{
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  "category": "other",
  "question": "What are the volume and surface area of a sphere with radius 3?",
  "answer": "volume 113.10, surface area 113.10",
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      "label": "Sphere and cylinder volume and surface area formulas",
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  "source": {
    "kind": "edition_page",
    "value": "Standard geometry reference (e.g. Coxeter, H.S.M. Introduction to Geometry, 2nd ed., Wiley, 1969).",
    "retrieved_utc": "2026-09-30T19:49:00Z"
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    "name": "Sphere and cylinder volume and surface area",
    "version": "1",
    "source_url_or_file": "methods/sphere-and-cylinder-volumes.cjs",
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  "method_deck": {
    "inputs": [
      {
        "name": "shape",
        "type": "string",
        "required": true,
        "description": "Which solid to measure.",
        "enum": [
          "sphere",
          "cylinder"
        ]
      },
      {
        "name": "radius",
        "type": "number",
        "unit": "length units",
        "required": true,
        "description": "Radius of the sphere or cylinder.",
        "min": 0
      },
      {
        "name": "height",
        "type": "number",
        "unit": "length units",
        "required": false,
        "description": "Height of the cylinder (ignored for sphere).",
        "min": 0
      }
    ],
    "outputs": [
      {
        "name": "volume",
        "unit": "cubic length units",
        "description": "Volume of the solid."
      },
      {
        "name": "surface_area",
        "unit": "square length units",
        "description": "Total surface area of the solid."
      }
    ],
    "steps": [
      "For a sphere of radius r: volume = (4/3) * pi * r^3; surface area = 4 * pi * r^2.",
      "For a right circular cylinder of radius r and height h: volume = pi * r^2 * h; total surface area = 2*pi*r*h + 2*pi*r^2 (the side plus both circular ends).",
      "Round both results to 2 decimals."
    ],
    "constants": [
      {
        "name": "pi",
        "value": 3.14159265358979,
        "unit": "dimensionless",
        "source": 0
      }
    ],
    "examples": [
      {
        "name": "worked example",
        "inputs": {
          "shape": "sphere",
          "radius": 3
        },
        "expected": {
          "answer": "volume 113.10, surface area 113.10"
        },
        "check": "By hand: volume=(4/3)*3.14159265*27=113.0973 rounds to 113.10; surface=4*3.14159265*9=113.0973 rounds to 113.10. It is an independently known fact that a sphere's volume and surface area are numerically equal precisely at radius 3, since (4/3)pi*r^3=4*pi*r^2 solves to r=3, giving a built-in cross-check that both values should match."
      },
      {
        "name": "test vector: cylinder",
        "inputs": {
          "shape": "cylinder",
          "radius": 2,
          "height": 5
        },
        "expected": {
          "answer": "volume 62.83, surface area 87.96"
        },
        "check": "By hand: volume=3.14159265*4*5=62.8319 rounds to 62.83; surface=2*3.14159265*2*5+2*3.14159265*4=62.8319+25.1327=87.9646 rounds to 87.96."
      }
    ],
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