{
    "name": "Law of Sines Calculator",
    "url": "https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Solve a triangle with the law of sines from two sides and a non-included angle (finding 0, 1 or 2 triangles) or from two angles and a side.",
    "summary": "The law of sines says a ÷ sin A = b ÷ sin B = c ÷ sin C. Given two sides and an angle that is not between them, there may be no triangle, one, or two, and this calculator finds them all.",
    "also_called": [
        "Sine rule calculator",
        "SSA triangle calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "mode",
            "label": "What you know",
            "type": "segmented",
            "default": "ssa",
            "options": {
                "ssa": "Two sides + angle (SSA)",
                "aas": "Two angles + side (AAS)"
            }
        },
        {
            "name": "ssa_A",
            "label": "Angle A",
            "type": "number",
            "default": "30",
            "unit": "°",
            "shown_when": {
                "mode": "ssa"
            }
        },
        {
            "name": "ssa_a",
            "label": "Side a (opposite angle A)",
            "type": "number",
            "default": "6",
            "shown_when": {
                "mode": "ssa"
            }
        },
        {
            "name": "ssa_b",
            "label": "Side b",
            "type": "number",
            "default": "10",
            "shown_when": {
                "mode": "ssa"
            }
        },
        {
            "name": "aas_A",
            "label": "Angle A",
            "type": "number",
            "default": "30",
            "unit": "°",
            "shown_when": {
                "mode": "aas"
            }
        },
        {
            "name": "aas_B",
            "label": "Angle B",
            "type": "number",
            "default": "45",
            "unit": "°",
            "shown_when": {
                "mode": "aas"
            }
        },
        {
            "name": "aas_a",
            "label": "Side a (opposite angle A)",
            "type": "number",
            "default": "10",
            "shown_when": {
                "mode": "aas"
            }
        }
    ],
    "outputs": [
        {
            "key": "solutions",
            "label": "Triangles possible",
            "format": "text"
        },
        {
            "key": "count",
            "label": "Number of triangles",
            "format": "integer"
        },
        {
            "key": "angle_b",
            "label": "Angle B (first triangle)",
            "format": "number"
        },
        {
            "key": "angle_c",
            "label": "Angle C (first triangle)",
            "format": "number"
        },
        {
            "key": "side_b",
            "label": "Side b",
            "format": "number"
        },
        {
            "key": "side_c",
            "label": "Side c (first triangle)",
            "format": "number"
        },
        {
            "key": "second",
            "label": "Second triangle",
            "format": "text"
        },
        {
            "key": "ratio",
            "label": "a ÷ sin A (= 2R)",
            "format": "number"
        }
    ],
    "formula": {
        "text": "a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R;  SSA: sin B = b·sin A ÷ a, and B may be B₁ or 180° − B₁;  C = 180° − A − B",
        "variables": {
            "a, b, c": "sides opposite angles A, B, C",
            "R": "the circumradius",
            "h = b·sin A": "the height that decides how many SSA triangles exist"
        }
    },
    "worked_example": {
        "inputs": {
            "mode": "ssa",
            "ssa_A": "30",
            "ssa_a": "6",
            "ssa_b": "10",
            "aas_A": "30",
            "aas_B": "45",
            "aas_a": "10"
        },
        "steps": [
            "Height h = b·sin A = 10 × 0.5 = 5. Since 5 < a = 6 < b = 10, two triangles exist.",
            "sin B = 10 × 0.5 ÷ 6 = 0.833333, so B₁ = 56.4427° and B₂ = 180° − 56.4427° = 123.5573°.",
            "Triangle 1: C = 180° − 30° − 56.4427° = 93.5573°, c = 6 × sin 93.5573° ÷ 0.5 ≈ 11.976879.",
            "Triangle 2: C = 26.4427°, c = 12 × sin 26.4427° ≈ 5.343629."
        ],
        "result": "With A = 30°, a = 6 and b = 10 there are two triangles: c ≈ 11.977 (B ≈ 56.44°) or c ≈ 5.344 (B ≈ 123.56°)."
    },
    "interpretation": "In the SSA case compare side a with the height h = b·sin A: a < h gives no triangle, a = h one right triangle, h < a < b two triangles, and a ≥ b one triangle. If a drawing or context tells you whether B is acute or obtuse, choose the matching triangle.",
    "assumptions": [
        "Angles are in degrees.",
        "Angle A is opposite side a; in SSA mode angle A is not between the two given sides."
    ],
    "limitations": [
        "When a is extremely close to the height h, rounding in the inputs can switch the answer between one and two triangles."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "When should I use the law of cosines instead?",
            "answer": "Use the law of cosines when you know all three sides (SSS) or two sides and the angle between them (SAS). The Triangle Calculator handles those cases."
        },
        {
            "question": "Why can SSA give two triangles?",
            "answer": "Side a can swing to meet the base line at two points when it is longer than the height but shorter than side b. Both positions satisfy the same sine value, because sin B = sin(180° − B)."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-calculator"
        },
        {
            "name": "Triangle Circumradius Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-circumradius-calculator"
        },
        {
            "name": "Right Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/right-triangle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}