{
  "id": "45-45-90-triangle",
  "version": "5251626c7307",
  "status": "published",
  "name": "45 45 90 Triangle Calculator",
  "question": "How do I solve a 45-45-90 triangle?",
  "summary": "Solves a 45-45-90 (isosceles right) triangle from a leg, the hypotenuse, the area or the perimeter: the sides with exact square roots, the area, the perimeter and the height.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/45-45-90-triangle-calculator",
  "markdown": "https://www.acalculator.org/math/45-45-90-triangle-calculator.md",
  "kind": "function",
  "method": "a : b : c = 1 : 1 : √2. Area = a² ÷ 2, perimeter = a(2 + √2), height to the hypotenuse = c ÷ 2.",
  "assumptions": [
    "a and b are the equal legs, 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": {
      "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": "Leg a",
      "description": "The side opposite the smallest angle (30° or 45°).",
      "format": "number"
    },
    "b": {
      "label": "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": {
      "know45": "leg",
      "v": 5
    },
    "outputs": {
      "c": 7.0710678118654755,
      "a": 5,
      "b": 5,
      "area": 12.5,
      "perimeter": 17.071067811865476,
      "height": 3.5355339059327373,
      "exact": "a = 5, b = 5, c = 5√2",
      "angles": "45°, 45°, 90°",
      "ratio": "1 : 1 : √2"
    },
    "text": "A 45-45-90 triangle with legs 5 has a hypotenuse of 7.071068 and an area of 12.5."
  },
  "examples": [
    {
      "given": {
        "know45": "leg",
        "v": 5
      },
      "expect": {
        "a": 5,
        "b": 5,
        "c": 7.0710678118654755,
        "area": 12.5,
        "perimeter": 17.071067811865476,
        "height": 3.5355339059327373,
        "exact": "a = 5, b = 5, c = 5√2"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (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": {
        "know45": "hyp",
        "v": 10
      },
      "expect": {
        "a": 7.071067811865475,
        "b": 7.071067811865475,
        "area": 24.999999999999996,
        "perimeter": 24.14213562373095,
        "exact": "a = 5√2, b = 5√2, c = 10"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (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 = 10 ÷ √2 = 5√2"
    },
    {
      "given": {
        "know45": "area",
        "v": 50
      },
      "expect": {
        "a": 10,
        "c": 14.142135623730951,
        "perimeter": 34.14213562373095,
        "height": 7.071067811865475
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (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": {
        "know45": "leg",
        "v": 1
      },
      "expect": {
        "c": 1.4142135623730951,
        "area": 0.5,
        "angles": "45°, 45°, 90°",
        "ratio": "1 : 1 : √2"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry: the 45-45-90 triangle with legs 1 has hypotenuse √2"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, section 7.2 Right Triangle Trigonometry: the 45-45-90 triangle has sides s, s, √2s. CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry"
  ],
  "related": [
    "special-right-triangles",
    "30-60-90-triangle",
    "right-triangle"
  ],
  "changelog": []
}