Volume follows material, not visible surface
Joined material volumes add; drilled, scooped or hollowed volumes subtract. Shared faces do not alter volume because they have zero thickness.
Learning goals
- Construct additive and subtractive volume ledgers.
- Infer component dimensions from total dimensions.
- Solve inverse combined-volume problems.
Concept lesson
Standard volume formulae
Recall cube, cuboid, cylinder, cone, sphere and hemisphere volumes, including the one-third cone factor and one-half sphere relation.
Total and component dimensions
Write a dimension equation before substitution: total height equals the sum of axial component heights; a hemisphere contributes one radius along the axis and two end hemispheres contribute one diameter.
Additive volume ledger
For a solid assembled from non-overlapping pieces, list each component with a plus sign and its own dimensions; joining faces are irrelevant to occupied volume.
Subtractive volume ledger
Start with the original solid volume, then subtract every non-overlapping cavity. A carved cone or hemisphere has negative sign even though its inner surface may be added in a surface-area problem.
Cone on a cylinder
When total height is 30 cm and cone height is 9 cm, cylinder height is 21 cm. Total volume is cylinder volume plus cone volume using their separate heights.
Cylinder with hemispherical ends
Two hemispheres form one sphere. If total axial length L includes both ends of radius r, cylinder length is L-2r; volume equals cylinder plus sphere.
Recover radius from volume
After isolating r squared in a cylinder relation take a positive square root; after isolating r cubed in a sphere relation take a positive cube root, then verify the supplied volume.
Dimension equation before volume
Label the object axis, write total length as a sum of component axial lengths, solve all missing heights, and only then build the volume ledger.
Material sign test
For every component ask whether material was attached or removed; assign plus or minus, check that cavities do not overlap, then calculate the signed total.
Inverse dimension with physical validation
Rearrange the combined-volume equation, isolate the squared or cubed dimension, take the positive root and test that it fits total dimensions and the diagram.
Attached-and-carved volume chain
Model a cylinder with one attached cone and one carved hemisphere: derive component heights, assign volume signs, simplify the expression and solve either total volume or an unknown dimension.
Volume-ledger answer protocol
Show the component-height equation, signed volume expression, substitution, cubic units and a reasonableness check for inverse results.
Combined-volume mastery
Solve an additive composite, a carved solid and an inverse-dimension problem with explicit dimension and signed-volume ledgers.