{
  "capability_id": "math.geometry.solid",
  "tool_id": "solid-geometry",
  "engine": "solid-geometry",
  "calculation_version": "1.11.0",
  "generated_at": "2026-09-20T11:19:38.023Z",
  "engine_build": "2026-09-20",
  "test_count": 66,
  "passed": 66,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 25,
      "passed": 25
    },
    "boundary": {
      "total": 13,
      "passed": 13
    },
    "property": {
      "total": 28,
      "passed": 28
    }
  },
  "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/solid-geometry",
  "api": "https://www.calculatorx.com/api/v1/calc/solid-geometry",
  "schema_input": "https://www.calculatorx.com/schemas/math.geometry.solid.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.geometry.solid.output.json",
  "assumptions": [
    "Euclidean 3-space",
    "Right solids; not oblique",
    "Ring torus R > r",
    "Not the circle-only page",
    "Not a vector",
    "Not ISO 5167-5",
    "Not a sphere page for ellipsoid",
    "Not CAS"
  ],
  "limitations": [
    "Right solids only; not oblique",
    "Not the circle-only page — 2-D circle stays on math.geometry.circle",
    "Not a vector — math.vector is R²/R³ algebra",
    "Not ISO 5167-5 cone meter (physics.fluid.iso_cone)",
    "Torus is a ring torus: major R > tube r; horn/spindle out of scope",
    "Right circular frustum needs R,r,h>0; r=R recovers a cylinder; not a cone page",
    "Right rectangular prism needs a,b,h>0; a=b=h is a cube; not a frustum page",
    "Right rectangular pyramid needs a,b,h>0; V=abh/3; apex above the base center; a=b is a square base; not a prism page",
    "Regular tetrahedron needs a>0; V=a³√2/12, A=a²√3; not a rectangular pyramid page",
    "Regular octahedron needs a>0; V=(√2/3)a³, A=2√3 a²; not a tetrahedron page",
    "Regular icosahedron needs a>0; V=(5/12)(3+√5)a³, A=5√3 a²; not an octahedron page",
    "Regular dodecahedron needs a>0; V=(1/4)(15+7√5)a³, A=3√(25+10√5) a²; not an icosahedron page",
    "Regular n-gon prism needs integer n≥3 and s,h>0; base from math.geometry.plane; n=4 matches a square rectangular prism; not /prism; not an n-gon pyramid",
    "Regular n-gon pyramid needs integer n≥3 and s,h>0; V=A h/3 with A from math.geometry.plane; n=4 matches a square rectangular pyramid; not /pyramid; not a tetrahedron; not /regular-frustum",
    "Regular n-gon frustum needs integer n≥3 and s,t,h>0; V=h/3(A+a+√(Aa)) with A,a from math.geometry.plane; s=t recovers a regular prism; lateral faces from the plane isosceles trapezoid; not /frustum; not /regular-pyramid",
    "Ellipsoid needs a,b,c>0; V=4/3 πabc; a=b=c is a sphere; not /sphere; not the 2-D ellipse; triaxial surface later",
    "unknown mode (including cas, vector, iso-cone) → INVALID_MODE"
  ],
  "boundary_conditions": [
    "r ≤ 0 or h ≤ 0 or R ≤ 0 or a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "torus R ≤ r → INVALID_INPUT",
    "frustum R ≤ 0 or r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "prism a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "pyramid a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "tetrahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "octahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "icosahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "dodecahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "regular_prism n < 3 → INVALID_INPUT",
    "regular_prism s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "regular_pyramid n < 3 → INVALID_INPUT",
    "regular_pyramid s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "regular_frustum n < 3 → INVALID_INPUT",
    "regular_frustum s ≤ 0 or t ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "ellipsoid a ≤ 0 or b ≤ 0 or c ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "unknown mode → INVALID_MODE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics"
  ],
  "methods": [],
  "fingerprint": "sha256:7f67821d0da3fe01663447846765c3965377fa7703eb4bb3af3f2a91b9746b4e",
  "tests": [
    {
      "id": "golden-unit-sphere",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "sphere",
        "r": 1
      },
      "expect": {
        "volume": 4.18879020479,
        "surface": 12.5663706144
      },
      "expectWarning": null,
      "actual": {
        "mode": "sphere",
        "r": 1,
        "volume": 4.18879020479,
        "surface": 12.5663706144,
        "formula": "V = (4/3)πr³,  A = 4πr²"
      },
      "tol": null
    },
    {
