Scale Factor: Definition, Diagram, Formulas, Examples

A scale factor is a number that may be used to adjust the size of any geometrical figure or object in relation to its original size. It’s used to draw any given figure’s enlarged or reduced shape, as well as to discover the missing length, area, or volume of an enlarged or reduced figure. It’s important to note that the scale factor only affects the size of the figure, not its form. On this page, let’s discuss everything about Scale Factor and its examples in detail. Read further to find more.

Scale

Definition: Scale is used to determine the ratio of the length of any shape or any object on the blueprint or model to the real size of a similar thing in the actual world.

When you draw a real-world object on a piece of paper, you use a scale to describe the measurements exactly.

Scale Factor

Definition: The scale factor may be defined as the ratio between the dimensions of the new shape and the actual shape.

A figure’s size is enlarged or reduced than the dimensions of the original shape, and the scale factor is known. 

Example: 

Scale Factor Formula

The basic formulas of the scale factor are given below:

\({\rm{Dimensions\;of\;Original\;shape\;}} \times {\rm{Scale\;factor}} = {\rm{Dimension\;of\;new\;shape}}\)

\({\rm{Scale\;factor\;}} = {\rm{\;}}\frac{{{\rm{Dimensions\;of\;new\;shape}}}}{{{\rm{Dimensions\;of\;Original\;shape}}}}\)

Example: If you see the below-given image, it shows how the scale factor can change the original shape to its larger and smaller versions.

In the following image, the original parallelogram has dimensions as \(5\) units and \(4\) units. Now, if you want to create a larger parallelogram, each of whose sides are \(4\) times larger, we shall use a scale factor of \(4\) so that the dimensions of the sides will now be \(20\) units and \(16\) units, respectively.

How to Identify the Scale Factor?

When you increase or enlarge the shape, you can say that the size has been scaled up and when you decrease or reduce, you can say that the measure has been scaled down.

Scale Up: It means that a more petite figure is enlarged to a bigger size. Here, the scale factor can be calculated using the basic formula given in the above section. 

\({\rm{Scale\;factor\;}} = {\rm{\;}}\frac{{{\rm{Dimensions\;of\;new\;shape}}}}{{{\rm{Dimensions\;of\;Original\;shape}}}}\)

The scale factor for scaling up is always greater than the number \(1\).

Example: If the dimension of the larger shape is \(30\) and the dimension of the smaller figure is \(15\), then the scale factor will be \(30÷15=2\). 

Hence, you can see that the scale factor is greater than the number \(1\).

Scale Down: It means that a more significant shape is reduced in a smaller size. Here, the scale factor can be calculated using the basic formula given in the previous section. 

\({\rm{Scale\;factor\;}} = {\rm{\;}}\frac{{{\rm{Dimensions\;of\;new\;shape}}}}{{{\rm{Dimensions\;of\;Original\;shape}}}}\)

The scale factor for scaling down is always less than the number \(1\). 

Example: If the dimension of the original bigger figure is \(24\) and we want to have a smaller figure of dimension \(8\), then the scale factor will be \(8 \div 24 = \frac{1}{3}\)

Hence, you can see that the scale factor is smaller than the number \(1\).

Observe the below-given image, which represents the scale-up and the scale down: 

Important:

The given points must be kept in mind while learning about the scale factor:

1. The scale factor of a dilated shape is denoted by \(r\) or \(k\).
2. The scale factor is more than the number \(1(k > 1)\) when the figure is enlarged.
3. The scale factor is smaller than the number \(1(0 < k < 1)\) when the shape is reduced.
4. The scale factor is equal to the number \(1(k = 1)\) when the shape remains the same.
5. The scale factor cannot be expressed as the number zero.

Scale Factor of a Triangle

Similar triangles have identical shapes, and the three angles are also identical. Sides is the only thing that differs. However, the ratio of the sides of one triangle is equal to the ratio of sides of another triangle. This is known as the scale factor.

