Properties of 3D solids: faces, edges, vertices
3D shapes have faces (flat surfaces), edges (where two faces meet) and vertices (corners where edges meet). For each common solid, you should know the count.
Common solids
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square-based pyramid | 5 | 8 | 5 |
| Tetrahedron (triangular pyramid) | 4 | 6 | 4 |
| Cylinder | 3 (2 circles + curved) | 2 (circular edges) | 0 |
| Cone | 2 (circle + curved) | 1 (circular edge) | 1 (apex) |
| Sphere | 1 (curved) | 0 | 0 |
Euler's formula
For any convex polyhedron: V − E + F = 2 (vertices − edges + faces = 2).
Worked example: a cube has V = 8, E = 12, F = 6. Check: 8 − 12 + 6 = 2. ✓
Prisms vs pyramids
- Prism — two identical parallel faces (the "ends") connected by rectangles.
- Pyramid — a base and triangular sides meeting at a single apex.
A prism is named by the shape of its end face: triangular prism, hexagonal prism, etc.
Curved surfaces
- Cylinder has 2 flat circular faces and 1 curved face. The "edge" is debatable — typically counted as 2 (the circles). 0 vertices.
- Cone has 1 flat circular face and 1 curved face. 1 edge (the circle). 1 vertex (the apex).
- Sphere has 1 curved face, 0 edges, 0 vertices.
Cross-sections
A cross-section is the 2D shape revealed when a 3D solid is sliced. For a prism, every cross-section parallel to the ends is identical.
⚠Common mistakes
- Counting edges of curved solids inconsistently — for GCSE: cylinder = 2, cone = 1.
- Confusing prism with pyramid — prism has parallel ends; pyramid tapers to a point.
- Forgetting Euler's formula — useful as a check.
- Misnaming a tetrahedron — it's a pyramid with 4 triangular faces (often equilateral).
- Treating a sphere as having vertices — it has none.
➜Try this— Quick check
A pentagonal prism has V = 10, F = 7. Use Euler's formula to find E.
- 10 − E + 7 = 2 → E = 15.
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