Scale – Definition with Examples

Have you ever observed how you can look at a map and it will tell you the exact location of a place? What would you do if you did not have a map? Well, you might have to fly high above the ground and see which way leads to your destination! But you don’t have to. Do you see how a builder takes the blueprint of a house and turns it into a real thing? 

All of this is possible because of the mathematical concept of the scale factor. The scale factor can be described as a parameter that is used to enlarge or reduce the sizes of shapes in two-dimensional and three-dimensional geometry. It can be used to create similar figures but with different dimensions.

What is a Scale Factor? 

A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).

For example, if we have a rectangle of sides 2 cm and 4 cm, we can enlarge it by multiplying each side by a number, say 2. The new figure we get will be similar to the original figure, but all its dimensions will be twice that of the original rectangle. Here, the number 2 will be called the scale factor.

Example of Scale Factor

Note that the scale factor only changes the dimension or side lengths of shapes but does not change the angle measures.

How Does the Scale Factor Work?

When describing enlargement, it is necessary to mention how much the shape has been enlarged. For example, scale factor 3 means that the new shape is thrice the size of the original shape.

If the scale factor is a fraction, the shape will be smaller. This is called reduction. Therefore, a 1/2 scaling factor means that the new shape is half of the original shape.

How Do You Find the Scale Factor?

The scale factor can be figured out by specifying the new and original dimensions.

  • Scale Factor = Dimension of New Shape/Dimension of Original Shape

However, there are two terms you need to understand when using scaling factors: scaling up and scaling down. Look at the figures below to understand this better.

Scale Up

Scale up means enlarging a small shape into a large one. The scale factor for upscaling is always greater than 1.

Example of Scaling Up

Scale Down

Scale down means that a large number is reduced to a small number. The scale factor for scaling down is always less than 1. 

Example of Scaling down

Uses of the Scale Factor

Scaling objects is a great way to visualize large real-world objects in a small space or magnify small objects to make them easier to see!

The scale factor is used to do the following:

  1. Draw a similar figure in geometry.
  2. Create a scale model.
  3. Create blueprints and scale plans for machinery and architecture.
  4. Shrink vast lands into small pieces of paper, like a map.
  5. Help architects, machine-makers, and designers work with models of objects that are too large to hold if they are their actual size.

Solved Examples

Example 1. Find the scale factor when a square of side 4 cm is enlarged to make a square of side 8 cm.

Solution: The formula for scale factor is:

Scale Factor = Dimensions of New Shape/Dimension of Original Shape

Therefore, the scale factor for the given enlargement is

Scale Factor = 8/4

Scale Factor = 2

Hence, the square has been enlarged by a scale factor of 2.

Example 2. A triangle with side lengths of 3 cm, 4 cm, and 5 cm has been enlarged by a scale factor of 4. What are the dimensions of the new triangle?

Solution:

Dimensions of the new shape = Scale factor ✕ Dimensions of original shape

Therefore, the dimensions of the new triangle will be 4 times the original.

So, the new dimensions are 12 cm, 16 cm, and 20 cm.

Example 3. If a circle of radius 3 cm was reduced to a circle of radius 1 cm, what is the scale factor for this reduction?

Solution: We know that,

Scale Factor = Dimension of new shape/Dimension of original shape

Radius of original circle = 3 cm

Radius of new circle = 1 cm

So, the scale factor for this reduction = 1/3

Practice Problems

What is a Scale Factor - Definition With Examples

Attend this quiz & Test your knowledge.

1If a cube of edge length 12 cm is enlarged to create a cube of edge length 36 cm. What is the scale factor?

2
3
4
5
CorrectIncorrect
Correct answer is: 3
Scale Factor = Dimension of New Shape/Dimension of Original Shape
Edge length of original cube = 12 cm,
Edge length of new cube = 36 cm.
So, the scale factor for this enlargement = 3

2If a sphere of radius 20 cm is reduced to create a sphere of radius 5 cm, what is the scale factor for this reduction?

$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{4}$
$\frac{1}{5}$
CorrectIncorrect
Correct answer is: $\frac{1}{4}$
We know that Scale Factor = Dimension of new shape/Dimension of original shape Radius of original sphere = 20 cm, Radius of new sphere = 5 cm. So, the scale factor for this reduction = $\frac{20}{5}$ = 4 cm

3If a square of side 5 cm is enlarged by a scale factor 2, what are the dimensions of the new square?

2 cm
1/5 cm
5 cm
10 cm
CorrectIncorrect
Correct answer is: 2 cm
Dimensions of new shape= Scale factor ✕ Dimensions of original shape
Therefore, the dimensions of the new square will be 2 times the original.
So, the side of the new square will be 10 cm.

4If a cuboid of dimensions 6 cm, 9 cm, and 12 cm is reduced by a scale factor of $\frac{1}{3}$, what will be its new dimensions?

2, 3, 4 cm
8, 12, 16 cm
4, 6, 8 cm
6, 9, 12 cm
CorrectIncorrect
Correct answer is: 2, 3, 4 cm
We know that,
Dimensions of new shape= Scale factor ✕ Dimensions of original shape
Therefore, the dimensions of the new cuboid will be $\frac{1}{3}$ times the original.
So, the dimensions of the new cuboid will be 2cm, 3 cm, 4 cm.

Frequently Asked Questions

The formula for calculating the scale factor is:

Scale Factor = Dimensions of new shape/Dimension of original shape

The scale factor can be used in the following ways:

  • To compare two 2D/3D geometric figures
  • To calculate ratios and proportions
  • To measure drawings of the same shape but with different dimensions
  • To transform the sizes in engineering and architectural fields

A scale drawing is an exact drawing of the object created using the scale factor to reduce or increase the dimensions of the original object.