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.

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.

## 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.

## 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:

- Draw a similar figure in geometry.
- Create a scale model.
- Create blueprints and scale plans for machinery and architecture.
- Shrink vast lands into small pieces of paper, like a map.
- 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

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

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?

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?

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?

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

**How is a scale factor calculated?**

The formula for calculating the scale factor is:

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

**How can we use the scale factor in real-life situations?**

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

**What is a scale drawing?**

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.