GCSE Maths Geometry and Measure

Faces, Edges and Vertices

# Faces, Edges and Vertices

Here we will learn about faces, edges and vertices including how to calculate the number of vertices, edges and faces of a 3D shape, and how to classify polyhedrons given the number of faces, edges and vertices.

There are also faces, edges and vertices worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are faces, edges and vertices?

Faces, edges and vertices are features of a 3D shape.

• Faces are flat surfaces on the shape.
• Edges are straight lines that define the sides of the polygons that make up each face of the shape.
• Vertices (or corners) are points where at least three edges meet.

To calculate the number of faces, edges and vertices of a 3D shape, we need to count the number of each using the 3D object.

Note, you need to be able to visualise the 3D object, you may not be given the shape to help you.

For example, a cube has 6 vertices, 12 edges and 6 faces.

There are several 3D shapes that we need to know the number of vertices, edges and faces of.

Below is a diagram of common 3D shapes (split into polyhedra and non-polyhedra) along with the number of vertices, edges and faces.

E.g. Polyhedrons – flat faces, straight edges and sharp vertices.

E.g. Non-polyhedradon’t have flat faces, straight edges and sharp vertices.

## Platonic solids

Some of the most famous polyhedra are called the Platonic solids named after the Greek philosopher and Mathematician, Plato. Each of the Platonic solids can be inscribed inside a sphere as they are considered to be regular 3D polyhedra.

The number of faces determines the name of the polyhedron:

## How to calculate the number of faces, edges and vertices

In order to calculate the number of faces, edges and vertices of a 3D shape:

1. Inspect the shape to visualise its faces / edges / vertices.
2. Count the number of faces / edges / vertices.

In order to count the number of faces, edges and vertices of a 3D shape:

1. Recall that a face is a flat surface and count the faces systematically going around the 3D shape. (TIPS: For prisms and pyramids imagine/sketch the net of the shape . For some of the platonic solids the number of faces is linked to the name of the shape.)
2. Recall that an edge is a straight line where two faces meet and count the edges systematically going around and between the faces. (TIPS: Be careful not to count the same edge twice.)
3. Recall that vertices are the corners of a 3D shape where 3 or more edges meet and count them systematically going around the shape.

*If there is an image of the 3D shape, note that some faces, edges and vertices may be hidden in the 2D representation.

## Faces, edges and vertices examples

### Example 1: cuboid

Below is a cuboid.

Calculate the number of vertices for the polyhedron.

1. Inspect the shape to visualise its faces / edges / vertices.

A vertex is a corner where at least 2 edges meet.

2Count the number of faces / edges / vertices.

A cuboid has 8 vertices.

### Example 2: triangular prism

A triangular prism has 5 faces and 6 vertices.

Calculate the number of edges of the polyhedron.

1. Inspect the shape to visualise its faces / edges / vertices.

An edge is a straight line that connects two vertices.

2Count the number of faces / edges / vertices.

The triangular prism has 9 edges.

### Example 3: dodecahedron

A dodecahedron has 12 faces and 30 edges.

Calculate the number of vertices for the polyhedron.

1. Inspect the shape to visualise its faces / edges / vertices.

A vertex is a corner where at least 2 edges meet.

2Count the number of faces / edges / vertices.

A dodecahedron has 20 vertices.

### Example 4: cone

Below is a cone.

Calculate the number of faces for the cone.

1. Inspect the shape to visualise its faces / edges / vertices.

A face is surrounded by vertices and edges.

2Count the number of faces / edges / vertices.

A cone has 2 faces.

### Example 5: möbius strip

A Möbius strip has 1 face and 0 vertices.

How many edges does it have?

1. Inspect the shape to visualise its faces / edges / vertices.

An edge surrounds the face of a polyhedron.

2Count the number of faces / edges / vertices.

Look at this shape very carefully. Trace the edge, starting at 1 .

The Möbius strip has one edge.

### Example 6: cylinder

A cylinder has 3 faces and 0 vertices.

Calculate the number of edges for the polyhedron.

1. Inspect the shape to visualise its faces / edges / vertices.

An edge surrounds the face of a polyhedron.

2Count the number of faces / edges / vertices.

Be careful here. The vertical sides of the cylinder are not edges, they just show the curved face of the cylinder.

A cylinder has 2 edges.

### Common misconceptions

• Incorrect values for a property

A possible error is that the incorrect number of faces, edges or vertices is calculated.
E.g.
The number of vertices for a cube is calculated as 8 whereas a cube has 6 vertices.

• Incorrect property

A possible error is that the faces, edges and vertices are mixed up and the wrong one is calculated.

