{
    "name": "Cone Surface Area Calculator",
    "url": "https://calculators.geekmidst.com/calculators/cone-surface-area-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/cone-surface-area-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find a cone’s lateral (curved) area, base area and total surface from the radius and either the vertical height or the slant height.",
    "summary": "The curved surface of a cone is πrl, where l is the slant height along the side. Add the circular base πr² for a closed cone.",
    "also_called": [
        "Cone slant height 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": "measure",
            "label": "Base given as",
            "type": "segmented",
            "default": "radius",
            "options": {
                "radius": "Radius",
                "diameter": "Diameter"
            }
        },
        {
            "name": "r",
            "label": "Base radius or diameter",
            "type": "number",
            "default": "3"
        },
        {
            "name": "known",
            "label": "I also know the",
            "type": "segmented",
            "default": "height",
            "options": {
                "height": "Vertical height",
                "slant": "Slant height"
            }
        },
        {
            "name": "h",
            "label": "Vertical height",
            "type": "number",
            "default": "4",
            "shown_when": {
                "known": "height"
            }
        },
        {
            "name": "l",
            "label": "Slant height",
            "type": "number",
            "default": "5",
            "shown_when": {
                "known": "slant"
            }
        },
        {
            "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": "total",
            "label": "Total surface area (with base)",
            "format": "number"
        },
        {
            "key": "lateral",
            "label": "Lateral (curved) area πrl",
            "format": "number"
        },
        {
            "key": "base",
            "label": "Base area πr²",
            "format": "number"
        },
        {
            "key": "slant",
            "label": "Slant height l",
            "format": "number"
        },
        {
            "key": "height",
            "label": "Vertical height h",
            "format": "number"
        },
        {
            "key": "volume",
            "label": "Volume ⅓πr²h",
            "format": "number"
        }
    ],
    "formula": {
        "text": "l = √(r² + h²);  Lateral = πrl;  Base = πr²;  Total = πr(r + l)",
        "variables": {
            "r": "base radius",
            "h": "vertical height",
            "l": "slant height"
        }
    },
    "worked_example": {
        "inputs": {
            "measure": "radius",
            "r": "3",
            "known": "height",
            "h": "4",
            "l": "5",
            "unit": "cm"
        },
        "steps": [
            "l = √(3² + 4²) = 5 cm.",
            "Lateral = π × 3 × 5 = 15π ≈ 47.12389 cm².",
            "Total = 15π + 9π = 24π ≈ 75.398224 cm²."
        ],
        "result": "A cone with base radius 3 cm and height 4 cm has a total surface area of about 75.40 cm²."
    },
    "interpretation": "Use the lateral area for an open cone such as a funnel, party hat or conical roof, and the total for a closed solid cone.",
    "assumptions": [
        "A right circular cone: the tip is directly above the centre of the base."
    ],
    "limitations": [
        "Seams, overlaps and material thickness are not included.",
        "A truncated cone (frustum) needs a different formula."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Why does the curved area use slant height, not height?",
            "answer": "Unrolled, the side is a sector of a circle whose radius is the slant height l, and its area works out to πrl."
        },
        {
            "question": "How do I get the slant height?",
            "answer": "By Pythagoras: l = √(r² + h²), because the radius, height and slant form a right triangle."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Surface Area Calculator",
            "url": "https://calculators.geekmidst.com/calculators/surface-area-calculator"
        },
        {
            "name": "Cylinder Surface Area Calculator",
            "url": "https://calculators.geekmidst.com/calculators/cylinder-surface-area-calculator"
        },
        {
            "name": "Volume Calculator",
            "url": "https://calculators.geekmidst.com/calculators/volume-calculator"
        },
        {
            "name": "Right Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/right-triangle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}