{
  "capability_id": "math.geometry.plane",
  "tool_id": "plane-geometry",
  "engine": "plane-geometry",
  "calculation_version": "1.3.0",
  "generated_at": "2026-09-20T09:07:29.782Z",
  "engine_build": "2026-09-20",
  "test_count": 25,
  "passed": 25,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 9,
      "passed": 9
    },
    "boundary": {
      "total": 5,
      "passed": 5
    },
    "property": {
      "total": 11,
      "passed": 11
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/math/plane-geometry",
  "api": "https://www.calculatorx.com/api/v1/calc/plane-geometry",
  "schema_input": "https://www.calculatorx.com/schemas/math.geometry.plane.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.geometry.plane.output.json",
  "assumptions": [
    "Euclidean plane",
    "Regular polygon integer n ≥ 3",
    "Annulus R > r",
    "Parallelogram θ in (0°, 180°)",
    "Isosceles trapezoid a, b, h > 0",
    "Irregular polygon shoelace 3–8 vertices",
    "Not the circle-only page",
    "Not a triangle solver",
    "Not solid geometry",
    "Not /trapezoidal-rule",
    "Not the fluid trapezoid",
    "Not CAS"
  ],
  "limitations": [
    "Euclidean plane figures only",
    "Not the circle-only page — 2-D circle stays on math.geometry.circle",
    "Not a triangle solver — SSS/SAS stay on math.geometry.triangle",
    "Not solid geometry (sphere / cylinder / cone)",
    "Regular polygon needs integer n ≥ 3",
    "Annulus needs outer R > inner r; R = r is a circle, not an annulus",
    "Parallelogram included angle θ must be in (0°, 180°); θ=90° recovers a rectangle",
    "Isosceles trapezoid needs bases a,b>0 and height h>0; a=b recovers a rectangle; not the fluid trapezoid; not /trapezoidal-rule",
    "Irregular polygon uses shoelace on 3–8 vertices; simple polygon assumed; area 0 is invalid; not /regular-polygon; not a triangle solver",
    "unknown mode (including cas, sphere, pythagorean) → INVALID_MODE"
  ],
  "boundary_conditions": [
    "a ≤ 0 or b ≤ 0 or s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "n < 3 or non-integer n → INVALID_INPUT",
    "annulus R ≤ r → INVALID_INPUT",
    "parallelogram θ ≤ 0° or θ ≥ 180° → INVALID_INPUT",
    "irregular n < 3 or n > 8 or A=0 → INVALID_INPUT",
    "unknown mode → INVALID_MODE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics"
  ],
  "methods": [],
  "fingerprint": "sha256:9fc1af0587cb9437fce2243bf5c517bd28f8ef8c9ee970ffb308e8a44017bf30",
  "tests": [
    {
      "id": "golden-rect-345",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "rectangle",
        "a": 3,
        "b": 4
      },
      "expect": {
        "area": 12,
        "perimeter": 14,
        "diagonal": 5
      },
      "expectWarning": null,
      "actual": {
        "mode": "rectangle",
        "a": 3,
        "b": 4,
        "area": 12,
        "perimeter": 14,
        "diagonal": 5,
        "square": false,
        "formula": "A = ab,  P = 2(a+b),  d = √(a²+b²)"
      },
      "tol": null
    },
    {
      "id": "golden-hexagon",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "regular_polygon",
        "n": 6,
        "s": 1
      },
      "expect": {
        "perimeter": 6,
        "interior_deg": 120,
        "area": 2.59807621135
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      "expectWarning": null,
      "actual": {
        "mode": "regular_polygon",
        "n": 6,
        "s": 1,
        "area": 2.59807621135,
        "perimeter": 6,
        "apothem": 0.866025403784,
        "circumradius": 1,
        "interior_deg": 120,
        "central_deg": 60,
        "formula": "A = (n s²) / (4 tan(π/n)),  P = n s"
      },
      "tol": null
    },
    {
      "id": "golden-annulus",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "annulus",
        "R": 2,
        "r": 1
      },
      "expect": {
        "area": 9.42477796077
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      "expectWarning": null,
      "actual": {
        "mode": "annulus",
        "r": 1,
        "R": 2,
        "area": 9.42477796077,
        "perimeter": 18.8495559215,
        "outer_circumference": 12.5663706144,
        "inner_circumference": 6.28318530718,
        "formula": "A = π(R² − r²),  P = 2π(R + r)"
      },
      "tol": null
    },
    {
      "id": "golden-parallelogram-30",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "parallelogram",
        "a": 2,
        "b": 3,
        "theta": 30
      },
      "expect": {
        "area": 3,
        "perimeter": 10
      },
      "expectWarning": null,
      "actual": {
        "mode": "parallelogram",
        "a": 2,
        "b": 3,
        "theta": 30,
        "area": 3,
        "perimeter": 10,
        "d1": 4.83655919486,
        "d2": 1.61483595284,
        "height": 1.5,
        "rectangle": false,
        "formula": "A = ab sin θ,  P = 2(a+b),  d² = a²+b²±2ab cos θ"
      },
      "tol": null
    },
    {
      "id": "golden-parallelogram-rect",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "parallelogram",
        "a": 3,
        "b": 4,
        "theta": 90
      },
      "expect": {
        "area": 12,
        "d1": 5,
        "d2": 5,
        "rectangle": true
      },
      "expectWarning": null,
      "actual": {
        "mode": "parallelogram",
        "a": 3,
        "b": 4,
        "theta": 90,
        "area": 12,
        "perimeter": 14,
        "d1": 5,
        "d2": 5,
        "height": 4,
        "rectangle": true,
        "formula": "A = ab sin θ,  P = 2(a+b),  d² = a²+b²±2ab cos θ"
      },
      "tol": null
    },
    {
      "id": "boundary-plane-mode",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "cas",
        "a": 3,
        "b": 4
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "mode must be rectangle, square, regular_polygon, annulus, parallelogram, trapezoid, or irregular_polygon. Not the circle-only page, not a triangle solver, not a solid, not CAS."
