1. Core Formulas
- Cube: s³
- Rect. Prism: l·w·h
- Sphere: (4/3)πr³
- Cylinder: πr²h
- Cone / Pyramid: (1/3)·(Base Area)·h
- Frustum (cone/pyramid): (h/3)(A₁ + A₂ + √(A₁A₂))
- Ellipsoid: (4/3)πabc
2. Decision Flow
- Identify shape & required minimal dimensions.
- Need slant? Only for surface area—ignore here.
- Composite? Sum parts – subtract voids.
- Irregular? Approximate via slicing/integration.
3. Sanity Checks
- All inputs positive; zero → zero volume.
- Cone / pyramid should be exactly 1/3 of prism with same base & height.
- Scaling dimensions by k → volume scales k³.
4. Shortcuts
- Reuse πr² when cylinder & cone share base.
- For tank fill %: actual volume = % × max volume.
- Frustum of cone: can also use difference of two full cones.
5. Pitfalls
- Using diameter where radius required.
- Forgetting 1/3 factor (pyramid / cone).
- Mixing unit systems (ft + in).
- Rounding too early—keep extra decimals until final.
6. Micro Examples
Cylinder r=4 h=6 → V = π·16·6 = 96π ≈ 301.5929
Cone r=3 h=8 → (1/3)π·9·8 = 24π ≈ 75.398
7. Mini FAQ
- Negative input? Not valid for length.
- Hollow object? Outer − inner volume.
- Units conversion? Convert dimensions first; then compute once.
8. Action Tip
When comparing capacity options, compute volume/footprint ratio to spot shapes giving more storage per floor area.