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      "assumptions",
      "limitations",
      "sources",
      "tests"
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    "Ix = Iy = π r⁴/4",
    "Sx = Sy = π r³/4",
    "kx = ky = r/2",
    "Not a rectangle, a hollow circle, bending stress, or a plastic modulus"
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    "r ≤ 0 → VALUE_MUST_BE_POSITIVE"
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      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Moments of inertia — circular area",
      "supports": "The centroidal moment of a solid circle about a diameter is π r⁴/4.",
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        "Ix": 12.566370614359172,
        "Iy": 12.566370614359172,
        "Sx": 6.283185307179586,
        "Sy": 6.283185307179586,
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        "ky": 1,
        "model": "circular_section"
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        "area": 12.566370614359172,
        "Ix": 12.566370614359172,
        "Iy": 12.566370614359172,
        "kx": 1,
        "ky": 1,
        "Sx": 6.283185307179586,
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        "convention": "Solid circle about centroidal diameters. Ix = Iy = π r⁴/4. Sx = Sy = π r³/4. kx = ky = r/2. A rectangle is the rectangular-section page. A hollow circle is its own page."
      },
      "tol": 1e-12
    },
    {
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      "ok": true,
      "kind": "golden",
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        "r": 1
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      "expect": {
        "area": 3.141592653589793,
        "Ix": 0.7853981633974483,
        "Iy": 0.7853981633974483,
        "Sx": 0.7853981633974483,
        "Sy": 0.7853981633974483,
        "kx": 0.5,
        "ky": 0.5
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      "actual": {
        "model": "circular_section",
        "r": 1,
        "area": 3.141592653589793,
        "Ix": 0.7853981633974483,
        "Iy": 0.7853981633974483,
        "kx": 0.5,
        "ky": 0.5,
        "Sx": 0.7853981633974483,
        "Sy": 0.7853981633974483,
        "solver": "area moment + section modulus",
        "convention": "Solid circle about centroidal diameters. Ix = Iy = π r⁴/4. Sx = Sy = π r³/4. kx = ky = r/2. A rectangle is the rectangular-section page. A hollow circle is its own page."
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      "ok": true,
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        "Ix": 201.06192982974676,
        "Iy": 201.06192982974676,
        "Sx": 50.26548245743669,
        "Sy": 50.26548245743669,
        "kx": 2,
        "ky": 2
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        "area": 50.26548245743669,
        "Ix": 201.06192982974676,
        "Iy": 201.06192982974676,
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        "ky": 2,
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        "convention": "Solid circle about centroidal diameters. Ix = Iy = π r⁴/4. Sx = Sy = π r³/4. kx = ky = r/2. A rectangle is the rectangular-section page. A hollow circle is its own page."
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      "inputs": {
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      "kind": "boundary",
      "inputs": {},
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "r is required."
    }
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