What is a scale factor example?

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A scale factor is a number which multiplies (“scales”) a quantity. For example,the “C” in y = Cx is the scale factor for x. If the equation were y = 5x, then the factor would be 5.

Scale Factor Error

These errors involve a proportional multiplier to a true measurement. For example, a digital scale that is 0.1 g higher than the true measurement will consistently give results that are 0.1 g higher than the true value.

Use in Graded Response Models

The D in the equation for graded response model probability is a scale factor:

Use in Confirmatory Factor Analysis

Confirmatory Factor Analysis allows a researcher to figure out if a relationship between a set of observed variables (“factors”) and their underlying constructs exists, through one or more hypothesis tests. In this sense, scale factors refers to factors that are associated with a particular scale. For example, the Emotional Autonomy Scale (Steinberg & Silverberg, 1986) measures four core components:

• Individuation,
• Perceives parents as people,
• Parental deidealization,
• Nondependency on parents.

CFA could theoretically be used to find underlying factors to this scale. Note that uncovering these factors is often a judgment call; With this particular scale, there isn’t a consensus about its factor structure (Stey et al., 2013).

References

Berkeljon, A. (2012). Multidimensional Item Response Theory inClinical Measurement: A Bifactor Graded Response Model Analysis of the Outcome Questionnaire-45.2. Brigham Young University – Provo.
Steinberg, L., & Silverberg, S. (1986). The Vicissitudes of Autonomy in Early Adolescence. Child Development, 57(4), 841-851. doi:10.2307/1130361
Stey, P. et al. (2013). Factor Structure and Psychometric Properties of a Brief Measure of Dysfunctional Individuation. SAGE. Retrieved March 29, 2019 from: https://www3.nd.edu/~dlapsle1/Lab/Scales_files/Stey%20Hill%20Lapsley%20Dysfunctional%20INdividuation%20Assessment.pdf

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A scale factor is the ratio of sizes of two similar figures. For example, the scale factor is 2 for the following two similar figures.

The scale factor is 2 because the triangle on the right is two times bigger than the triangle on the left.

How to find the scale factor

1. Decide which figure is the original figure.


2. Make a ratio of corresponding sides. Scale factor is equal new figure divided by original figure.

      3.Simplify, if possible.

For the two similar figures above, the original figure is the triangle on the left.

A ratio of corresponding sides is equal to 16/8 or 2.

If on the other hand, we make the triangle on the right our original figure, the scale factor is 8/16 or 1/2.

If the triangles are not similar, then you cannot find the scale factor

When two figures are identical in size, the scale factor is equal to 1.

When a figure is shrunk, the scale factor is less than 1.

When a figure is enlarged, the scale factor is bigger than 1.

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Suppose you have two similar figures , one larger than the other. The scale factor is the ratio of the length of a side of one figure to the length of the corresponding side of the other figure.

Example:

Here, X Y U V = 12 3 = 4 . So, the scale factor is 4 .

Note that when a two-dimensional figure is enlarged ( dilated ) by a scale factor of k , the area of the figure is changed by a factor of k 2 . In case of three-dimensional figures, when a figure is dilated by a scale factor of k , the volume is changed by a factor of k 3 .

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Scale factor is a number by which the size of any geometrical figure or shape can be changed with respect to its original size. It is used to draw the enlarged or reduced shape of any given figure and to find the missing length, area, or volume of an enlarged or reduced figure. It should be noted that the scale factor helps in changing the size of the figure and not its shape.

What is a Scale Factor?

Scale factor is defined as the number or the conversion factor which is used to change the size of a figure without changing its shape. It is used to increase or decrease the size of an object. The scale factor can be calculated if the dimensions of the original figure and the dimensions of the dilated (increased or decreased) figure are known. For example, a rectangle has a length of 5 units and a width of 2 units. Now, if we increase the size of this rectangle by a scale factor of 2, the sides will become 10 units and 4 units, respectively. Hence, we can use the scale factor to get the dimensions of the changed figures.

