Scale math

Here you will learn about scale math, including scale diagrams, scale drawing, scale factors and real life applications.

Students will first learn about scale math as part of geometry in $7$th grade.

What is scale math?

Scale math involves enlarging or reducing objects. It refers to the process of resizing an object or drawing by a certain ratio, known as the scale factor, while maintaining its proportions.

If you have two similar geometric figures, one will be a scale diagram of the other.

You can calculate the scale factors for length, area and volume.

Let’s look at this example,

Length scale factor

The length scale factor can be calculated by comparing two lengths.

You can compare the length of $A$ (the smaller figure) with the length of $B$ (the larger figure).

This gives the ratio $3\text{:}6$ which simplifies to $1\text{:}2.$

The length scale factor from $A$ to $B$ is $\bf{2}.$

Area scale factor

The area scale factor can be calculated by comparing two areas.

You can compare the area of $A$ with the area of $B.$

Area $A=3 \mathrm{~cm} \times 3 \mathrm{~cm}=9 \mathrm{~cm}^2$

Area $B=6 \mathrm{~cm} \times 6 \mathrm{~cm}=36 \mathrm{~cm}^2$

This gives the ratio $9\text{:}36$ which simplifies to $1\text{:}4.$

The area scale factor from $A$ to $B$ is $\bf{4}.$

Volume scale factor

The volume scale factor can be calculated by comparing two volumes.

Volume $A=3 \mathrm{~cm} \times 3 \mathrm{~cm} \times 3 \mathrm{~cm}=27 \mathrm{~cm}^3$

Volume $B=6 \mathrm{~cm} \times 6 \mathrm{~cm} \times 6 \mathrm{~cm}=216 \mathrm{~cm}^3$

This gives the ratio $27\text{:}216$ which simplifies to $1\text{:}8.$

The volume scale factor from $A$ to $B$ is $\bf{8}.$

Alternatively you can calculate the area and volume scale factors by starting with the scale factor for length.

You can square the length scale factor to calculate the area scale factor.

You can cube the length scale factor to calculate the volume scale factor.

In the above example,

Scale factor for length $=\bf{2}$

Scale factor for area $=2^{2}=\bf{4}$

Scale factor for volume $=2^{3}=\bf{8}$

Scale math is used for real world scale drawings as it is much easier to create a scale drawing of an object, than to draw the object using actual distances. The ratio usually takes the form $1\text{:}n$ of the model/plan to the actual distance.

Step-by-step guide: Scale drawing

For example, below is a scale diagram of a floor plan of a house where $1$ square is equal to $2$ meters.

Writing this as a ratio you get $\bf{1}$ square$\textbf{:}\bf{2} \, \textbf{m}.$

From this diagram, you can calculate the width of the living room by counting squares:

As the living room is $4$ squares wide and you have the ratio $1$ square:$2m,$ multiplying the number of squares by $2$ you get the actual width of the living room.

$4 \times 2=8 \, m$, so the living room is $8 \, m$ wide.

What is scale math?

Common Core State Standards

How does this relate to $7$th grade math?

• Grade 7 – Geometry (7.G.A.1)
  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

How to use scale math

In order to calculate the actual / real life distance from a scale:

1. State the scale factor as a ratio in the form $\bf{1}\textbf{:}\textbf{n}.$
2. Multiply $\textbf{n}$ by the length given from the model.
3. Write the units.

In order to calculate the model length from a scale:

1. Divide the real life distance by the map scale ratio.
2. Write the units.

In order to calculate a ratio from a scale diagram:

1. Identify two lines that are mathematically similar.
2. Express the lengths as a ratio $\textbf{A}\textbf{:}\textbf{B}.$
3. Simplify the ratio.

Scale math examples

Example 1: map (find the actual distance)

A map has a scale of $1 \, cm\text{:}5 \, km.$ Find the actual distance represented by $3 \, cm$ on the map.

1. State the scale factor as a ratio in the form $\bf{1}\textbf{:}\textbf{n}.$

2Multiply $\textbf{n}$ by the length given from the model.

