{
    "name": "Sector Area Calculator",
    "url": "https://calculators.geekmidst.com/calculators/sector-area-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/sector-area-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Work out the area of a circle sector from the radius and either the central angle or the arc length, with the angle, arc and perimeter.",
    "summary": "A sector is a pie slice of a circle. Its area is the same fraction of πr² as its angle is of a full turn, or simply ½ × radius × arc length.",
    "also_called": [
        "Circle sector 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": "angle",
            "options": {
                "angle": "Radius and angle",
                "arc": "Radius and arc length"
            }
        },
        {
            "name": "r",
            "label": "Radius",
            "type": "number",
            "default": "6"
        },
        {
            "name": "angle",
            "label": "Central angle",
            "type": "number",
            "default": "90",
            "shown_when": {
                "known": "angle"
            }
        },
        {
            "name": "angle_unit",
            "label": "Angle in",
            "type": "segmented",
            "default": "deg",
            "options": {
                "deg": "Degrees",
                "rad": "Radians"
            },
            "shown_when": {
                "known": "angle"
            }
        },
        {
            "name": "arc",
            "label": "Arc length",
            "type": "number",
            "default": "6",
            "shown_when": {
                "known": "arc"
            }
        },
        {
            "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": "area",
            "label": "Sector area",
            "format": "number"
        },
        {
            "key": "arc_length",
            "label": "Arc length",
            "format": "number"
        },
        {
            "key": "perimeter",
            "label": "Sector perimeter (2r + arc)",
            "format": "number"
        },
        {
            "key": "angle_deg",
            "label": "Central angle",
            "format": "number"
        },
        {
            "key": "angle_rad",
            "label": "Central angle in radians",
            "format": "number"
        },
        {
            "key": "fraction",
            "label": "Share of the full circle",
            "format": "percent"
        }
    ],
    "formula": {
        "text": "Area = (θ/360°) × πr² = ½r²θ (θ in radians) = ½ × r × s;  s = rθ",
        "variables": {
            "r": "radius",
            "θ": "central angle",
            "s": "arc length"
        }
    },
    "worked_example": {
        "inputs": {
            "known": "angle",
            "r": "6",
            "angle": "90",
            "angle_unit": "deg",
            "arc": "6",
            "unit": "cm"
        },
        "steps": [
            "90° is 90/360 = 25% of the circle.",
            "Area = 0.25 × π × 6² = 9π ≈ 28.274334 cm².",
            "Arc = 0.25 × 2π × 6 = 3π ≈ 9.424778 cm."
        ],
        "result": "A quarter circle of radius 6 cm has an area of about 28.27 cm²."
    },
    "interpretation": "Use it for pie-chart slices, fan-shaped rooms, sprinkler coverage or a slice of a round cake. Doubling the radius quadruples the area at the same angle.",
    "assumptions": [
        "The sector is cut from a perfect circle by two radii."
    ],
    "limitations": [
        "The arc cannot be longer than the full circumference, and the angle cannot exceed 360°.",
        "For the area between the arc and its chord, use the segment area on the arc length calculator."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "What is the difference between a sector and a segment?",
            "answer": "A sector is bounded by two radii and the arc (a pie slice). A segment is bounded by a chord and the arc."
        },
        {
            "question": "Why is area ½ × r × arc?",
            "answer": "A sector behaves like a triangle with base equal to the arc and height equal to the radius, just as a circle’s area is ½ × r × circumference."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Arc Length Calculator",
            "url": "https://calculators.geekmidst.com/calculators/arc-length-calculator"
        },
        {
            "name": "Circle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/circle-calculator"
        },
        {
            "name": "Area Calculator",
            "url": "https://calculators.geekmidst.com/calculators/area-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}