Scale Factor: Definition, Formula & Uses

Scale Factor is defined as the ratio of the corresponding sides of two similar figures. The method is used to determine the enlarged or reduced shape of any closed figure.

• Scale factor can easily measure the scale of any 2D and 3D geometric figure.
• It can be used to calculate the length, breadth and volume of the required image.
• The factor is represented in the form of a fraction.
• If the scale factor is a whole number, then the image will be enlarged.
• In case the scale factor is a fraction, then the image will be reduced.
• Navigators can use the scale factor to draw a map for navigational purposes.
• For example, the scale factor of 1/2 means that the shape will be 1/2 of its original size. 

Read More: Standard Measurement Units

Key Terms: Scale Factor, Whole Number, Fraction, Geometrical Shapes, Length, Breadth, Volume, Factor

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Scale Factor: Definition

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Scale Factor is a factor that determines the ratio between the scale of the original object in comparison with that of the new object. With the help of the factor, similar images of different scales can be constructed easily.

• The concept is used when individuals want to increase the size of the 2D or 3D image.
• To calculate the scale factor, first determine the value of the two corresponding sides.
• The factor does not change the value of the angles used in the figure.

Consider an equation x=kz where k is the scale factor of z.

• It can also be written as x directly proportional to z.
• Here, k is called the constant of proportionality.

Scale Factor

Read More: Multiplication and Division of Integers

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Types of Scale Factor

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Scale Factor is divided into two categories, which are as follows:

Scaling Up Factor

Scaling Up is a process in which the size of the smaller image is enlarged to the bigger image. In this method, the size of the original image is multiplied by a number larger than 1. 

Scaling Down Factor

Scaling Down is a process in which the size of the larger image is reduced to a smaller image. In this method, the size of the original image is either multiplied by a number less than one or divided by a number larger than 1.

Read More: Circle

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Scale Factor Formula

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Scale Factor is used to determine the scale of different dimensional images. The formula used to calculate the scale factor is as follows:

• Scale Factor = Dimensions of the Original Shape / Dimensions of the new image

Scale Factor formula for types of scale factor are as follows:

• Scaling Up Scale Factor = Larger Figure Dimension / Smaller Figure Dimension
• Scaling Down Scale Factor = Smaller Figure Dimension / Larger Figure Dimension

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How to find the scale factors for different geometric figures?

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The steps to calculate the scale factor for different geometric figures are as follows:

• First, determine the similarity between the corresponding figures.
• Next, calculate the value of the corresponding length of each figure by flipping or rotating the figure.
• Analyze whether the image needs to be scaled up or scaled down by setting up the required ratio.
• Simply, the ratio obtained in the above steps to smaller values.

Scale factors for different geometric figures

Read More: Areas related to Circles

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Solved Examples of Scale Factor

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Given below are some examples of Scale Factor

Example 1: Consider two squares having their sides equivalent to 10 cm and 5 cm respectively. Calculate the scale factor for a given figure to draw a smaller square.

Ans.  Given,

• Side of the Larger Square = 10 cm
• Side of the Smaller Square = 5 cm
• Formula for Scale Factor is as follows:

Scale Factor = Dimensions of the Original Shape / Dimensions of the new image

• Scale Factor = 5 / 10 = 1 / 2

Hence the scale factor for two given squares is calculated to ½.

Example 2: Consider two dimensional figures having their sides equivalent to 16 cm and 4 cm respectively. Calculate the scaling up factor for a given figure.

Ans.  Given,

• Side of the Larger Square = 16 cm
• Side of the Smaller Square = 4 cm
• Formula for Scale Factor is as follows:

Scaling Up Scale Factor = Larger Figure Dimension / Smaller Figure Dimension

• Scale Factor = 16 / 4 = 4 / 1

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Scale Factor of a Rectangle

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The scale factor of a rectangle is calculated by multiplying the length and width of the rectangle with similar quantities of the older rectangle.

• Scale Factor of length = new length / old length
• Scale Factor of width = new width / old width
• New Area = Area Scale Factor x Current Area

Scale Factor of a Rectangle​

Read More: Relation between HCF and LCM

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Scale Factor of a Triangle

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To determine the scale factor of a triangle first determine whether the given triangles are similar. In case of triangle, two scenario arises:

• The value of the enlarged triangle can be calculated by multiplying the scale factor with the value of the smaller triangle.
• The value of the smaller triangle can be calculated by dividing the scale factor with the value of the larger triangle.

Scale Factor of a Triangle

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Uses of Scale Factor 

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Various uses of scaling factor are as follows:

• Dimensions of the map can be calculated using the scale factor.
• Architects used the scale factor to determine the spacing in the house.
• Scale Factor can be used in cooking recipes and procedures. 
• The factor can be used to build the model of real world design.

Also Read:

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Things To Remember

• Scale Factor is the amount of which the size of the geometric figure is enlarged or reduced.
• It depends upon the size of the figure.
• The scale factor of a figure is denoted by k.
• If the value of k is greater than 1, then the size of the image is enlarged.
• If the value of k is less than one but greater than 0, then the size of the image is reduced.
• In case the scale factor is equal to 1, then the size of the image remains the same.

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