{
  "id": "special-right-triangles",
  "version": "4ffae590ce9a",
  "status": "published",
  "name": "Special Right Triangles Calculator",
  "question": "How do I solve special right triangles?",
  "summary": "Solves a 30-60-90 or 45-45-90 triangle from one side, the area or the perimeter: all three sides with exact square roots, the area, the perimeter and the height.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/special-right-triangles-calculator",
  "markdown": "https://www.acalculator.org/math/special-right-triangles-calculator.md",
  "kind": "function",
  "method": "30-60-90: a : b : c = 1 : √3 : 2, area = a²√3 ÷ 2, perimeter = a(3 + √3). 45-45-90: a : b : c = 1 : 1 : √2, area = a² ÷ 2, perimeter = a(2 + √2). Height to the hypotenuse = a × b ÷ c.",
  "assumptions": [
    "a is the side opposite the smallest angle (30° or 45°), b the other leg, c the hypotenuse.",
    "The known value is from 0.000001 to 1,000,000,000, in any one unit; the area is in that unit squared.",
    "Exact sides show when a side is known as a decimal with at most 6 places."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "type": {
        "title": "Triangle",
        "description": "The 30-60-90 triangle (half an equilateral triangle) or the 45-45-90 triangle (half a square).",
        "type": "string",
        "enum": [
          "30-60-90",
          "45-45-90"
        ]
      },
      "know": {
        "title": "You know the",
        "description": "Which part of the triangle you know.",
        "type": "string",
        "enum": [
          "short",
          "long",
          "hyp",
          "area",
          "perimeter"
        ]
      },
      "know45": {
        "title": "You know the",
        "description": "Which part of the triangle you know.",
        "type": "string",
        "enum": [
          "leg",
          "hyp",
          "area",
          "perimeter"
        ]
      },
      "v": {
        "title": "Its value",
        "description": "The length of the known side, or the area or perimeter, in any one unit.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000000
      }
    }
  },
  "outputs": {
    "c": {
      "label": "Hypotenuse c",
      "description": "The longest side, opposite the 90° angle.",
      "format": "number"
    },
    "a": {
      "label": "Short leg a",
      "description": "The side opposite the smallest angle (30° or 45°).",
      "format": "number"
    },
    "b": {
      "label": "Long leg b",
      "description": "The other leg: a × √3 in a 30-60-90 triangle, equal to a in a 45-45-90 triangle.",
      "format": "number"
    },
    "area": {
      "label": "Area",
      "description": "Half the product of the legs: a × b ÷ 2.",
      "format": "number"
    },
    "perimeter": {
      "label": "Perimeter",
      "description": "The three sides added: a + b + c.",
      "format": "number"
    },
    "height": {
      "label": "Height to the hypotenuse",
      "description": "The distance from the right angle to the hypotenuse: a × b ÷ c.",
      "format": "number"
    },
    "exact": {
      "label": "Exact sides",
      "description": "The sides with square roots, when a side is known as a decimal with up to 6 places.",
      "format": "text"
    },
    "angles": {
      "label": "Angles",
      "description": "The three angles of the triangle.",
      "format": "text"
    },
    "ratio": {
      "label": "Side ratio a : b : c",
      "description": "The ratio of the three sides.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "type": "30-60-90",
      "know": "short",
      "know45": "leg",
      "v": 5
    },
    "outputs": {
      "c": 10,
      "a": 5,
      "b": 8.660254037844386,
      "area": 21.650635094610962,
      "perimeter": 23.660254037844386,
      "height": 4.330127018922193,
      "exact": "a = 5, b = 5√3, c = 10",
      "angles": "30°, 60°, 90°",
      "ratio": "1 : √3 : 2"
    },
    "text": "A 30°, 60°, 90° triangle with sides 5, 8.660254 and 10 has an area of 21.650635."
  },
  "examples": [
    {
      "given": {
        "type": "30-60-90",
        "know": "short",
        "v": 7
      },
      "expect": {
        "a": 7,
        "b": 12.12435565298214,
        "c": 14,
        "area": 42.43524478543749,
        "perimeter": 33.12435565298214,
        "height": 6.06217782649107,
        "exact": "a = 7, b = 7√3, c = 14"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry, Example 5: the side opposite 30° is 7, so the hypotenuse is 14 and the adjacent side 7√3 ≈ 12.1; hand calculation in content.mdx"
    },
    {
      "given": {
        "type": "45-45-90",
        "know45": "leg",
        "v": 5
      },
      "expect": {
        "a": 5,
        "b": 5,
        "c": 7.0710678118654755,
        "area": 12.5,
        "perimeter": 17.071067811865476,
        "exact": "a = 5, b = 5, c = 5√2",
        "angles": "45°, 45°, 90°"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry; hand calculation in content.mdx: c = 5√2"
    },
    {
      "given": {
        "type": "30-60-90",
        "know": "long",
        "v": 6
      },
      "expect": {
        "a": 3.464101615137755,
        "c": 6.92820323027551,
        "area": 10.392304845413264,
        "height": 3,
        "exact": "a = 2√3, b = 6, c = 4√3"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry; hand calculation in content.mdx: a = 6 ÷ √3 = 2√3, c = 2a = 4√3"
    },
    {
      "given": {
        "type": "45-45-90",
        "know45": "area",
        "v": 50
      },
      "expect": {
        "a": 10,
        "b": 10,
        "c": 14.142135623730951,
        "perimeter": 34.14213562373095
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry; hand calculation in content.mdx: a² ÷ 2 = 50, so a = 10"
    },
    {
      "given": {
        "type": "30-60-90",
        "know": "perimeter",
        "v": 30
      },
      "expect": {
        "a": 6.3397459621556145,
        "b": 10.98076211353316,
        "c": 12.679491924311229,
        "area": 34.8076211353316
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry; hand calculation in content.mdx: a = 30 ÷ (3 + √3)"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, section 7.2 Right Triangle Trigonometry: the 30-60-90 triangle has sides s, √3s, 2s and the 45-45-90 triangle sides s, s, √2s; Example 5 (side opposite 30° is 7, hypotenuse 14). CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry"
  ],
  "related": [
    "30-60-90-triangle",
    "45-45-90-triangle",
    "right-triangle",
    "pythagorean-theorem"
  ],
  "changelog": []
}