Less common M5.11

Faces Edges and Vertices

This topic covers the fundamental vocabulary used to describe 3D shapes. Mastering this allows you to 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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