3D Shape Properties
3D Shape Properties
Three-dimensional shapes are described by their faces (flat or curved surfaces), edges (lines where faces meet) and vertices (points where edges meet). Knowing the precise count and nature of these features — and the relationship between them — is the foundation for surface area and volume work.
Standard 3D Shapes
| Shape | Faces | Edges | Vertices | Cross-section |
|---|---|---|---|---|
| Cube | 6 (all squares) | 12 | 8 | Square |
| Cuboid | 6 (rectangles) | 12 | 8 | Rectangle |
| Triangular prism | 5 (2 triangles, 3 rectangles) | 9 | 6 | Triangle |
| Square-based pyramid | 5 (1 square, 4 triangles) | 8 | 5 | — |
| Tetrahedron | 4 (all triangles) | 6 | 4 | — |
| Cylinder | 3 (2 circles, 1 curved) | 2 (curved) | 0 | Circle |
| Cone | 2 (1 circle, 1 curved) | 1 (curved) | 1 (apex) | — |
| Sphere | 1 (curved) | 0 | 0 | — |
Euler's Formula
For any convex polyhedron (a solid with only flat faces and no holes):
\[ F + V - E = 2 \]where \( F \) = faces, \( V \) = vertices, \( E \) = edges. This can be used to verify counts or find a missing value.
Example — triangular prism: \( 5 + 6 - 9 = 2 \). ✓
Prisms and Pyramids
- A prism has two identical parallel faces (the cross-sections) joined by rectangles. It is named by the shape of its cross-section.
- A pyramid has a polygonal base and triangular faces meeting at an apex. It is named by the shape of its base.
- A right prism or right pyramid has its axis perpendicular to its base.
Key Takeaways
- Euler's formula: \( F+V-E=2 \) for all convex polyhedra.
- Prisms: uniform cross-section throughout their length; all non-base faces are rectangles.
- Pyramids: one polygonal base, triangular lateral faces meeting at an apex.
- Curved solids (cylinder, cone, sphere) are not polyhedra — Euler's formula does not apply.
- The cross-section of a prism is constant — slicing parallel to the base always gives the same shape and size.