Scale Diagram

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In order to access this I need to be confident with:

Place value Ratio scale Enlargement Metric units

This topic is relevant for:

GCSE Maths Ratio and Proportion Scale

Scale Diagram

Here we will learn about a scale diagram, including similar shapes, using scale factors, and geometric problems.

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

What is a scale diagram?

A scale diagram is an accurate enlargement of an object that has scaled lengths of the original. Scale diagrams are mathematically similar to the original object.

Enlargements can make an object larger or smaller.

In order to draw a scale diagram we need to know the scale factor of enlargement and the actual measurements of the object.

E.g.

Below are two similar triangles, $\mathrm{A}$ and $\mathrm{B}.$

Triangle $\mathrm{A}$ has a width of $3cm$ , and a slant height of $5cm.$

Triangle $\mathrm{B}$ has a width of $6cm$ and a slant height of $x.$

To find the value of $x$ we need a scale factor (which we can express as a ratio).

Comparing two similar dimensions, the ratio of the width of $\mathrm{A}$ to the width of $\mathrm{B}$ is $3:6.$ Simplifying this ratio, we get $1:2.$ This means that each length in triangle $\mathrm{B}$ is twice as long as the length in triangle $\mathrm{A}.$

The value of $x$ is therefore equal to $5 \times 2=10cm.$

What is a scale diagram?

How to calculate a ratio from a scale diagram

In order to calculate a ratio from a scale diagram:

Identify two lines that are mathematically similar.
Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .
Simplify the ratio.

Explain how to calculate a ratio from a scale diagram

Scale drawing worksheet (includes scale diagram)

Get your free scale diagram worksheet of 20+ scale drawing questions and answers. Includes reasoning and applied questions.

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Scale drawing worksheet (includes scale diagram)

Get your free scale diagram worksheet of 20+ scale drawing questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Related lessons on scale

Scale diagram is part of our series of lessons to support revision on scale. You may find it helpful to start with the main scale lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Scale diagram examples

Example 1: a straight line

A straight line on a road is measured as $5.7m.$ The diagram shows a scaled version of the line is represented below.

State the ratio of the diagram to the actual distance of the line.

Identify two lines that are mathematically similar.

Here we have the two lines with the lengths $5.7m$ and $15cm.$

2Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .

The ratio is of the form $\text{diagram : actual}$ so we have the ratio

15cm:5.7m

3Simplify the ratio.

Converting the right hand side of the ratio to be in centimetres we have

15cm:570cm

Since the units are the same we can rewrite the ratio without them

15:570

Writing the ratio in the simplest form we have:

1:38

Example 2: 2D polygon

The scale diagram of a parallelogram is $12cm$ wide and has a vertical height of $10cm.$ The real life structure is $42cm$ wide. State the ratio of the scale diagram to the real life structure.

Identify two lines that are mathematically similar.

Here we have the two widths of the parallelograms with the lengths $12cm$ and $42cm.$

Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .

The ratio is of the form $\text{diagram : actual}$ so we have the ratio

$12cm:42cm$

Since the units are the same we can rewrite the ratio without them

$12:42$

Simplify the ratio.

Writing the ratio in the simplest form (divide throughout by $6$ ) we have:

$2:7$

Example 3: 3D object

The diagram of Cube $\text{A}$ has a volume of $64cm^3$ . Cube $\text{A}$ has an actual volume of $27cm^{3}.$ State the ratio of the side lengths of the model to the actual size.

Identify two lines that are mathematically similar.

As we know the volume of each cube and the formula for the volume of a cube is $V=L^3$ where $L$ is the side length of the cube, we can calculate the side length of each cube

$\begin{aligned} &L_{model}=\sqrt[3]{64}=4cm \\\\ &L_{actual}=\sqrt[3]{27}=3cm \end{aligned}$

Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .

The ratio is of the form $\text{model : actual}$ distance so we have the ratio

$4:3$

Simplify the ratio.

This ratio is already in the simplest form.

Example 4: map

The distance between two points on a map is $8.6cm.$ The actual distance between the two points is $2.15km.$ Calculate the map scale in the form $1:n$ where both values are in centimetres.

Identify two lines that are mathematically similar.

Here we have the two distances $8.6cm$ and $2.15km.$

Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .

The ratio is of the form $\text{diagram : actual}$ so we have the ratio

$8.6cm:2.15km$

Converting $2.15km$ to centimetres (multiply by $100 \ 000$ ) we get

$8.6cm:215 \ 000cm$

Simplify the ratio.

We want the ratio in the form $1:n$ so we divide both sides by $8.6.$

$1cm:25000 cm$

Since the units are the same we can rewrite the ratio without them.

The map has a scale of $1:25000.$

Example 5: floor plan

Below is the scale diagram of a floor plan of a bedroom. The length of the bed is $190cm.$ State the ratio of the floor plan to the actual distances in its simplest form.

Identify two lines that are mathematically similar.

We need to measure the length of the bed on the floor plan so we can compare this to the actual length.

The length of the bed on the floor plan is $8.5cm.$

So the two similar lengths are $190cm$ and $8.5cm.$

Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .

The ratio is of the form $\text{plan : actual}$ so we have the ratio

$8.5cm:190cm$

Simplify the ratio.

