Volume and surface area of 3D solids
Prisms
A prism has the same cross-section all the way through.
Volume of a prism = (area of cross-section) × length.
| Prism type | Volume |
|---|---|
| Cuboid | l × w × h |
| Triangular prism | (½ × b × h) × length |
| Cylinder | πr² × h |
Surface area = 2 × cross-section area + lateral surface (perimeter of cross-section × length). For a cylinder, total SA = 2πr² + 2πrh.
Pyramids and cones (CCEA formula sheet provides these for Higher)
| Solid | Volume | Surface area |
|---|---|---|
| Pyramid (any base) | ⅓ × base area × height | base + sum of triangular faces |
| Cone (radius r, slant l, height h) | ⅓πr²h | πr² + πrl |
| Sphere | ⁴⁄₃ πr³ | 4πr² |
The slant length l for a cone (right circular): l = √(r² + h²).
Composite solids
Common: cylinder + hemisphere; cone + cylinder; cuboid with hole drilled.
- Add volumes when solids are joined.
- Subtract when a solid has a hole or a cavity.
Volume conversions
1 cm³ = 1 ml. 1000 cm³ = 1 L. 1 m³ = 1,000,000 cm³.
Surface area subtleties
When two solids are joined:
- The internal touching faces are no longer surfaces — subtract them.
- Example: cone on top of a cylinder of the same radius — exclude the top of the cylinder and the base of the cone.
Common CCEA exam tip
Always show the formula being used (with values substituted) before evaluating. The M1 mark is on the formula; A1 on the evaluated answer with correct units (cm³, m³ etc.).
AI-generated · claude-opus-4-7 · v3-ccea-maths-leaves