Less common M5.11

Faces Edges and Vertices

The fundamental vocabulary for describing 3D shapes lets you accurately identify and count the component parts of common solids, a key skill for spatial reasoning problems.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Polyhedra (shapes with flat polygonal sides like cubes, prisms, pyramids) have faces, edges, and vertices.
  • Shapes with curved surfaces (cylinders, cones, spheres) do not have faces or edges in the strict sense. They have flat or curved surfaces.
  • A prism always has two identical end-faces (the bases) and rectangular side-faces connecting them. The number of side-faces equals the number of sides of the base.
  • A pyramid has one base face and triangular side-faces that all meet at a single point (the apex).
  • Euler's Formula, V - E + F = 2, relates the number of Vertices, Edges, and Faces for any simple polyhedron. It's a powerful tool for checking your counts.
  • Be precise with terminology: a cylinder has 2 flat surfaces and 1 curved surface, not 3 'faces'.

Formulae

V - E + F = 2

To verify your counts of vertices (V), edges (E), and faces (F) for any simple polyhedron (e.g., cubes, prisms, pyramids). This formula does not apply to shapes with curves.

Definitions

Face
A flat, two-dimensional surface that forms part of the boundary of a solid object. For polyhedra, faces are polygons.
Edge
A straight line segment where two faces of a polyhedron meet.
Vertex (plural: Vertices)
A point where two or more edges meet; a corner. The pointed tip of a cone is also considered a vertex.
Surface
The outer boundary of a 3D shape. A surface can be flat (like a face) or curved (like the side of a cone).

Worked example

A pentagonal prism and a hexagonal pyramid are placed on a table. What is the total number of edges on both shapes combined?

  1. 1

    First, analyze the pentagonal prism.

    A pentagon has 5 sides.

  2. 2

    The prism has two pentagonal bases, each with 5 edges.

    That's 5 + 5 = 10 edges.

  3. 3

    There are 5 rectangular faces connecting the vertices of the two bases, adding 5 more edges.

    Total edges for the prism = 10 + 5 = 15.

  4. 4

    Alternatively for the prism:

    Edges = (Edges on base) x 2 + (Vertices on base) = 5 x 2 + 5 = 15
  5. 5

    Next, analyze the hexagonal pyramid.

    A hexagon has 6 sides.

  6. 6

    The hexagonal base has 6 edges.

  7. 7

    There are 6 edges connecting the 6 vertices of the base to the single apex.

    Total edges for the pyramid = 6 + 6 = 12.

  8. 8

    Finally, sum the edges from both shapes:

    15 (prism) + 12 (pyramid) = 27.

  9. 9

    Check with Euler's formula.

    Prism:

    V=10, F=7 → 10-15+7=2

    Correct.

    Pyramid:

    V=7, F=7 → 7-12+7=2

    Correct.

Answer: 27

Common mistakes

  • ×Incorrectly applying the term 'face' to curved surfaces. A cylinder has zero faces, but two flat surfaces and one curved surface.
  • ×Miscounting by forgetting to include the base(s). For example, only counting the triangular faces of a pyramid and forgetting the base.
  • ×Double-counting edges, especially on more complex shapes. A systematic approach (e.g., count top, count bottom, count connecting edges) prevents this.

No-calculator tips

  • ✓Draw a quick, simple sketch of the shape to help visualize and count its components.
  • ✓For prisms: Faces = (sides of base) + 2; Vertices = (sides of base) x 2; Edges = (sides of base) x 3.
  • ✓For pyramids: Faces = (sides of base) + 1; Vertices = (sides of base) + 1; Edges = (sides of base) x 2.
  • ✓Use Euler's Formula (V - E + F = 2) to quickly double-check your counts for any polyhedron before moving on.

Read this topic in the official UAT-UK ESAT guide →

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