    },
    {
      "id": "boundary-polygon-n2",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "regular_polygon",
        "n": 2,
        "s": 1
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "n must be an integer ≥ 3."
    },
    {
      "id": "boundary-para-deg",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "parallelogram",
        "a": 2,
        "b": 3,
        "theta": 0
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Included angle θ must be in (0°, 180°)."
    },
    {
      "id": "golden-trapezoid-243",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "trapezoid",
        "a": 2,
        "b": 4,
        "h": 3
      },
      "expect": {
        "area": 9,
        "isosceles": true
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      "expectWarning": null,
      "actual": {
        "mode": "trapezoid",
        "a": 2,
        "b": 4,
        "h": 3,
        "height": 3,
        "leg": 3.16227766017,
        "midsegment": 3,
        "area": 9,
        "perimeter": 12.3245553203,
        "isosceles": true,
        "rectangle": false,
        "formula": "A = (a+b)h/2,  P = a+b+2√(((a−b)/2)²+h²)  (isosceles)"
      },
      "tol": null
    },
    {
      "id": "golden-trapezoid-rect",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "trapezoid",
        "a": 3,
        "b": 3,
        "h": 4
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      "expect": {
        "area": 12,
        "perimeter": 14,
        "rectangle": true
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      "expectWarning": null,
      "actual": {
        "mode": "trapezoid",
        "a": 3,
        "b": 3,
        "h": 4,
        "height": 4,
        "leg": 4,
        "midsegment": 3,
        "area": 12,
        "perimeter": 14,
        "isosceles": true,
        "rectangle": true,
        "formula": "A = (a+b)h/2,  P = a+b+2√(((a−b)/2)²+h²)  (isosceles)"
      },
      "tol": null
    },
    {
      "id": "boundary-trap-h",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "trapezoid",
        "a": 2,
        "b": 4,
        "h": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "h must be positive."
    },
    {
      "id": "golden-irreg-16",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "irregular_polygon",
        "vertices": "0,0; 5,0; 4,4; 0,3"
      },
      "expect": {
        "area": 16,
        "n": 4
      },
      "expectWarning": null,
      "actual": {
        "mode": "irregular_polygon",
        "n": 4,
        "vertices": [
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            0
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          [
            5,
            0
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          [
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            4
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        "area": 16,
        "perimeter": 16.2462112512,
        "formula": "A = (1/2)|Σ (x_i y_{i+1} − x_{i+1} y_i)|  (shoelace)"
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      "tol": null
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    {
      "id": "golden-irreg-rect",
      "ok": true,
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      "inputs": {
        "mode": "irregular_polygon",
        "vertices": [
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            0
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          [
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            0
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          [
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            4
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      "expect": {
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        "vertices": [
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            4
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        "area": 12,
        "perimeter": 14,
        "formula": "A = (1/2)|Σ (x_i y_{i+1} − x_{i+1} y_i)|  (shoelace)"
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      "id": "boundary-irreg-n2",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "irregular_polygon",
        "vertices": "0,0; 1,0"
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Need at least 3 vertices."
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    {
      "id": "property-plane-rect-d",
      "ok": true,
      "kind": "property",
      "detail": "3-4 rectangle",
      "expect": 5,
      "actual": 5
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    {
      "id": "property-plane-hex",
      "ok": true,
      "kind": "property",
      "detail": "hexagon 120°",
      "expect": 120,
      "actual": 120
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    {
      "id": "property-plane-hex-a",
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      "kind": "property",
      "detail": "hexagon area",
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      "actual": 2.59807621135
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    {
      "id": "property-plane-annulus",
      "ok": true,
      "kind": "property",
      "detail": "annulus 2-1",
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      "id": "property-plane-square",
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      "kind": "property",
      "detail": "square s=2",
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      "ok": true,
      "kind": "property",
      "detail": "parallelogram 2-3-30°",
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      "detail": "shoelace 0,0;5,0;4,4;0,3",
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      "kind": "property",
      "detail": "rectangle vertices",
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  "tier": "engine_tested",
  "verification": {
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      "total": 25,
      "golden": 9,
      "boundary": 5,
      "property": 11
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