Three-dimensional geometry & measurement
The idea
Three-dimensional measurement extends area thinking by one dimension: prisms and cylinders obey volume = base area × height because they are a base stacked straight upward; pyramids and cones hold exactly one third of the matching prism or cylinder; and a sphere of radius r encloses (4/3)πr³. Surface area, by contrast, stays two-dimensional — it unrolls a solid into flat or curved pieces and totals them. Real objects are usually composites — a silo is a cylinder wearing a hemispherical cap — so the key skill is decomposing a solid into pieces with known formulas.
Keep volume and surface area in separate mental drawers: volume measures filling, carries cubic units, and scales by k³ under a length scale factor k, while surface area measures wrapping, carries square units, and scales by k². The frequent error with composites is double-counting hidden faces — where a hemisphere sits on a cylinder, neither the flat disk of the cap nor the cylinder's top belongs in the surface area, since both are interior. Volumes of pieces, though, simply add. Rough bounds catch most slips: a capped silo must hold more than its bare cylinder but less than a full-height cylinder.
Worked example
A grain silo consists of a cylinder of radius 4 m and height 10 m topped by a hemisphere of the same radius. Find the silo's total volume, exactly and to the nearest cubic meter.
- Decompose the solid into a cylinder and a hemisphere sharing radius 4 m. The cylinder's volume is πr²h = π × 4² × 10 = 160π m³.
- A full sphere of radius 4 holds (4/3)π × 4³ = 256π/3 m³, so the hemisphere holds half of that: 128π/3 m³.
- Add the pieces over a common denominator: 160π = 480π/3, so the total is 480π/3 + 128π/3 = 608π/3 m³.
- Convert to a decimal: 608π/3 ≈ 636.7, so the silo holds about 637 m³.
- Bound-check the result: a full 14 m cylinder would hold 224π ≈ 703.7 m³ and the bare 10 m cylinder holds 160π ≈ 502.7 m³; the answer lands between them, exactly where a dome-capped silo should.
Answer. The silo's volume is 608π/3 m³, approximately 637 m³.
Check your understanding
- Why does a cone hold exactly one third of the cylinder with the same base and height, and how could you demonstrate that with sand or water?
- Which surfaces vanish when two solids are joined into a composite, and how do you keep them out of a surface-area total?
- If every length of a solid is scaled by 3, what happens to its surface area and its volume, and why do the two answers differ?
- When a problem wants both an exact answer in π and a decimal, at which point should the rounding happen, and what goes wrong if you round early?
Build the foundations first
Three-dimensional geometry & measurement builds on these concepts. If any feel shaky, start there.