Card 7 of 8
3D shapes
Faces, edges and vertices, Euler's formula, volume and surface area.
| Solid | Faces | Vertices | Edges | F + V − E |
|---|---|---|---|---|
| Cube | 6 | 8 | 12 | 2 |
| Cuboid | 6 | 8 | 12 | 2 |
| Triangular prism | 5 | 6 | 9 | 2 |
| Square-based pyramid | 5 | 5 | 8 | 2 |
| Triangular pyramid (tetrahedron) | 4 | 4 | 6 | 2 |
| Pentagonal prism | 7 | 10 | 15 | 2 |
| Hexagonal prism | 8 | 12 | 18 | 2 |
| Octahedron | 8 | 6 | 12 | 2 |
Euler's rule: for any solid with flat faces, Faces + Vertices − Edges = 2. Always. Use it to check your counting, or to find a missing number in the exam.
Curved solids don't obey it (they're the exception examiners like):
| Solid | Faces | Vertices | Edges |
|---|---|---|---|
| Cylinder | 3 (2 flat, 1 curved) | 0 | 2 circular edges |
| Cone | 2 (1 flat, 1 curved) | 1 apex | 1 circular edge |
| Sphere | 1 curved | 0 | 0 |
Vocabulary that must be exact:
- Face = a flat (or curved) surface
- Edge = where two faces meet (a line)
- Vertex = where edges meet (a corner). Plural: vertices
Prism vs pyramid: a prism has the same cross-section all the way through (two identical ends); a pyramid rises from one base to a single point (apex).
Volume and surface area
| Solid | Volume | Surface area |
|---|---|---|
| Cube | side³ | 6 × side² |
| Cuboid | l × w × h | 2(lw + lh + wh) |
| Prism | area of cross-section × length | add all the faces |
| Cylinder | π r² h | 2π r² + 2π r h |
Working backwards is the exam's favourite: "a cube has surface area 294 cm² — what's its volume?" → one face = 294 ÷ 6 = 49 → side = √49 = 7 → volume = 7³ = 343 cm³.