Solid Geometry
Core concepts
Know formulas for volume and surface area of prisms, pyramids, cylinders, cones, and spheres. Understand cross-sections, similarity in 3D, and the relationship between linear, area, and volume scaling. Be comfortable with coordinates in space and the distance formula in 3D.
Traps
- Slant height vs height: the cone's lateral surface uses slant height ℓ, not h.
- Volume ratio under scaling: if lengths scale by k, volumes scale by k³ and surface areas by k².
- Sphere formulas: V = (4/3)πr³; S = 4πr²; do not swap them.
- Pyramid/cone volume: one-third of the corresponding prism/cylinder.
- Units: mixing cm and m gives wrong answers.
Rule of thumb
For any solid with a uniform cross-section, V = (base area) × height; for pointed solids, divide by 3.
Worked example
A cone has radius 3 and height 4. The slant height is ℓ = √(3² + 4²) = 5. Volume: V = (1/3)πr²h = (1/3)π·9·4 = 12π. Lateral surface area: πrℓ = π·3·5 = 15π.