Paint what can be touched
Combined-solid surface area is not the sum of component TSAs. Imagine painting or touching the finished object: joined faces disappear, openings remain absent and cavity walls become exposed.
Learning goals
- Identify exposed, joined, open and hollowed surfaces.
- Construct surface ledgers for joined solids and cavities.
- Calculate material or painting cost from exposed area.
Concept lesson
Component surface formulae
Recall curved/lateral and base areas separately for cylinders, cones and hemispheres. Component-level pieces are safer than adding total surface formulae.
Radius and slant-height readiness
Convert diameter to radius and calculate cone slant height l from r and h before writing the surface ledger.
Visible-surface classification
Mark every component face E for exposed or H for hidden. A common joining circle is hidden from the outside and contributes neither component base area.
Open, closed and grounded objects
A tent usually excludes its floor, an open vessel excludes its mouth, and a closed object includes its exposed end faces. Follow the material or painting wording rather than a memorised template.
Cone mounted on a hemisphere
For equal joining radii, the common circular face is hidden. Exposed area equals cone CSA plus hemisphere CSA; first derive the cone height or slant height if required.
Cylinder with hemispherical cavity
The cavity removes the circular end region but exposes the hemisphere's curved inner surface. Add cylinder CSA, remaining exposed base if any, and cavity CSA according to the diagram.
Capsule boundary
A capsule exposes cylinder CSA plus two hemisphere CSAs, equivalent to cylinder CSA plus one sphere surface area. The stated total length includes both rounded ends, so cylinder length is total length minus the diameter.
Exposed-face ledger
List each curved and plane face, mark exposed/hidden/open, then translate only exposed rows into formula terms before substituting numbers.
Cavity sign versus surface exposure
Explain why carving subtracts volume but normally adds the cavity's inner curved area to exposed surface; then complete the correct surface ledger.
Area-to-cost conversion
After finding exposed area, align its unit with the quoted rate, calculate cost, and state whether the rate is per square unit or per block of square units.
Joined-and-hollowed surface chain
For a cylinder with an attached cone and carved hemisphere, infer component dimensions, build a face-status ledger, calculate exposed area and convert it to painting or material cost.
Surface-ledger answer protocol
Show component dimensions, exposed-surface expression, substitution and units. The expression earns evidence even if later arithmetic slips.
Combined-surface mastery
Solve joined, open and hollowed surface problems using face-status ledgers, then complete one cost conversion.