      "id": "golden-cone-345",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "cone",
        "r": 3,
        "h": 4
      },
      "expect": {
        "slant": 5,
        "volume": 37.6991118431
      },
      "expectWarning": null,
      "actual": {
        "mode": "cone",
        "r": 3,
        "h": 4,
        "slant": 5,
        "volume": 37.6991118431,
        "lateral": 47.1238898038,
        "surface": 75.3982236862,
        "formula": "V = (1/3)πr²h,  l = √(r²+h²)"
      },
      "tol": null
    },
    {
      "id": "golden-unit-cylinder",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "cylinder",
        "r": 1,
        "h": 1
      },
      "expect": {
        "volume": 3.14159265359
      },
      "expectWarning": null,
      "actual": {
        "mode": "cylinder",
        "r": 1,
        "h": 1,
        "volume": 3.14159265359,
        "lateral": 6.28318530718,
        "surface": 12.5663706144,
        "formula": "V = πr²h,  A = 2πr(r+h)"
      },
      "tol": null
    },
    {
      "id": "boundary-solid-mode",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "cas",
        "r": 1
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "mode must be sphere, cylinder, cone, torus, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular_prism, regular_pyramid, regular_frustum, or ellipsoid. Not the circle-only page, not a vector, not ISO 5167-5, not CAS."
    },
    {
      "id": "boundary-torus-horn",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "torus",
        "R": 1,
        "r": 1
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Ring torus needs major radius R greater than tube radius r."
    },
    {
      "id": "golden-frustum-213",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "frustum",
        "R": 2,
        "r": 1,
        "h": 3
      },
      "expect": {
        "volume": 21.9911485751
      },
      "expectWarning": null,
      "actual": {
        "mode": "frustum",
        "R": 2,
        "r": 1,
        "h": 3,
        "slant": 3.16227766017,
        "volume": 21.9911485751,
        "lateral": 29.8037647974,
        "surface": 45.5117280653,
        "cylinder": false,
        "formula": "V = (1/3)πh(R²+Rr+r²),  l = √(h²+(R−r)²)"
      },
      "tol": null
    },
    {
      "id": "golden-frustum-cyl",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "frustum",
        "R": 1,
        "r": 1,
        "h": 1
      },
      "expect": {
        "volume": 3.14159265359,
        "cylinder": true
      },
      "expectWarning": null,
      "actual": {
        "mode": "frustum",
        "R": 1,
        "r": 1,
        "h": 1,
        "slant": 1,
        "volume": 3.14159265359,
        "lateral": 6.28318530718,
        "surface": 12.5663706144,
        "cylinder": true,
        "formula": "V = (1/3)πh(R²+Rr+r²),  l = √(h²+(R−r)²)"
      },
      "tol": null
    },
    {
      "id": "boundary-frustum-h",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "frustum",
        "R": 2,
        "r": 1,
        "h": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "h must be positive."
    },
    {
      "id": "golden-prism-234",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "prism",
        "a": 2,
        "b": 3,
        "h": 4
      },
      "expect": {
        "volume": 24,
        "surface": 52,
        "lateral": 40,
        "cube": false
      },
      "expectWarning": null,
      "actual": {
        "mode": "prism",
        "a": 2,
        "b": 3,
        "h": 4,
        "volume": 24,
        "lateral": 40,
        "surface": 52,
        "diagonal": 5.38516480713,
        "cube": false,
        "formula": "V = abh,  A = 2(ab+ah+bh)"
      },
      "tol": null
    },
    {
      "id": "golden-prism-cube",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "prism",
        "a": 1,
        "b": 1,
        "h": 1
      },
      "expect": {
        "volume": 1,
        "surface": 6,
        "cube": true
      },
      "expectWarning": null,
      "actual": {
        "mode": "prism",
        "a": 1,
        "b": 1,
        "h": 1,
        "volume": 1,
        "lateral": 4,
        "surface": 6,
        "diagonal": 1.73205080757,
        "cube": true,
        "formula": "V = abh,  A = 2(ab+ah+bh)"
      },
      "tol": null
    },
    {
      "id": "boundary-prism-h",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "prism",
        "a": 2,
        "b": 3,
        "h": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "h must be positive."
