Three-dimensional (3-D) shapes are solids that take up space. Every 3-D shape is described by three geometric properties: its faces (flat or curved surfaces), its edges (lines where two faces meet) and its vertices (corner points where edges meet).
Prisms and pyramids
A prism has two identical parallel bases joined by rectangles. A pyramid has one base and triangular faces that meet at an apex. If the base is an n-sided polygon, a prism has n + 2 faces, 3n edges and 2n vertices, while a pyramid has n + 1 faces, 2n edges and n + 1 vertices.
Worked example
A cuboid (rectangular prism) has a 4-sided base, so n = 4. Faces = n + 2 = 6, edges = 3n = 12, vertices = 2n = 8.
Curved solids and Euler's rule
A cylinder has 2 flat circular faces and 1 curved surface (3 surfaces, 2 edges, 0 vertices). A cone has 1 flat circular face and 1 curved surface with 1 apex (1 vertex, 1 edge). A sphere has just 1 curved surface, no edges and no vertices.
Key idea
For any polyhedron (a solid with only flat faces), Euler's formula holds: V − E + F = 2, where V = vertices, E = edges, F = faces. Check the cuboid: 8 − 12 + 6 = 2. ✓
Remember
- Faces are surfaces, edges are lines, vertices are corners.
- A cone has 1 vertex (apex); a cylinder and a sphere have none.
- Euler's rule V − E + F = 2 works only for flat-faced solids, not for cones, cylinders or spheres.