3Write the units.

Example 2: plan of a structure (find the actual distance)

The construction plans for a structure has the scale $2 \, cm\text{:}3 \, m.$ What is the actual distance of $62 \, cm$ on the plan?

Example 3: map of the UK (find the model length)

A map of the UK is drawn using the scale $30 \, cm\text{:}1,500 \, km.$ Calculate how far $480 \, km$ would be on the map.

Example 4: floor plan (find the model length)

A plan of a kitchen uses the scale $5 \, cm\text{:}20 \, cm.$ Calculate the distance on the plan for the actual distance of $8.7 \, m.$

Example 5: 2D polygon (calculate the ratio)

A kite has a vertical height of $28 \, cm$ and a width of $12 \, cm.$ The scale diagram of the same kite has a width of $8 \, cm.$ What is the ratio of the real kite to the scale diagram? Write your answer in its simplest form.

Example 6: scale model

The circumference of Jupiter is $439,624 \, km.$ A scale model of the planet has a diameter of $11.2 \, cm.$ Calculate the ratio of the scale diagram to the real planet Jupiter. Give your answer if the form $a \, cm\text{:} \, b \, km.$ Write $a$ and $b$ correct to $2$ significant figures.

Teaching tips for scale math

• Use real-world objects, like maps, blueprints, or model cars, to introduce the concept of scale. Discuss how scale represents a ratio between the measurements on the model and the actual size of the object.

• Engage students in activities where they create their own scaled drawings or models on a piece of paper. For example, have them draw a simple object at a different scale or create a scale model of a room using graph paper.

• Clearly explain the scale factor as the ratio of any two corresponding side lengths in the original figure and the scaled figure. Provide worksheets to practice calculating the scale factor with different examples.

• Discuss how scaling can involve both enlarging and reducing objects. Use examples to illustrate how the scale factor greater than $1$ indicates enlargement (scale up), while a scale factor less than $1$ indicates reduction (scale down).

Easy mistakes to make

• Multiplying the real life distance by the ratio scale
  Multiplying the actual real-life distance by the ratio scale to find the distance on a plan will lead to an incorrect result.

• Dividing the model / plan / map distance by the ratio scale
  A common misconception is to divide the distance on the model / plan / map by the ratio scale resulting in an incorrect real life distance.

• 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 expected.
  
  For example, a ratio represents the scale diagram to the real life distance. The ratio $1\text{:}2$ means that the real life distance is twice the length of the scale diagram. The ratio $2\text{:}1,$ on the other hand, means that the real life distance is half of the scale diagram.

• Ratio scale not simplified
  If given the ratio $2 \, cm\text{:}5 \, km,$ it is easier to calculate when the ratio is in the form $1\text{:}n$ and so you must find an equivalent ratio before using the scale. Here the ratio would be $1 \, cm\text{:}2.5 \, km$ so $1 \, cm$ on the map would be equal to $2.5 \, km$ in real life.

• Incorrect units in the solution
  A common misconception is to mix up the units for the model with the units for the real life distance.
  
  For example, if you were calculating the distance of $10 \, cm$ on a map with the scale ratio of $1 \, cm\text{:}5 \, km, \, 10 \times 5=50 \, cm$ could be incorrectly stated. Whereas the correct solution would be $10 \times 5=50 \, km.$

• Converting units
  Sometimes the units need to be converted, so it is important to be able to confidently convert between different metric units.
  
  For example, the map scale is given as $1\text{:}25,000$ which means that $1 \, cm$ on the map is equivalent to $25,000 \, cm$ in real life. If the answer needs to be written in kilometers, the real life value in centimeters must be divided by $100,000$ to get the same measurement in kilometers.
  
  $1\text{:}25,000=1 \, cm\text{:} 0.25 \, km.$

Related congruence and similarity lessons

• Congruence and similarity
• Congruent shapes
• Congruent triangles
• Similar shapes

The next lessons are

• Transformations
• Mathematical proof
• Area

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