Every 3D shape has three key properties: faces (the surfaces), edges (the lines where two faces meet), and vertices (the corner points where edges join). Understanding these helps you describe and compare shapes, and unlocks Euler's remarkable formula, which connects all three properties in one elegant equation.
What do vertices, edges, and faces mean?
Before you can count them, you need a precise definition for each term.
Face: A face is a flat surface on a 3D shape. A cube has six flat, square faces. Shapes like cylinders and cones also have curved surfaces — exam questions sometimes ask you to count flat faces and curved surfaces separately, so always read the question carefully.
Edge: An edge is the line segment where two faces meet. On a cube, for example, where two square faces join along one side, that join is an edge. A cube has 12 edges. For curved shapes, a curved edge (such as the circular rim at the base of a cone) forms where a flat face meets a curved surface.
Vertex (plural: vertices): A vertex is a corner point where three or more edges meet. The word vertex is used in both 2D and 3D geometry, and the plural is always vertices, never "vertexes". A cube has 8 vertices — one at each of its corners.
A quick memory prompt: Faces are the flat front panels, Edges are where panels end and join, Vertices are the very pointy corners.
What is Euler's formula for polyhedra?
The Swiss mathematician Leonhard Euler discovered that for any convex polyhedron — a 3D shape with entirely flat, polygonal faces and no holes through it — the number of faces (F), vertices (V), and edges (E) always satisfies:
F + V − E = 2
This is one of the most celebrated results in all of geometry. Let us verify it for four familiar shapes.
- Cube: F = 6, V = 8, E = 12 → 6 + 8 − 12 = 2 ✓
- Triangular prism: F = 5, V = 6, E = 9 → 5 + 6 − 9 = 2 ✓
- Square-based pyramid: F = 5, V = 5, E = 8 → 5 + 5 − 8 = 2 ✓
- Tetrahedron (triangular-based pyramid): F = 4, V = 4, E = 6 → 4 + 4 − 6 = 2 ✓
Euler's formula applies to polyhedra only — shapes whose faces are all flat polygons. It does not apply to cylinders, cones, or spheres because these have curved surfaces.
What are the properties of common 3D shapes?
Use this reference table for revision. Memorise the cube and triangular prism in particular — they appear most frequently in KS3 tests.
| Shape | Faces | Vertices | Edges |
|---|---|---|---|
| Cube | 6 | 8 | 12 |
| Cuboid | 6 | 8 | 12 |
| Triangular prism | 5 | 6 | 9 |
| Square-based pyramid | 5 | 5 | 8 |
| Tetrahedron (triangular-based pyramid) | 4 | 4 | 6 |
| Pentagonal prism | 7 | 10 | 15 |
| Hexagonal prism | 8 | 12 | 18 |
| Cylinder | 2 flat + 1 curved | 0 | 2 |
| Cone | 1 flat + 1 curved | 1 | 1 |
| Sphere | 0 flat + 1 curved | 0 | 0 |
Notice that a cube and a cuboid share the same F, V, E counts — they differ in the lengths of their edges, but not in the way their faces connect. For any n-sided prism there is a useful general rule: F = n + 2, V = 2n, E = 3n. You can verify this for the triangular prism (n = 3: F = 5, V = 6, E = 9) and the pentagonal prism (n = 5: F = 7, V = 10, E = 15).
How do curved surfaces affect the count?
Cylinders, cones, and spheres are not polyhedra because they have curved surfaces, and Euler's formula does not apply to them in the usual sense. Nevertheless, KS3 exams do expect you to count their faces, vertices, and edges.
Cylinder: It has two flat circular faces (the top and bottom discs) and one curved surface (the tube). Where each flat face meets the curved surface, there is a circular edge — so a cylinder has 2 edges. No edges meet at a sharp corner, so there are 0 vertices.
Cone: It has one flat circular face (the base) and one curved surface. There is one circular edge at the base, so 1 edge. The apex (the pointed tip) is the only place where the curved surface comes to a point — that is 1 vertex.
Sphere: The entire surface is curved. There are 0 flat faces, 0 edges, and 0 vertices.
When an exam asks "how many faces does a cylinder have?" the expected KS3 answer is 3 (2 flat + 1 curved). Be guided by exactly what the question asks.
How do I use Euler's formula to find a missing value?
Rearrange F + V − E = 2 to isolate whichever quantity you need:
- To find the number of edges: E = F + V − 2
- To find the number of faces: F = E − V + 2
- To find the number of vertices: V = E − F + 2
Worked example 1: A polyhedron has 7 faces and 10 vertices. How many edges does it have?
E = F + V − 2 E = 7 + 10 − 2 E = 15
Cross-check: this matches the pentagonal prism in the reference table above. ✓
Worked example 2: A polyhedron has 9 edges and 5 faces. How many vertices does it have?
V = E − F + 2 V = 9 − 5 + 2 V = 6
Cross-check: this matches the triangular prism (F = 5, V = 6, E = 9). ✓
What are the most common mistakes with 3D shape properties?
Keeping these pitfalls in mind will save marks in tests.
Counting curved surfaces as zero faces. A cylinder has three surfaces in total (two flat faces and one curved surface). Saying "a cylinder has 0 faces" is a very common error. In most KS3 questions, each surface counts — curved or not.
Applying Euler's formula to curved shapes. Euler's formula applies to polyhedra only. It does not apply to cylinders, cones, or spheres because those shapes include curved surfaces. Always check that a shape is a polyhedron (all flat faces) before applying F + V − E = 2.
Confusing vertices with edges. Edges are lines; vertices are points. On a cube, 12 edges connect 8 vertices — keep these two properties clearly separate in your revision notes.
Frequently Asked Questions
What is the difference between a face and a surface?
In KS3 maths, "face" usually refers specifically to a flat, polygonal surface on a 3D shape, while "surface" is a broader term that includes curved sides as well. A cylinder has two flat faces and one curved surface — three surfaces in total. Some textbooks and exam boards use the terms interchangeably, so always follow the precise wording of the question in front of you.
Why does Euler's formula always equal 2 for convex polyhedra?
The value 2 comes from the topological structure of a sphere: any convex polyhedron can be continuously stretched into a sphere without tearing or joining edges, and this transformation preserves the relationship F + V − E. Shapes with holes through them — like a torus (doughnut shape) — satisfy a different version of the formula. At KS3, the focus is on applying the formula correctly, not on proving it.
Do a cube and a cuboid have the same number of faces, vertices, and edges?
Yes — both have 6 faces, 8 vertices, and 12 edges. They differ in shape: a cube has all edges the same length, while a cuboid has three different edge lengths. Euler's formula measures how the faces connect, not their size or shape, so any rectangular box (no matter its proportions) always gives F + V − E = 6 + 8 − 12 = 2.
How do I remember which property is which?
Think of building a cardboard box. The flat panels are the faces. The folds where two panels crease together are the edges. The little corner points where multiple folds meet are the vertices. For a cube (like a standard dice): 6 faces (the six numbered sides), 12 edges (the wire-frame struts of the box), and 8 vertices (the eight corners of the box). These three numbers are worth memorising cold.
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