Scale Factor

What is a Scale Factor?

Wondering what is a scale factor? Well, the scale factor definition is –

It is a multiplying entity that lies in the range of (0, infinity). An original figure can be increased or decreased depending on the scale factor.

Suppose you have two circles that look identical. The term identical doesn't ensure that both the circles are equal. Their radii might be different. The one figure would be the scaled figure of the other one.

Scale factors are closely related to the concepts of ratio and proportions. In ratio and proportions, we multiply or divide a given amount to form a new quantity.

Scale Factor Formula -

The formula of conversion -

Suppose, a rectangle of the dimension of 3x4 cm is given. To find a new rectangle with scaling factor 2. We multiply both length and breadth by the scaling factor, which gives the new dimension as 6x8 cm. 

From this example, we can calculate the scale factor formula for the new dimension as:

Old dimension* scaling factor = New dimension

From the same example, if we need to find the scaling factor, provided old and new dimensions.

Scaling factor= New dimension/old dimension. In this case.

Scaling factor= New length/Old length =6/3=2.

                                  or

Scaling factor = New breadth/Old breadth = 8/4 =2. 

Scale Factor Problems

To generalize -

Let Old dimension, be represented by x

The new dimension, be represented by y

The scaling factor, be represented by k

Then x*k=y

Which implies x is proportional to y.

Assessing the Scaling Factor

If K>1 - It becomes the scale factor for enlargement formula.

If k<1 - It becomes the scale factor for reduction formula.

If k=0 - Implies the figure doesn't exist, making it irrelevant.

When two shapes are similar, their volume and area are related by a factor k or the scaling factor.

Finding Scale Factor of Similar Figures

Scaling Factor of the Rectangle.

Enlarging or Reducing the dimensions of the rectangle results in the formation of a new rectangle. For scaling both the length and breadth of the rectangle, is changed.

For enlarging, both the length and breadth of the rectangle, are multiplied by a factor k, which is greater than 1. In this case, the perimeter increases by k times, and the area increases by k2 times.

For reducing, both the length and breadth of the rectangle are multiplied by a factor k, which is less than 1, but not zero. In this case, the perimeter decreases by k times, and the area decreases by k2 times.

How to Find the Scale Factor of a Circle?

Scaling factor of Circles - The perimeter and area of a circle depends only on its radius. So, changing the radius of a circle will scale it up or down depending on the scaling factor.

How to Find the Scale Factor of a Rectangle?

Similar to Rectangle,

For enlarging, the radius of the circle is multiplied by a factor k, which is greater than 1. In this case, the perimeter becomes k times the original perimeter and area become k2 times the original area.

For enlarging, the radius of the circle is multiplied by a factor k, which is less than 1. In this case, the perimeter becomes k times the original perimeter and area become k2 times the original area.

How to Find the Scale Factor of Triangle?

Scaling Factor of Triangles.

We all know that triangles have three sides. For similar triangles, they have the same shape and angle measurements.

Scale Factor of Enlargement Formula

For enlarging, the sides of the triangle are multiplied by a factor k. If K>1 it forms a bigger triangle with perimeter k times the original perimeter.

For reducing, the sides of the triangle are multiplied by a factor k, which is less than 1. and the perimeter decreases by k times.

The Practicality of Scale Factor.

Scale factor is similar to the concept of ratios and proportions. They have a lot of real-life applications, understanding them, will give us a lot more insight into the concepts of scaling factor. So, let us go through some scale factor problems.

  • Scaled drawing and sculpturing- Is an important real-life application of scaling. From a miniature, a new enlarged drawing or sculpture can be easily obtained. Because of scaling, the new figure so obtained will be very similar.

  • In the introductory example, we calculated the ingredient requirement of tea for two people from one person. Similarly, this idea could be extended to find the requirements of n people from any given information. Hence, we can calculate the increase in ingredients by multiplying the same with the number of people to be fed.

FAQ (Frequently Asked Questions)

1. How to Figure out the Scale Factor?

A. Scaling Factor is used to measure shape dimensions in geometry. The scaling factor may be used to enlarge or reduce the size of a figure. This is a central principle in geometry, working with both 2D and 3D figures.

Finding a scale factor of similar figures- Scaling factors helps in forming a new figure, with respect to the original figure. If the data of the original figure, such perimeter, and area are given, then finding such information of the new figure becomes easy. They can be calculated using scaling factor k.

In two identical geometric figures, the scale factor is the ratio of their respective sides, and dividing the two corresponding lengths of the sides gives the ratio.

2. What is the Scaling Factor of Triangles and what is the Scale Factor Between 4 cm and 10 cm?

A. We all know that triangles have three sides. For similar triangles, they have the same shape and angle measurements. For enlarging, the sides of the triangle are multiplied by a factor k. If K>1 it forms a bigger triangle with perimeter k times the original perimeter.

For reducing, the sides of the triangle are multiplied by a factor k, which is less than 1. and the perimeter decreases by k times.

To find out the scale factor between 4 and 10, we need - 

4 and 10 to be divided by 2.

Length A: 4  of 2 = 2

Length B: 10 of 2 = 5 

And the A to B scale ratio is 2:5.