{
    "name": "Regular Polygon Calculator",
    "url": "https://calculators.geekmidst.com/calculators/regular-polygon-area-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/regular-polygon-area-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the area and perimeter of any regular polygon (hexagon, octagon, n-gon) from its side, circumradius or apothem, plus its interior angle.",
    "summary": "A regular polygon has equal sides and equal angles. Knowing the number of sides and one length (side, circumradius or apothem) fixes its area, perimeter and every angle.",
    "also_called": [
        "Regular polygon perimeter calculator",
        "Hexagon calculator",
        "Octagon 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": "n",
            "label": "Number of sides n",
            "type": "number",
            "default": "6"
        },
        {
            "name": "known",
            "label": "I know the",
            "type": "select",
            "default": "side",
            "options": {
                "side": "Side length",
                "circumradius": "Circumradius (centre to corner)",
                "apothem": "Apothem (centre to middle of a side)"
            }
        },
        {
            "name": "value",
            "label": "Length",
            "type": "number",
            "default": "2"
        },
        {
            "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": "Area",
            "format": "number"
        },
        {
            "key": "perimeter",
            "label": "Perimeter",
            "format": "number"
        },
        {
            "key": "side",
            "label": "Side length",
            "format": "number"
        },
        {
            "key": "apothem",
            "label": "Apothem",
            "format": "number"
        },
        {
            "key": "circumradius",
            "label": "Circumradius",
            "format": "number"
        },
        {
            "key": "interior_angle",
            "label": "Each interior angle",
            "format": "number"
        },
        {
            "key": "angle_sum",
            "label": "Sum of interior angles",
            "format": "number"
        }
    ],
    "formula": {
        "text": "P = n·s;  apothem a = s ÷ (2 tan(π/n));  R = s ÷ (2 sin(π/n));  Area = ½·P·a = n·s² ÷ (4 tan(π/n));  interior angle = (n − 2) × 180° ÷ n",
        "variables": {
            "n": "number of sides",
            "s": "side length",
            "a": "apothem",
            "R": "circumradius",
            "P": "perimeter"
        }
    },
    "worked_example": {
        "inputs": {
            "n": "6",
            "known": "side",
            "value": "2",
            "unit": "cm"
        },
        "steps": [
            "Perimeter = 6 × 2 = 12 cm.",
            "Apothem = 2 ÷ (2 tan 30°) = √3 ≈ 1.732051 cm.",
            "Area = ½ × 12 × √3 = 6√3 ≈ 10.392305 cm²."
        ],
        "result": "A regular hexagon with 2 cm sides has an area of about 10.39 cm² and a perimeter of 12 cm."
    },
    "interpretation": "As n grows, the polygon approaches its circumscribed circle: a 1,000-gon’s area is within a few millionths of πR².",
    "assumptions": [
        "The polygon is regular and convex."
    ],
    "limitations": [
        "Irregular polygons need their vertex coordinates (shoelace formula) instead."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "What is the apothem?",
            "answer": "The distance from the centre to the midpoint of any side. It is the radius of the inscribed circle."
        },
        {
            "question": "How wide is a hexagon across the flats?",
            "answer": "Twice the apothem; across the corners it is twice the circumradius, which for a hexagon equals two side lengths."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Area Calculator",
            "url": "https://calculators.geekmidst.com/calculators/area-calculator"
        },
        {
            "name": "Rhombus Area Calculator",
            "url": "https://calculators.geekmidst.com/calculators/rhombus-area-calculator"
        },
        {
            "name": "Circle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/circle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}