38"
  ],
  "name": "Reuleaux triangle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KMN/KMNV11B91V.md",
  "status": null,
  "tag": "KMNV11B91V",
  "uri": "m2://KMNV11B91V",
  "url": "https://tamnd.github.io/math2/wiki/reuleaux-triangle/",
  "writing": {
    "tag": "KMNV11B91V",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}