Measure from the correct reference points
A radius runs from the center of a circular section to its boundary. A diameter crosses the entire circle through its center, so it is twice the radius. A cylinder labeled diameter 6 cm has radius 3 cm. Entering 6 as its radius multiplies its volume by four. For a sphere the same mistake multiplies volume by eight. Select the diameter method when that is the measurement shown; the interpretation and formula then make the halving explicit.
Diameter crosses the circle; radius starts at its center
The arrow d crosses the entire sphere through its center. With diameter 6 cm, radius is 3 cm, not 6 cm. All drawings are direction schematics, not to scale.
| Measurement | Worked value | Interpretation |
|---|---|---|
| Diameter d | 6 cm | Boundary to opposite boundary through center |
| Radius r = d/2 | 3 cm | Center to boundary |
| Sphere volume | 36π cm³ ≈ 113.097 cm³ | 4π × 3³/3; using 6 as radius gives eight times this volume |
Read diagram directions and numbered surface cues
- d: Diameter. Boundary to opposite boundary through the center; d = 2r.
- 1 ✓ Included: Entire spherical surface.
Solid outlines are schematic. Arrow endpoints identify the measurement; letters map to the table. ✓ included; × excluded; ? undetermined. Hidden edges use dashes.
A hemisphere adds one radius to the straight section
H runs from the bottom disk to the top pole. The matching hemisphere and cylinder share one axis. The dashed internal interface is excluded from the exterior.
| Measurement | Worked value | Reference |
|---|---|---|
| Total height H | 13 cm | Bottom to top pole |
| Radius r | 2 cm | Axis to circular rim |
| Straight height h = H − r | 11 cm | Bottom disk to hemisphere join |
| Preset volume | 148π/3 cm³ ≈ 154.985 cm³ | Cylinder plus hemisphere; matching circular interface is internal |
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.
Full axes extend both ways from the center
Each a, b and c arrow spans a full principal axis. The depth axis is projected obliquely on the page; all three principal axes are perpendicular in the model.
| Principal axis | Full axis | Semiaxis used in V = 4πabc/3 |
|---|---|---|
| Horizontal | 4 cm | 2 cm |
| Vertical | 6 cm | 3 cm |
| Depth | 8 cm | 4 cm |
| Volume | 32π cm³ ≈ 100.531 cm³ | Surface is not supplied by this volume model |
Read diagram directions and numbered surface cues
- a: Full axis a. Full horizontal principal axis through the center.
- b: Full axis b. Full vertical principal axis through the center.
- c: Full axis c. Full depth principal axis; shown obliquely in this projection.
- 1 ? Undetermined: Ellipsoid surface (not supplied).
Solid outlines are schematic. Arrow endpoints identify the measurement; letters map to the table. ✓ included; × excluded; ? undetermined. Hidden edges use dashes.
Separate straight, total and perpendicular height
A cylinder topped by one same-radius hemisphere has total height equal to the straight cylinder height plus the radius. For total height 13 cm and radius 2 cm, the cylinder section is 11 cm, not 13 cm. The coaxial preset uses that difference and excludes the shared circular face. For a prism, perpendicular height is the separation of the parallel base planes. A slanted joining edge may be longer and cannot replace that perpendicular measurement.
Check axes and model assumptions
An ellipsoid with full axes 4, 6 and 8 cm has semiaxes 2, 3 and 4 cm. Its volume is 32π cm³. The input method explicitly distinguishes full axes from semiaxes. Likewise, a right rectangular pyramid means its apex is directly above the base center. A base rectangle alone does not establish that location. Read the declared model and the accessible measurement table together; the schematic is not to scale and cannot prove that your physical object fits the model.
Choose the next measurement
When the drawing does not determine an exterior surface, keep the useful volume task separate. An offset pyramid with known base area and perpendicular height still has volume Gh/3. Its total surface needs additional face information, which this bounded workbench does not infer. Use the general-base method and carry the surface-not-determined explanation with your volume report. If your drawing instead explicitly states a centered apex, select the centered method and enter the corresponding independent dimensions.