    },
    {
      "id": "golden-pyramid-234",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "pyramid",
        "a": 2,
        "b": 3,
        "h": 4
      },
      "expect": {
        "volume": 8,
        "square_base": false
      },
      "expectWarning": null,
      "actual": {
        "mode": "pyramid",
        "a": 2,
        "b": 3,
        "h": 4,
        "volume": 8,
        "lateral": 20.9133206222,
        "surface": 26.9133206222,
        "slant_a": 4.27200187266,
        "slant_b": 4.12310562562,
        "square_base": false,
        "formula": "V = abh/3,  A = ab + a√(h²+(b/2)²) + b√(h²+(a/2)²)"
      },
      "tol": null
    },
    {
      "id": "golden-pyramid-square",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "pyramid",
        "a": 6,
        "b": 6,
        "h": 4
      },
      "expect": {
        "volume": 48,
        "surface": 96,
        "lateral": 60,
        "slant_a": 5,
        "slant_b": 5,
        "square_base": true
      },
      "expectWarning": null,
      "actual": {
        "mode": "pyramid",
        "a": 6,
        "b": 6,
        "h": 4,
        "volume": 48,
        "lateral": 60,
        "surface": 96,
        "slant_a": 5,
        "slant_b": 5,
        "square_base": true,
        "formula": "V = abh/3,  A = ab + a√(h²+(b/2)²) + b√(h²+(a/2)²)"
      },
      "tol": null
    },
    {
      "id": "boundary-pyramid-h",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "pyramid",
        "a": 2,
        "b": 3,
        "h": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "h must be positive."
    },
    {
      "id": "golden-tetra-unit",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "tetrahedron",
        "a": 1
      },
      "expect": {
        "volume": 0.117851130198,
        "surface": 1.73205080757
      },
      "expectWarning": null,
      "actual": {
        "mode": "tetrahedron",
        "a": 1,
        "volume": 0.117851130198,
        "surface": 1.73205080757,
        "height": 0.816496580928,
        "face": 0.433012701892,
        "formula": "V = a³√2/12,  A = a²√3"
      },
      "tol": null
    },
    {
      "id": "golden-tetra-sqrt2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "tetrahedron",
        "a": 1.4142135623730951
      },
      "expect": {
        "volume": 0.333333333333
      },
      "expectWarning": null,
      "actual": {
        "mode": "tetrahedron",
        "a": 1.4142135623730951,
        "volume": 0.333333333333,
        "surface": 3.46410161514,
        "height": 1.15470053838,
        "face": 0.866025403784,
        "formula": "V = a³√2/12,  A = a²√3"
      },
      "tol": null
    },
    {
      "id": "boundary-tetra-a",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "tetrahedron",
        "a": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "a must be positive."
    },
    {
      "id": "golden-octa-unit",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "octahedron",
        "a": 1
      },
      "expect": {
        "volume": 0.471404520791,
        "surface": 3.46410161514
      },
      "expectWarning": null,
      "actual": {
        "mode": "octahedron",
        "a": 1,
        "volume": 0.471404520791,
        "surface": 3.46410161514,
        "height": 1.41421356237,
        "face": 0.433012701892,
        "formula": "V = (√2/3)a³,  A = 2√3 a²"
      },
      "tol": null
    },
    {
      "id": "golden-octa-sqrt2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "octahedron",
        "a": 1.4142135623730951
      },
      "expect": {
        "volume": 1.33333333333
      },
      "expectWarning": null,
      "actual": {
        "mode": "octahedron",
        "a": 1.4142135623730951,
        "volume": 1.33333333333,
        "surface": 6.92820323028,
        "height": 2,
        "face": 0.866025403784,
        "formula": "V = (√2/3)a³,  A = 2√3 a²"
      },
      "tol": null
    },
    {
      "id": "boundary-octa-a",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "octahedron",
        "a": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "a must be positive."
