Solids made of solids
A composite solid combines simple solids: a house shape (prism + triangular prism), a silo (cylinder + hemisphere), an ice cream cone (cone + hemisphere), a pencil (cylinder + cone).
Volume: add (or subtract) the parts
Decompose the solid, find each volume, and add.
Example: A grain silo is a cylinder (r = 3 m, h = 10 m) topped with a hemisphere (r = 3 m).
- Cylinder: π(9)(10) = 90π
- Hemisphere: ½ × (4/3)π(27) = 18π
- Total: 108π ≈ 339 m³
Subtracting: A concrete block 40 × 20 × 20 cm with a cylindrical hole (r = 5 cm) through its 20 cm height.
- Block: 16,000 cm³
- Hole: π(25)(20) = 500π ≈ 1,571 cm³
- Concrete: ≈ 14,429 cm³
Surface area: watch the hidden faces
When solids join, the surfaces where they touch are hidden faces. They’re inside, so they’re not part of the outside surface area.
Example: A cube (side 4) sits centred on top of a larger cube (side 6).
- Big cube: 6 × 36 = 216
- Small cube: 6 × 16 = 96
- Hidden: the small cube’s bottom (16) and the matching patch of the big cube’s top (16)
- Surface area: 216 + 96 − 32 = 280 units²
(Alternatively: the big cube’s top loses 16 but the small cube’s top replaces it, so you can also think of it as 216 + 4 × 16 = 280.)
Silo surface area
Cylinder side + hemisphere top + bottom circle (no circle between them):
- Side: 2π(3)(10) = 60π
- Hemisphere: ½ × 4π(9) = 18π
- Bottom: 9π
- Total: 87π ≈ 273 m²
Strategy
- Sketch and label each part.
- For volume: add parts; subtract holes.
- For surface area: list every outside surface; leave out hidden ones; add inside surfaces of holes.
- Keep units consistent.
Worked examples
Example 1: Ice cream cone: cone (r = 3 cm, h = 10 cm) + hemisphere (r = 3 cm). Volume?
30π + 18π = 48π ≈ 151 cm³.
Example 2: A shed: rectangular prism 4 × 3 × 2.5 m with a triangular prism roof (triangle base 3 m, height 1 m, length 4 m). Volume?
30 + ½(3)(1)(4) = 30 + 6 = 36 m³.
Example 3: Two cubes of side 2 glued face to face. Surface area?
2 × 24 − 2 × 4 = 40 units².
Common mistakes
- Counting hidden faces in surface area.
- Forgetting to halve for hemispheres.
- Forgetting the inside surface of a hole in surface area problems.
Try it yourself
- A cylinder (r = 2, h = 6) with a cone (r = 2, h = 3) on top. Volume (exact)?
- A 10 × 10 × 10 cube with a 2 × 2 × 10 square hole straight through. Volume?
- Three cubes of side 1 in a row. Surface area?
Answers: 1) 24π + 4π = 28π 2) 960 units³ 3) 14 units²
Quick recap
- Decompose a composite solid into simple solids.
- Volume: add parts, subtract holes.
- Surface area: leave out hidden faces.