{
  "id": "hexagon",
  "version": "11b28589c725",
  "status": "published",
  "name": "Hexagon Calculator",
  "question": "What is the hexagon’s area?",
  "summary": "Finds every measure of a regular hexagon from any one of them: side, perimeter, area, apothem, long diagonal and short diagonal.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/hexagon-calculator",
  "markdown": "https://www.acalculator.org/math/hexagon-calculator.md",
  "kind": "formulas",
  "method": "P = 6s; apothem = s√3 ÷ 2; A = ½ × apothem × P = (3√3 ÷ 2)s²; long diagonal D = 2s; short diagonal d = s√3.",
  "assumptions": [
    "The hexagon is regular: six equal sides and six equal angles of 120°.",
    "The radius (center to a corner) equals the side, so the long diagonal is two sides.",
    "Type one measure; the side follows from it, and every other measure from the side."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "s": {
        "title": "Side (s)",
        "description": "The length of one side of the regular hexagon.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000000
      },
      "p": {
        "title": "Perimeter (P)",
        "description": "The distance around the hexagon: six sides.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 6000000000
      },
      "a": {
        "title": "Area (A)",
        "description": "The space inside the hexagon: half the apothem times the perimeter.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "area",
        "exclusiveMinimum": 0,
        "maximum": 3000000000000000000
      },
      "ap": {
        "title": "Apothem (inradius)",
        "description": "The distance from the center to the middle of a side, at a right angle to the side.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 1000000000
      },
      "d": {
        "title": "Long diagonal (D)",
        "description": "The distance between opposite corners, through the center: two sides.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 2000000000
      },
      "ds": {
        "title": "Short diagonal (d)",
        "description": "The distance between two corners with one corner between them: the side times √3.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0,
        "maximum": 2000000000
      }
    }
  },
  "solve": {
    "fillAny": 1,
    "variables": [
      "s",
      "p",
      "a",
      "ap",
      "d",
      "ds"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "s": {
      "label": "Side (s)",
      "description": "The length of one side of the regular hexagon.",
      "format": "quantity"
    },
    "p": {
      "label": "Perimeter (P)",
      "description": "The distance around the hexagon: six sides.",
      "format": "quantity"
    },
    "a": {
      "label": "Area (A)",
      "description": "The space inside the hexagon: half the apothem times the perimeter.",
      "format": "quantity",
      "unit": "in2"
    },
    "ap": {
      "label": "Apothem (inradius)",
      "description": "The distance from the center to the middle of a side, at a right angle to the side.",
      "format": "quantity"
    },
    "d": {
      "label": "Long diagonal (D)",
      "description": "The distance between opposite corners, through the center: two sides.",
      "format": "quantity"
    },
    "ds": {
      "label": "Short diagonal (d)",
      "description": "The distance between two corners with one corner between them: the side times √3.",
      "format": "quantity"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "s": 0.1016
    },
    "outputs": {
      "p": 0.6095999999999999,
      "a": 0.026818797576267282,
      "ap": 0.08798818102449896,
      "d": 0.2032,
      "ds": 0.17597636204899791
    },
    "text": "A regular hexagon with side 4 in has an area of 41.5692 in² and a perimeter of 24 in."
  },
  "examples": [
    {
      "given": {
        "s": 4
      },
      "expect": {
        "p": 24,
        "ap": 3.4641016151377544,
        "a": 41.569219381653056,
        "d": 8,
        "ds": 6.928203230275509
      },
      "source": "OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area; hand calculation in content.mdx: apothem 4√3 ÷ 2 = 2√3, A = ½ × 2√3 × 24 = 24√3"
    },
    {
      "given": {
        "p": 30
      },
      "expect": {
        "s": 5,
        "a": 64.9519052838329,
        "ap": 4.330127018922193
      },
      "source": "hand calculation in content.mdx: s = 30 ÷ 6 = 5, A = (3√3 ÷ 2) × 25 = 64.9519; OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; each angle of a regular hexagon is 120°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference; OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area"
    },
    {
      "given": {
        "a": 100
      },
      "expect": {
        "s": 6.204032394013997,
        "p": 37.22419436408398
      },
      "source": "hand calculation in content.mdx: s = √(2 × 100 ÷ (3√3)) = 6.2040; OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area"
    },
    {
      "given": {
        "d": 10
      },
      "expect": {
        "s": 5,
        "ds": 8.660254037844386
      },
      "source": "hand calculation in content.mdx: s = 10 ÷ 2 = 5, d = 5√3 = 8.6603; OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; each angle of a regular hexagon is 120°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference"
    }
  ],
  "sources": [
    "OpenStax, Contemporary Mathematics, §10.6 Area (A = ½ap for a regular polygon; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area (retrieved 2026-10-02)",
    "OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; a regular hexagon’s angles add to 720°, 120° each). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference (retrieved 2026-10-02)"
  ],
  "related": [
    "square",
    "area",
    "triangle-area"
  ],
  "changelog": []
}