If you have to find the enlarged triangle similar to the smaller triangle, you need to multiply the lengths of the sides of the smaller triangle by the scale factor.

In the same way, if you have to draw a smaller triangle similar to the giant triangle, you need to divide the length of the sides of the original triangle by using the scale factor formula.

Scale Factor Facts

The scale factor is a ratio of the corresponding sides of similar shapes.

A scale factor tells the amount in which the shape has been enlarged or has been reduced.

The original shape has been reduced if the scale factor is less than the number \(1\).

The original shape has been enlarged if the scale factor is greater than the number \(1\).

Solved Examples – Scale Factor 

Q.1. Write the correct formulas for the given situations: If the shape has to be enlarged and if the shape has to be reduced.
Ans: If the shape has to be enlarged:
The original shape has been enlarged if the scale factor is greater than the number \(1\). This means the shape is scaled up, and the formula is given below:
The scale factor is more than the number \(1(k > 1)\) when the figure is enlarged.
\({\rm{The\;scale\;factor\;formula}} = {\rm{Greater\;shape\;dimensions}} \div {\rm{Smaller\;shape\;dimensions\;}}\)
If the shape has to be reduced:
The scale factor is smaller than the number \(1 (0<k<1)\) when the shape is reduced. This means the shape is scaled down, and the formula is given below:
\({\rm{The}}\,{\rm{scale}}\,{\rm{factor}}\,{\rm{formula}} = {\rm{Smaller}}\,{\rm{shape}}\,{\rm{dimensions}} \div {\rm{Greater}}\,{\rm{shape}}\,{\rm{dimensions}}\)

Q.2. Identify the scale factor used to make the smaller rectangle.

Ans: We have the given dimensions of the giant rectangle are \({\rm{40\,cm}}\) and the dimensions of the smaller rectangle are \({\rm{20\,cm}}\).
In this diagram, the shape is scaled down.
\({\rm{The\;scale\;factor\;formula}} = {\rm{Greater\;shape\;dimensions}} \div {\rm{Smaller\;shape\;dimensions\;}}\)
So, \(\frac{{20}}{{40}} = \frac{1}{2}\)
Hence, the scale factor used to create the smaller rectangle is \(\frac{1}{2}\).

Q.3. Increase the dimensions of the given figure using scale factor \(4\).

Ans: We have the length of the shape \({\rm{ = 6\,cm}}\) and the width or breadth of the shape \({\rm{ = 3\,cm}}\).
Now, you have to increase the size of the given shape by the scale factor \(4\).
So, you have to multiply the given dimensions by the number \(4\).
Length \({\rm{ = 6 \times 4 = 24\;cm}}\)
Breadth \({\rm{ = 3 \times 4 = 12\;cm}}\)
Hence, the new increased dimensions of the shape are \({\rm{24\;cm}}\) and \({\rm{12\;cm}}\).

Q.4. A \(2D\) shape of a triangle is increased by the scale factor of \(2\), which gives new dimensions as \(6\) units by \(10\) units by \(12\) units. Find the dimensions of the original triangle.
Ans: Given that the new dimensions of the triangle are \(6\) units, \(10\) units, and \(12\) units.
The scale factor is \(2\).
\({\rm{Scale\;factor\;}} = {\rm{\;}}\frac{{{\rm{Dimensions\;of\;new\;shape}}}}{{{\rm{Dimensions\;of\;Original\;shape}}}}\)
Now, \(2 = 6 \div {\rm{Dimensions\;of\;the\;original\;shape}}\)
So, \({\rm{Dimensions\;of\;the\;original\;shape}} = 6 \div 2 = 3{\rm{\;units}}.\)
\(2 = 10 \div {\rm{Dimensions\;of\;the\;original\;shape}}\)
Now, \({\rm{Dimensions\;of\;the\;original\;shape}} = 10 \div 2 = 5{\rm{\;units}}\)
\(2 = 12 \div {\rm{Dimensions\;of\;the\;original\;shape}}\)
Next, \({\rm{Dimensions\;of\;the\;original\;shape}} = 12 \div 2 = 6{\rm{\;units}}\)
Hence the dimensions of the original triangle are \(3\) units, \(5\) units and \(6\) units.