    },
    {
      "id": "golden-ico-unit",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "icosahedron",
        "a": 1
      },
      "expect": {
        "volume": 2.18169499062,
        "surface": 8.66025403784
      },
      "expectWarning": null,
      "actual": {
        "mode": "icosahedron",
        "a": 1,
        "volume": 2.18169499062,
        "surface": 8.66025403784,
        "height": 1.90211303259,
        "face": 0.433012701892,
        "formula": "V = (5/12)(3+√5)a³,  A = 5√3 a²"
      },
      "tol": null
    },
    {
      "id": "golden-ico-a2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "icosahedron",
        "a": 2
      },
      "expect": {
        "volume": 17.453559925
      },
      "expectWarning": null,
      "actual": {
        "mode": "icosahedron",
        "a": 2,
        "volume": 17.453559925,
        "surface": 34.6410161514,
        "height": 3.80422606518,
        "face": 1.73205080757,
        "formula": "V = (5/12)(3+√5)a³,  A = 5√3 a²"
      },
      "tol": null
    },
    {
      "id": "boundary-ico-a",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "icosahedron",
        "a": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "a must be positive."
    },
    {
      "id": "golden-dodeca-unit",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "dodecahedron",
        "a": 1
      },
      "expect": {
        "volume": 7.66311896062,
        "surface": 20.6457288071
      },
      "expectWarning": null,
      "actual": {
        "mode": "dodecahedron",
        "a": 1,
        "volume": 7.66311896062,
        "surface": 20.6457288071,
        "height": 2.80251707689,
        "face": 1.72047740059,
        "formula": "V = (1/4)(15+7√5)a³,  A = 3√(25+10√5) a²"
      },
      "tol": null
    },
    {
      "id": "golden-dodeca-a2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "dodecahedron",
        "a": 2
      },
      "expect": {
        "volume": 61.304951685
      },
      "expectWarning": null,
      "actual": {
        "mode": "dodecahedron",
        "a": 2,
        "volume": 61.304951685,
        "surface": 82.5829152283,
        "height": 5.60503415378,
        "face": 6.88190960236,
        "formula": "V = (1/4)(15+7√5)a³,  A = 3√(25+10√5) a²"
      },
      "tol": null
    },
    {
      "id": "boundary-dodeca-a",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "dodecahedron",
        "a": 0
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "a must be positive."
    },
    {
      "id": "golden-regular-prism-hex",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "regular_prism",
        "n": 6,
        "s": 1,
        "h": 1
      },
      "expect": {
        "volume": 2.59807621135,
        "surface": 11.1961524227
      },
      "expectWarning": null,
      "actual": {
        "mode": "regular_prism",
        "n": 6,
        "s": 1,
        "h": 1,
        "base": 2.59807621135,
        "apothem": 0.866025403784,
        "volume": 2.59807621135,
        "lateral": 6,
        "surface": 11.1961524227,
        "formula": "V = A h,  A = (n s²)/(4 tan(π/n)),  S = 2A + n s h"
      },
      "tol": null
    },
    {
      "id": "golden-regular-prism-tri",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "regular_prism",
        "n": 3,
        "s": 2,
        "h": 1
      },
      "expect": {
        "volume": 1.73205080757
      },
      "expectWarning": null,
      "actual": {
        "mode": "regular_prism",
        "n": 3,
        "s": 2,
        "h": 1,
        "base": 1.73205080757,
        "apothem": 0.57735026919,
        "volume": 1.73205080757,
        "lateral": 6,
        "surface": 9.46410161514,
        "formula": "V = A h,  A = (n s²)/(4 tan(π/n)),  S = 2A + n s h"
      },
      "tol": null
    },
    {
      "id": "boundary-regular-prism-n",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "regular_prism",
        "n": 2,
        "s": 1,
        "h": 1
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "n must be an integer ≥ 3."
    },
    {
      "id": "golden-regular-pyramid-hex",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "regular_pyramid",
        "n": 6,
        "s": 1,
        "h": 1
      },
      "expect": {
        "volume": 0.8660254037844386
      },
      "expectWarning": null,
      "actual": {
        "mode": "regular_pyramid",
        "n": 6,
        "s": 1,
        "h": 1,
        "base": 2.59807621135,
        "apothem": 0.866025403784,
        "slant": 1.32287565553,
        "volume": 0.866025403783,
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