Choose nonoverlapping parts
Split the solid into primitive pieces whose interiors do not overlap. Add each piece with its own shape, dimensions and units, then save it to the component audit. For a 2 × 3 × 4 m cuboid, volume is 24 m³. A contained 1 × 1 × 4 m cutout removes 4 m³, leaving 20 m³. The calculator can check the signed arithmetic but cannot determine spatial containment from lengths alone. You must confirm the decomposition rather than treating a positive total as proof of fit.
Keep an added part and a contained cutout separate
The box diagram identifies each part’s perpendicular L, W and H directions. The signed table supplies the decomposition: the 1 × 1 × 4 m cutout must be contained in the 2 × 3 × 4 m box. The drawing does not infer placement or containment.
| Part / sign | Dimensions | Volume |
|---|---|---|
| Outer box (+) | L 2 m × W 3 m × H 4 m | +24 m³ |
| Contained cutout (−) | L 1 m × W 1 m × H 4 m | −4 m³ |
| Signed total | 24 − 4 | 20 m³ |
| Exterior surface | Depends on cutout placement and exposed faces | Undetermined; never add component surfaces blindly |
Read diagram directions and numbered surface cues
- L: Length. Front horizontal edge; one of three perpendicular directions.
- W: Width. Receding base edge perpendicular to length and height.
- H: Height. Vertical edge perpendicular to both base edges.
- 1 ✓ Included: Side faces (four).
- 2 ✓ Included: Top face.
- 3 ✓ Included: Bottom face.
Solid outlines are schematic. Arrow endpoints identify the measurement; letters map to the table. ✓ included; × excluded; ? undetermined. Hidden edges use dashes.
A defined coaxial join has a known hidden interface
The numbered internal circle is marked × excluded. It occurs once on each component if they are measured separately, but neither copy is exposed in the joined exterior. Matching radius, shared axis and full rim contact are required.
| Joined part | Worked contribution | Exterior selection |
|---|---|---|
| Cylinder: r 2 cm; h 11 cm | Volume 44π cm³ | Lateral 44π cm² |
| Hemisphere: r 2 cm | Volume 16π/3 cm³ | Curved 8π cm² |
| Cylinder bottom | Volume contribution none | Bottom 4π cm² |
| Shared circle | Internal interface | Exclude both copies |
| Joined total | 148π/3 cm³ ≈ 154.985 cm³ | 56π cm² ≈ 175.929 cm² |
Read diagram directions and numbered surface cues
- r: Radius. Axis to circular rim.
- H: Total height. Total height, from bottom to top of hemisphere (H = h + r).
- 1 ✓ Included: Cylinder lateral surface.
- 2 ✓ Included: Curved hemisphere.
- 3 ✓ Included: Bottom disk.
- 4 × Excluded: Shared internal circle.
Solid outlines are schematic. Arrow endpoints identify the measurement; letters map to the table. ✓ included; × excluded; ? undetermined. Hidden edges use dashes. Curved cap and straight section share the declared axis; interface circles are not exterior faces.
Audit every sign and subtotal
A plus row adds a volume; a minus row subtracts a contained cutout. Up to 12 labeled rows are supported. Remove an incorrect row and add a corrected primitive instead of silently editing a stale subtotal. The report lists each sign, model, entered dimensions, formula and volume. A negative total is rejected. A zero total is explicitly an empty or fully removed mathematical model, not a claim about physical capacity. User labels are text only and are not interpreted as expressions.
Exclude a known shared face
For a right cylinder with one same-radius coaxial hemisphere, total height 13 cm and radius 2 cm give straight cylinder height 11 cm. Volume is π×2²×11 + 2π×2³/3 = 148π/3 cm³. The defined exterior includes the cylinder’s lateral surface, the curved hemisphere and the cylinder bottom: 2π×2×11 + 3π×2² = 56π cm². The matching circular interface is internal and is excluded. Displaced or tilted pieces do not meet this preset’s declared geometry.
Do not blindly add exterior areas
Arbitrary joining hides contact faces, and cutouts can expose new faces. The sum of complete component surfaces therefore does not determine the joined exterior. A surface-led composite request explicitly says that the surface is not determined; the valid signed volume remains a separate usable audit. Copy or download that volume-only report with its limitation. This workbench does not infer intersections, optimize packing, estimate material purchases or evaluate structural safety. Use its signed audit only for the decomposition you can justify.