Nets, Faces, Edges and Vertices

What you will learn

• Count faces, edges and vertices of common solids.
• Recognise nets of a cube, cuboid, cylinder and pyramid.
• Use $F + V - E = 2$ to check your counts.

Before you start: you should know the names of common 3D shapes. See Recognising 3D Shapes.

The idea

A solid shape has three kinds of parts.

• A face is a flat or curved surface.
• An edge is where two faces meet.
• A vertex (plural vertices) is a corner where edges meet.

Nets

A net is a flat pattern that folds up into a solid. Imagine cutting along some edges of a box and opening it out flat. The faces stay joined along the edges you did not cut.

Many different nets can fold into the same solid. A cube has $11$ different nets. To test one, picture folding it: no two faces may land on top of each other, and nothing may be left over.

Euler’s formula

For solids with flat faces, faces plus vertices minus edges is always $2$. For a cube, $6 + 8 - 12 = 2$. If your counts do not give $2$, count again.

Worked examples

Example 1: Counting a pyramid

Count the faces, edges and vertices of a square-based pyramid.

1. Faces: $1$ square base and $4$ triangles, so $5$.
2. Edges: $4$ round the base and $4$ sloping up, so $8$.
3. Vertices: $4$ at the base and $1$ at the top, so $5$.
4. Check: $5 + 5 - 8 = 2$.

Answer: $5$ faces, $8$ edges, $5$ vertices

Example 2: Missing edges

A triangular prism has $5$ faces and $6$ vertices. How many edges?

1. $F + V - E = 2$.
2. $5 + 6 - E = 2$, so $E = 11 - 2 = 9$.

Answer: $9$ edges

Example 3: Which net?

A net has $5$ squares in a row and $1$ more square attached at one end. Will it fold into a cube?

1. A cube needs the four side faces to wrap round. A row of $5$ would need two squares to overlap.
2. So the squares would land on top of each other.

Answer: no, it cannot fold into a cube.

Tip: a curved face is not counted the same way as a flat one. A cylinder has $3$ faces (2 flat, 1 curved), $2$ edges and no vertices.

Why it matters

Packaging designers draw nets before cutting cardboard. Every cereal box, pizza box and carton starts as a flat net. Nets also lead to surface area, the total area of all the faces, which is the next step in Surface Area of Cubes and Cuboids.