{
    "name": "Triangle Circumradius Calculator",
    "url": "https://calculators.geekmidst.com/calculators/triangle-circumradius-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/triangle-circumradius-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the radius of the circle through a triangle’s three corners from its three sides, one side and the opposite angle, or vertex coordinates.",
    "summary": "The circumradius R is the radius of the one circle that passes through all three vertices. Every side divided by the sine of its opposite angle equals 2R.",
    "also_called": [
        "Circumcircle calculator",
        "Circumcentre 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": "select",
            "default": "sss",
            "options": {
                "sss": "Three sides",
                "side_angle": "One side and the angle opposite it",
                "coords": "Coordinates of the three vertices"
            }
        },
        {
            "name": "a",
            "label": "Side a",
            "type": "number",
            "default": "5",
            "shown_when": {
                "mode": "sss"
            }
        },
        {
            "name": "b",
            "label": "Side b",
            "type": "number",
            "default": "6",
            "shown_when": {
                "mode": "sss"
            }
        },
        {
            "name": "c",
            "label": "Side c",
            "type": "number",
            "default": "7",
            "shown_when": {
                "mode": "sss"
            }
        },
        {
            "name": "side",
            "label": "Side",
            "type": "number",
            "default": "10",
            "shown_when": {
                "mode": "side_angle"
            }
        },
        {
            "name": "angle",
            "label": "Angle opposite that side",
            "type": "number",
            "default": "30",
            "unit": "°",
            "shown_when": {
                "mode": "side_angle"
            }
        },
        {
            "name": "x1",
            "label": "Vertex 1: x",
            "type": "number",
            "default": "0",
            "shown_when": {
                "mode": "coords"
            }
        },
        {
            "name": "y1",
            "label": "Vertex 1: y",
            "type": "number",
            "default": "0",
            "shown_when": {
                "mode": "coords"
            }
        },
        {
            "name": "x2",
            "label": "Vertex 2: x",
            "type": "number",
            "default": "4",
            "shown_when": {
                "mode": "coords"
            }
        },
        {
            "name": "y2",
            "label": "Vertex 2: y",
            "type": "number",
            "default": "0",
            "shown_when": {
                "mode": "coords"
            }
        },
        {
            "name": "x3",
            "label": "Vertex 3: x",
            "type": "number",
            "default": "0",
            "shown_when": {
                "mode": "coords"
            }
        },
        {
            "name": "y3",
            "label": "Vertex 3: y",
            "type": "number",
            "default": "3",
            "shown_when": {
                "mode": "coords"
            }
        }
    ],
    "outputs": [
        {
            "key": "circumradius",
            "label": "Circumradius R",
            "format": "number"
        },
        {
            "key": "diameter",
            "label": "Circumcircle diameter (2R)",
            "format": "number"
        },
        {
            "key": "circumference",
            "label": "Circumcircle circumference",
            "format": "number"
        },
        {
            "key": "circle_area",
            "label": "Circumcircle area",
            "format": "number"
        },
        {
            "key": "triangle_area",
            "label": "Triangle area",
            "format": "text"
        },
        {
            "key": "centre",
            "label": "Circumcentre",
            "format": "text"
        },
        {
            "key": "position",
            "label": "Where the centre lies",
            "format": "text"
        }
    ],
    "formula": {
        "text": "R = abc ÷ (4K), with K from Heron’s formula;  R = a ÷ (2 sin A);  circumcentre = point equidistant from the three vertices",
        "variables": {
            "a, b, c": "the side lengths",
            "K": "the triangle’s area",
            "A": "the angle opposite side a"
        }
    },
    "worked_example": {
        "inputs": {
            "mode": "sss",
            "a": "5",
            "b": "6",
            "c": "7",
            "side": "10",
            "angle": "30",
            "x1": "0",
            "y1": "0",
            "x2": "4",
            "y2": "0",
            "x3": "0",
            "y3": "3"
        },
        "steps": [
            "Semi-perimeter s = (5 + 6 + 7) ÷ 2 = 9.",
            "Heron: K = √(9 × 4 × 3 × 2) = √216 ≈ 14.696938.",
            "R = (5 × 6 × 7) ÷ (4 × 14.696938) = 210 ÷ 58.787754 ≈ 3.572173."
        ],
        "result": "A triangle with sides 5, 6 and 7 has a circumradius of about 3.5722, so its circumcircle is about 7.1443 across."
    },
    "interpretation": "For an acute triangle the circumcentre sits inside it; for a right triangle it is the midpoint of the hypotenuse; for an obtuse triangle it lies outside. One side and its opposite angle fix R but not the rest of the triangle.",
    "assumptions": [
        "Angles are in degrees.",
        "Side lengths and coordinates share one unit; R is in that unit."
    ],
    "limitations": [
        "Nearly flat triangles have a very large circumradius that is sensitive to rounding in the inputs."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Is the circumradius the same as the inradius?",
            "answer": "No. The circumradius belongs to the circle through the corners (outside the triangle). The inradius belongs to the circle touching all three sides (inside). R ≥ 2r always, with equality only for an equilateral triangle."
        },
        {
            "question": "Why does one side and its opposite angle give R?",
            "answer": "By the extended law of sines, a ÷ sin A = 2R for every side. Any triangle with that side and opposite angle fits the same circle."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-calculator"
        },
        {
            "name": "Right Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/right-triangle-calculator"
        },
        {
            "name": "Law of Sines Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}