Writing the ratio in the simplest form we have:

$17cm:380cm$

Since the units are the same we can rewrite the ratio without them

$17:380$

Remember, the simplest form shouldn’t contain decimals.

Example 6: scale model

A scale model of the planets in the solar system is given below. The minimum distance from Earth to Mars is $54.6 \times 10^{6}km$ from the surface of each planet. State the ratio in the form $1cm:n \ km$ of the scale model to the real distance.

Write $n$ in standard form correct to $3$ significant figures.

Identify two lines that are mathematically similar.

Measuring the distance between Earth and Mars, we get

The distance from Earth to Mars in the diagram is $4.9cm.$

The distance from Earth to Mars in real life is $54.6 \times 10^{6}km$

Express the lengths as a ratio $\bf{\mathrm{A:B}}$ .

The ratio is of the form $\text{model : actual}$ so we have the ratio

$4.9cm:54.6 \times 10^{6}km$

Simplify the ratio.

Simplifying the ratio to the form $1:n$ , we divide both sides by $4.9$ to get

$1cm:11142857.14km$

$1cm:1.11 \times 10^{7}km$

(the equivalent to walking around the equator of the Earth $278$ times!)

Common misconceptions

The order of the ratio is incorrect

Stating the ratio in the incorrect order will mean that the scale diagram is much larger / smaller than what is expected.

E.g. Let a ratio represent the scale diagram to the real life distance. The ratio $1:2$ means that the real life distance is twice the length of the scale diagram.

The ratio $2:1$ means that the real life distance is half of the scale diagram.

Incorrect units in the solution

The units for the model can be mixed up with the units for the real life distance.

E.g.

If we were calculating the distance of 10cm on a map with the scale ratio of $1cm:5km, \ 10 \times 5=50cm$ is stated whereas the correct solution would be $10 \times 5=50km.$

Converting units

Sometimes the units in a question need to be converted so that they are all the same, so it is important to be confident in converting between different metric units.

E.g. The map scale is given as $1:25000$ which means that $1cm$ on the map is equivalent to $25000cm$ in real life.

If the answer is asked to be written in kilometres, the real life value in centimetres must be divided by $100 \ 000$ to get the same measurement in kilometres.

Practice scale diagram questions

1cm:2m

2:1

10cm:20cm

1:20

First we need to write down the ratio with the given information.

10cm:2m

Then we can convert so that they have the same units.

10cm:200cm

Finally, we can simplify the ratio.

10:200=1:20

1cm:2.05m

1:2.05

2:1

1m:20.5cm

\begin{aligned} &8cm:16.4cm \\\\ &1cm:2.05cm=1:2.05 \end{aligned}

50cm:6cm

5:6

6:500

6:5

A cube with a volume of $125m^3$ has a side length of $\sqrt[3]{125}=5m$

A cube with a cross sectional area of $36cm^2$ has a side length of $\sqrt{36}=6cm$

6cm:5m=6cm:500cm

1:25000

1:0.5

1:2

1:5000

\begin{aligned} &2.65km=265000 cm \\\\ &265000 \div 5.3=50000 \\\\ &1:50000 \end{aligned}

5:1

20:1

2:5

40cm:1m

\begin{aligned} &5.5 \times 2=11cm \quad \text{(desk length on diagram)} \\\\ &2.2m:11cm \\\\ &220cm:11cm \\\\ &220:11 \\\\ &20:1 \end{aligned}

5:1

1:300

2:500

1.5cm:25m

\begin{aligned} &25 \times 0.18=4.5m \\\\ &1.5cm:4.5m \\\\ &1.5cm:450cm \\\\ &1.5:450 \\\\ &1:300 \end{aligned}

Scale diagram GCSE questions

1. The length of a path is measured at $26.2$ metres. A scale drawing of the same path has a length of $14 \ cm.$ State the ratio of the diagram to the actual path in the form $1:n.$

(2 marks)

Show answer

14cm:26.2 m =14:2620

(1)

1:187

(1)

2. A camera screen has a height of $7cm$ and a width of $5cm.$

A photo frame is $19cm$ wide.

(a) What is the scale ratio of the camera screen to the photo frame?

Write your answer in the form $1:n.$

(b) Calculate the height of the photo frame.

(4 marks)

Show answer

(a)

5:19

(1)

1:3.8

(1)

(b)

7 \times 3.8

(1)

26.6cm

(1)

3. A box of chocolates has the following dimensions:

Scale Diagram GCSE Question 3

A scale model is created. The ratio of the real object to the model is $1:0.2.$ Calculate the volume of the model chocolate box. State the units in your answer.

(3 marks)

Show answer

(3 \times 0.2) \times (8 \times 0.2) \times (5 \times 0.2)

(1)

0.96

(1)

cm^3

(1)

Learning checklist

You have now learned how to:

Use scale factors, scale diagrams and maps

Solve problems involving similar shapes where the scale factor is known or can be found

The next lessons are

Still stuck?

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Introduction

What is a scale diagram?

How to calculate a ratio from a scale diagram

Scale diagram worksheets

Scale diagram examples

↓

Example 1: a straight line Example 2: 2D polygon Example 3: 3D object Example 4: map Example 5: floor plan Example 6: scale model

Common misconceptions

Practice scale diagram questions

Scale diagram GCSE questions

Learning checklist

Next lessons

Still stuck