{
  "id": "isosceles-triangle",
  "version": "b8bbe3e3bc4d",
  "status": "published",
  "name": "Isosceles Triangle Calculator",
  "question": "How do I solve an isosceles triangle?",
  "summary": "Solves an isosceles triangle from its leg and base, base and height, leg and height, leg and vertex angle, or base and base angle: every side, the height, the angles, the area and the perimeter.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/isosceles-triangle-calculator",
  "markdown": "https://www.acalculator.org/math/isosceles-triangle-calculator.md",
  "kind": "function",
  "method": "With half the base w = b ÷ 2 and the height h: a² = h² + w², area = w × h, perimeter = 2a + b, vertex angle = 2·atan(w ÷ h), base angle = atan(h ÷ w).",
  "assumptions": [
    "The two legs are equal; the base is the third side.",
    "Lengths are in any one unit, and the area is in that unit squared. Angles are in degrees."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "known": {
        "title": "I know",
        "description": "Which two measures of the triangle you know.",
        "type": "string",
        "enum": [
          "lb",
          "bh",
          "lh",
          "lv",
          "ba"
        ]
      },
      "a": {
        "title": "Leg (a)",
        "description": "The length of each of the two equal sides.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "b": {
        "title": "Base (b)",
        "description": "The length of the third, unequal side.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "h": {
        "title": "Height (h)",
        "description": "The distance from the vertex angle straight down to the base.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "v": {
        "title": "Vertex angle",
        "description": "The angle between the two equal legs, in degrees.",
        "type": "number",
        "exclusiveMinimum": 0,
        "exclusiveMaximum": 180
      },
      "ba": {
        "title": "Base angle",
        "description": "Each of the two equal angles at the base, in degrees.",
        "type": "number",
        "exclusiveMinimum": 0,
        "exclusiveMaximum": 90
      }
    }
  },
  "outputs": {
    "area": {
      "label": "Area",
      "description": "Half the base times the height.",
      "format": "number"
    },
    "leg": {
      "label": "Leg (a)",
      "description": "Each of the two equal sides.",
      "format": "number"
    },
    "base": {
      "label": "Base (b)",
      "description": "The unequal side.",
      "format": "number"
    },
    "height": {
      "label": "Height (h)",
      "description": "The height to the base.",
      "format": "number"
    },
    "perimeter": {
      "label": "Perimeter",
      "description": "Two legs plus the base.",
      "format": "number"
    },
    "vertex": {
      "label": "Vertex angle (°)",
      "description": "The angle between the legs, in degrees.",
      "format": "number"
    },
    "baseAngle": {
      "label": "Base angles (°)",
      "description": "Each of the two equal base angles, in degrees.",
      "format": "number"
    },
    "legHeight": {
      "label": "Height to a leg",
      "description": "The distance from a base corner straight to the opposite leg: 2 × area ÷ leg.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "known": "lb",
      "a": 5,
      "b": 6,
      "h": 4,
      "v": 60,
      "ba": 45
    },
    "outputs": {
      "area": 12,
      "leg": 5,
      "base": 6,
      "height": 4,
      "perimeter": 16,
      "vertex": 73.73979529168804,
      "baseAngle": 53.13010235415598,
      "legHeight": 4.8
    },
    "text": "An isosceles triangle with legs 5 and base 6 has an area of 12 and a perimeter of 16."
  },
  "examples": [
    {
      "given": {
        "known": "lb",
        "a": 5,
        "b": 6
      },
      "expect": {
        "height": 4,
        "area": 12,
        "perimeter": 16,
        "vertex": 73.73979529168804,
        "baseAngle": 53.13010235415598,
        "legHeight": 4.8
      },
      "source": "hand calculation in content.mdx: h = √(5² − 3²) = 4, A = ½ × 6 × 4 = 12; OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; the angles of a triangle add to 180°), https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05); OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh), https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids (retrieved 2026-10-05)"
    },
    {
      "given": {
        "known": "lv",
        "a": 10,
        "v": 60
      },
      "expect": {
        "base": 10,
        "height": 8.660254037844387,
        "area": 43.30127018922193,
        "baseAngle": 60
      },
      "source": "hand calculation in content.mdx: b = 2 × 10 × sin 30° = 10, h = 10 cos 30° = 5√3; OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (sin, cos and tan as side ratios), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry (retrieved 2026-10-05)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "known": "bh",
        "b": 8,
        "h": 3
      },
      "expect": {
        "leg": 5,
        "area": 12,
        "perimeter": 18,
        "vertex": 106.26020470831196
      },
      "source": "hand calculation in content.mdx: a = √(4² + 3²) = 5; OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; the angles of a triangle add to 180°), https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "known": "ba",
        "b": 6,
        "ba": 45
      },
      "expect": {
        "height": 3,
        "leg": 4.242640687119285,
        "vertex": 90,
        "area": 9
      },
      "source": "hand calculation in content.mdx: h = 3 tan 45° = 3, a = 3√2; OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (sin, cos and tan as side ratios), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry (retrieved 2026-10-05)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "known": "lh",
        "a": 13,
        "h": 12
      },
      "expect": {
        "base": 10,
        "area": 60,
        "perimeter": 36
      },
      "source": "hand calculation in content.mdx: b ÷ 2 = √(13² − 12²) = 5; OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; the angles of a triangle add to 180°), https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05)"
    }
  ],
  "sources": [
    "OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (the Pythagorean theorem; the angles of a triangle add to 180°), CC BY 4.0. https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05)",
    "OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids (the area of a triangle, A = ½bh), CC BY 4.0. https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids (retrieved 2026-10-05)",
    "OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (sine, cosine and tangent as ratios of sides), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry (retrieved 2026-10-05)"
  ],
  "related": [
    "triangle",
    "triangle-area",
    "right-triangle",
    "45-45-90-triangle"
  ],
  "changelog": []
}