Q.5. Increase the dimensions of the given figure using scale factor \(5\).

Ans: Given that the equal sides of the isosceles triangle \({\rm{ = 8\,cm}}\) and that its base is \({\rm{4\,cm}}\)
Now, you have to increase the size of the given triangle by the scale factor \(5\).
So, you have to multiply the given dimensions by the number \(5\).
Length of the equal sides \({\rm{ = 8 \times 5 = 40\;cm}}\)
Base \({\rm{ = 4 \times 5 = 20\;cm}}\)
Hence, the new increased dimensions of the shape are \({\rm{40\;cm}}\) and \({\rm{20\;cm}}\).

Summary

Scale Factor is utilised to scale the shapes in various dimensions. In geometry, you learn about different geometrical shapes, including both two-dimensional and three-dimensional shapes. The scale factor is a measure for the identical shapes, which look identical but have various scales or measures. Assume that the two circle looks similar, but they can have different radii. The scale factor defines the scale by which a figure is larger or smaller than the original shape. You can draw the enlarged shape or reduce the shape of any original form with the help of the scale factor.

In this article about the scale factor, we explained the term scale and the scale factor with an example. Then, we moved to calculate the scale factor, followed by identifying the scale factor and scale up and scale down, provided with a model in each and a diagram. Finally, we talked about a few essential points on the scale factor and then discussed the scale factor of a triangle and glanced at facts of the scale factor. Then we have given solved examples along with a few FAQs.

Frequently Asked Questions (FAQs) – Scale Factor

Q.1. Explain Scale Factor with an example.
Ans: The scale factor is defined as the ratio between the dimensions of the new figure and the actual figure of a similar type.
The figure’s size can be enlarged or reduced than the original shape using the scale factor. 
Example: You have a shape of a square that has \(16\) units side, and you want to reduce the size by \(25\% \) or, say, one-quarter \(\frac{1}{4}\).
Here, the scale factor that is to be used is \(\frac{1}{4}\).
So, the new side of the square will be \(= 16 \times \frac{1}{4} = 4\) units.

Q.2. How do you find the scale factor?
Ans: We find the scale factor by dividing the dimensions of the new figure by that of the original figure of a similar type.
\({\rm{Scale\;factor\;}} = {\rm{\;}}\frac{{{\rm{Dimensions\;of\;new\;shape}}}}{{{\rm{Dimensions\;of\;Original\;shape}}}}\)

Q.3. What is a scale factor of \(2\)?
Ans: The scale factor \(2\) means the new shape obtained after scaling up the original figure by \(2\) or making the dimensions of the original figure’s shape double.
Below is an example for a better understanding of the scale factor of \(2\).

You have to divide the measurement of the new triangle with the original triangle by using the scale factor formula to get the scale factor.
\({\rm{Scale\;factor}} = \frac{{{\rm{Dimensions\;of\;new\;shape}}}}{{{\rm{Dimensions\;of\;Original\;shape}}}} \Rightarrow \frac{{6{\rm{cm}}}}{{3{\rm{cm}}}} = 2\)

Q.4. What is a scale factor \({{\rm{7}}^{{\rm{th}}}}\) Grade?
Ans: The scale factor is defined as the ratio between the dimensions of the new figure and the actual figure of a similar type.
You can draw the enlarged or reduced shape of any similar figures with the help of the scale factor.

Q.5. What does scale factor \(1.5\) mean?
Ans: When you want to make a scaled copy, you have to multiply all the lengths of the original figure by a number called the scale factor.
In this case, the scale factor \(1.5\) means, while drawing the new figure, we have to multiply all the dimensions of the original figure by \(1.5\).

We hope you find this article on Scale Factor helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.