3D Solid Volume/Surface Area Calculator mechanism

Model scope

The explicitly supplied sphere and cone volume formulas are standard and correct under the stated assumptions, but the catalog claim is broader than the formulas provided. The exact ambiguity requiring review is whether 'surface area' means total closed surface area, lateral area, or exposed area, and which dimensions define prism, pyramid, frustum, and torus inputs.

• The sphere volume formula V=(4/3)πr³ is correct when r is the radius.
• The cone volume formula V=(1/3)πr²h is correct for a right circular cone when r is base radius and h is perpendicular height.
• SA formulas are unspecified: the specification does not define whether surface area means total, closed, lateral, or exposed area.
• Formulas are missing for prism, pyramid, cylinder, frustum, and torus despite those solids being claimed outputs.
• Prism and pyramid calculations require base shape and dimensions; 'dimensions' alone is ambiguous.
• A torus requires major radius R and minor-tube radius r, plus a constraint such as R>r for a standard ring torus.

Units and conventions

• All dimensions in each calculation must use the same length unit.
• Volume outputs use cubic length units and surface-area outputs use square length units.
• r means radius, not diameter; h means perpendicular height.
• π is assumed to be the mathematical constant, evaluated here as 3.141592653589793.
• Test vectors for cone and frustum use total closed surface area, including the base or bases.
• The frustum test uses the standard right-circular-frustum formulas V=(πh/3)(R²+Rr+r²) and SA=π(R²+r²+(R+r)s), where s=sqrt((R-r)²+h²).

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "solid_type": "sphere",
    "radius": 3,
    "length_unit": "m"
  },
  "expected_outputs": {
    "volume": 113.09733552923255,
    "volume_unit": "m^3",
    "surface_area": 113.09733552923255,
    "surface_area_unit": "m^2"
  },
  "working": "V=(4/3)πr³=(4/3)π(3³)=(4/3)π(27)=36π=113.09733552923255 m^3. SA=4πr²=4π(3²)=36π=113.09733552923255 m^2."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification does not state torus validity regimes or spindle-torus handling, so inputs with R<r can produce a plausible but geometrically inappropriate result. Extreme sphere radii also overflow the volume in ordinary JavaScript Number arithmetic, and negative cone heights expose an unstated sign convention.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json