{
    "name": "Triangle Inradius Calculator",
    "url": "https://calculators.geekmidst.com/calculators/triangle-inradius-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/triangle-inradius-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the radius of the largest circle that fits inside a triangle (the incircle) from its three sides or from its area and perimeter.",
    "summary": "The inradius is the radius of the circle that touches all three sides of a triangle. It equals the area divided by the semiperimeter: r = A ÷ s.",
    "also_called": [
        "Incircle calculator",
        "Inscribed circle radius 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": "known",
            "label": "I know",
            "type": "segmented",
            "default": "sides",
            "options": {
                "sides": "Three sides",
                "area_perimeter": "Area and perimeter"
            }
        },
        {
            "name": "a",
            "label": "Side a",
            "type": "number",
            "default": "3",
            "shown_when": {
                "known": "sides"
            }
        },
        {
            "name": "b",
            "label": "Side b",
            "type": "number",
            "default": "4",
            "shown_when": {
                "known": "sides"
            }
        },
        {
            "name": "c",
            "label": "Side c",
            "type": "number",
            "default": "5",
            "shown_when": {
                "known": "sides"
            }
        },
        {
            "name": "area",
            "label": "Area",
            "type": "number",
            "default": "84",
            "shown_when": {
                "known": "area_perimeter"
            }
        },
        {
            "name": "perimeter",
            "label": "Perimeter",
            "type": "number",
            "default": "42",
            "shown_when": {
                "known": "area_perimeter"
            }
        },
        {
            "name": "unit",
            "label": "Length unit",
            "type": "select",
            "default": "cm",
            "options": {
                "mm": "Millimetres (mm)",
                "cm": "Centimetres (cm)",
                "m": "Metres (m)",
                "km": "Kilometres (km)",
                "in": "Inches (in)",
                "ft": "Feet (ft)",
                "yd": "Yards (yd)",
                "u": "No unit (plain numbers)"
            }
        }
    ],
    "outputs": [
        {
            "key": "inradius",
            "label": "Inradius r",
            "format": "number"
        },
        {
            "key": "incircle_area",
            "label": "Area of the incircle",
            "format": "number"
        },
        {
            "key": "incircle_circumference",
            "label": "Circumference of the incircle",
            "format": "number"
        },
        {
            "key": "area",
            "label": "Triangle area",
            "format": "number"
        },
        {
            "key": "semiperimeter",
            "label": "Semiperimeter s",
            "format": "number"
        },
        {
            "key": "share",
            "label": "Share of the triangle covered by the incircle",
            "format": "percent"
        }
    ],
    "formula": {
        "text": "r = A ÷ s;  s = (a + b + c) ÷ 2;  A = √(s(s − a)(s − b)(s − c)) (Heron)",
        "variables": {
            "r": "inradius",
            "A": "triangle area",
            "s": "semiperimeter",
            "a, b, c": "side lengths"
        }
    },
    "worked_example": {
        "inputs": {
            "known": "sides",
            "a": "3",
            "b": "4",
            "c": "5",
            "area": "84",
            "perimeter": "42",
            "unit": "cm"
        },
        "steps": [
            "s = (3 + 4 + 5) ÷ 2 = 6.",
            "Area = √(6 × 3 × 2 × 1) = 6 cm².",
            "r = 6 ÷ 6 = 1 cm."
        ],
        "result": "The 3-4-5 right triangle has an inscribed circle of radius exactly 1 cm."
    },
    "interpretation": "The incircle’s centre (the incentre) is where the three angle bisectors meet. The inradius is the largest round piece you can cut from a triangular sheet.",
    "assumptions": [
        "The sides form a valid, non-degenerate triangle."
    ],
    "limitations": [
        "In area-and-perimeter mode the tool only checks that the area is possible for the perimeter (at most P² ÷ (12√3)); it cannot tell which triangle you have.",
        "The circumscribed circle (circumradius) is a different measure, R = abc ÷ 4A."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Why does r = A ÷ s work?",
            "answer": "Lines from the incentre to the corners cut the triangle into three triangles of height r on bases a, b and c, so A = ½r(a + b + c) = r·s."
        },
        {
            "question": "Is there a shortcut for right triangles?",
            "answer": "Yes: r = (a + b − c) ÷ 2, where c is the hypotenuse. For 3-4-5 that gives (3 + 4 − 5) ÷ 2 = 1."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Heron’s Formula Calculator",
            "url": "https://calculators.geekmidst.com/calculators/heron-triangle-area-calculator"
        },
        {
            "name": "Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-calculator"
        },
        {
            "name": "Circle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/circle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}