E.g.

The number of edges is counted on a shape instead of the number of faces.

### Practice faces, edges and vertices questions

1. A cube has 6 faces and 8 vertices.

Calculate the number of edges for the polyhedron.

6

12

8

14

Counting the number of edges, we have

4 vertical edges + 8 horizontal edges = 12 edges

2. An icosahedron has 12 vertices and 30 edges.

Use the fact that V-E+F=2 to calculate the number of faces for the polyhedron where V is the number of vertices, E is the number of edges and F is the number of faces of the polyhedron.

20

12

30

28
V-E+F=2\\ 12-30+F=2\\ F=20

3. An octagonal pyramid has 9 faces and 9 vertices.

Calculate the number of edges for the polyhedron.

9

18

20

16

The base has 8 edges as it is an octagon. Each vertex is joined to the apex and so 8+8=16 edges.

4. An L-shaped prism has 12 vertices and 18 edges.

Calculate the number of faces for the prism.

18

12

6

8

Count each face separately, remember the front and the back faces.

5. A hexagonal pyramid has 12 edges and 7 vertices.

Calculate the number of faces of the polyhedron.

6

7

14

1

Count each face separately, remember the base of the pyramid

6. How many vertices does a sphere have?

1

0

2

4

A sphere does not have any vertices.

### Faces, edges, and vertices GCSE questions

1. The trapezoidal prism below has a constant cross section. The front face has 4 sides, and all the adjacent faces are rectangles. Calculate the difference between the number of vertices and the number of faces for the prism.

(3 marks)

Show answer

Number of vertices = 8

(1)

Number of faces = 6

(1)

8-6=2

(1)

2. (a) The frustum below is made from slicing a cone horizontally, removing the top part of the cone.

Determine the number of faces for the frustum.
(b) The formula for the volume of a cone is equal to V=\frac{1}{3}\pi{r}^{2}h . If the radius of the base of the frustum is equal to 3cm and the height of the original cone is 5cm , calculate the volume of the cone. Write your answer to 1 decimal place. State the units.

(4 marks)

Show answer

(a)
3

(1)

(b)
V=\frac{1}{3}\pi\times {3}^{2}\times 5

(1)

V=47.1

(1)

cm^3

(1)

3. (a) Two cuboids are placed together to form an L-shaped prism. The front and back faces are flat, congruent, 2D shapes. Calculate the number of vertices, edges and faces of the new prism.
(b) Calculate the value of V-E+F if:

• V represents the number of vertices
• E represents the number of edges
• F represents the number of faces.

(4 marks)

Show answer

(a)
Vertices = 12

(1)

Edges = 18

(1)

Faces = 8

(1)

(b)
12-18+8=2

(1)

## Learning checklist

You have now learned how to:

• Identify and describe the properties of 3-D shapes, including the number of edges, vertices and faces

## Interesting 3D shapes

There are a range of 3D shapes that are mathematically interesting.

Below is a diagram of a Möbius strip:

The Möbius strip is a continuous loop with one edge and one face. Imagine tracing a line around the loop. The start point would also be the end point, yet the line appears to exist on “both sides” of the Möbius strip.

Another interesting 3D shape is the standard football. The 3D shape is based on a truncated icosahedron with 12 pentagons and 20 hexagons. The shape has 90 edges and 60 vertices. Below is an example of the net of a truncated icosahedron.

## Euler’s polyhedra formula

For a polyhedron to be classified as a cube, a dodecahedron or a tetrahedron for example, we can use Euler’s polyhedra formula to calculate the number of vertices, edges or faces of the shape:

V-E+F=2

where F is the number of faces, V the number of vertices, and E the number of edges of a polyhedron. This means that if you know two of the three characteristics of the polyhedron, you can calculate the missing value.

E.g. We know that a cube has 4 vertices and 4 faces:

Substituting these values into the formula, we get:

$4-E+4=2\\ 8-E=2\\ E=6$

So a cube has 6 edges.

This is only for polyhedrons.

The Euler characteristic

So far, we have looked at V-E+F=2 for any convex polyhedron including the Platonic solids. However, this is not true for all polyhedra. The Euler characteristic is the value of the expression V-E+F , so for a cube, the Euler characteristic is 2 . We represent the Euler characteristic using the symbol, so we can say for any polyhedra:

V-E+F=\chi

For example, a sphere has the following features: V=0, E=0, F=1 .

The Euler characteristic for a sphere is:

$\chi=V-E+F\\ =0-0+1\\ =1$

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#### FREE GCSE maths practice papers (Edexcel, AQA & OCR)

8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.

Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.