Observe the following figure which shows how the scale factor can change the original figure to its larger and smaller versions. In the following figure, the original rectangle has the dimensions given as 3 units and 2 units. To create an enlarged figure, the dimensions are multiplied by the scale factor of 3 using the formula: Dimensions of the new shape = Dimensions of the original shape × Scale factor. This gives the new dimensions as 9 units and 6 units respectively. Similarly, to create a reduced figure, we multiply the original dimensions with the scale factor of 1/2. This gives the reduced dimensions as 1.5 units and 1 unit.

Scale Factor Formula

The basic formula to find the scale factor of a figure is expressed as,

Scale factor = Dimensions of the new shape ÷ Dimensions of the original shape.

This formula can also be used to calculate the dimensions of the new figure or the original figure by simply substituting the values in the formula.

How to Find the Scale Factor?

The scale factor can be calculated when the new dimensions and the original dimensions are given. However, there are two terms that need to be understood when using the scale factor. When the size of a figure is increased, we say that it has been scaled up and when it is decreased, we say that it has scaled down.

Scale Up

Scaling up means that a smaller figure is enlarged to a bigger one. In this case, the scale factor can be calculated by a formula, which is another version of the basic formula given in the previous section.

Scale factor = Larger figure dimensions ÷ Smaller figure dimensions

The scale factor for scaling up is always greater than 1. For example, if the dimension of the larger figure is 15 and that of the smaller one is 5, let us place this in the formula which makes it: 15 ÷ 5 = 3. Thus, we can see that the scale factor is greater than 1.

Scale Down

Scaling down means that a larger figure is reduced to a smaller one. Even in this case, the scale factor can be calculated by a formula, which is another version of the basic formula.

Scale factor = Smaller figure dimensions ÷ Larger figure dimensions

The scale factor for scaling down is always less than 1. For example, if the dimension of the smaller figure is 8 and that of the larger one is 24, let us place this in the formula which makes it: 8 ÷ 24 = 1/3. Thus, we can see that the scale factor is less than 1.

Observe the following triangles which explain the concept of a scaled-up figure and a scaled-down figure.

Important Notes

The following points should be remembered while studying about the scale factor:

• The scale factor of a dilated figure is denoted by 'r' or 'k'.
• If the scale factor is more than 1 (k > 1), the image is enlarged.
• If the scale factor is less than 1 (0< k <1), the image is contracted.
• If the scale factor is 1 (k = 1), the image remains the same.
• The scale factor cannot be zero.

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1. Example 1. Write the correct scale factor formulas for the following situations. a) If the image has to be enlarged.

   b) If the image has to be reduced.

   Solution:

   a) If the image has to be enlarged, the scale factor is more than 1 (k > 1). This means the image is scaled up.
   The scale factor formula = Larger figure dimensions ÷ Smaller figure dimensions

   b) If the image needs to be reduced, the scale factor is less than 1 (0<k<1). This means the image is scaled down.
   The scale factor formula = Smaller figure dimensions ÷ Larger figure dimensions

2. Example 2. There are two similar polygons as shown below. Find the scale factor used to create the smaller polygon.

   

   Solution: The given dimension of the bigger polygon is 40 units. The given dimension of the smaller polygon is 20 units. Here, the figure is scaled down. The scale factor = Smaller figure dimensions ÷ Larger figure dimensions = 20/40 = 1/2.

   Therefore, the scale factor used to create the smaller polygon is 1/2.

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FAQs on Scale Factor

Scale factor is a number that is used to draw the enlarged or reduced shape of any given figure. It is a number by which the size of any geometrical figure or shape can be changed with respect to its original size. It helps in changing the size of the figure but not its shape.

What Happens if the Scale Factor is Greater Than 1?

If the scale factor is more than 1 (k > 1), it means that the given figure needs to be enlarged.

What Does a Scale Factor of 0.5 Mean?

A scale factor of 0.5 means that the changed image will be scaled down. For example, the original figure of a square has one of its sides as 6 units. Now, let us use the scale factor of 0.5, to change its size. We will use the formula: Dimensions of the new shape = Dimensions of the original shape × Scale factor. Substituting the values in the formula: the dimensions of the new square will be = 6 × 0.5 = 3 units. This shows that a scale factor of 0.5 changed the figure to a smaller one.

How do you Find the Scale Factor?

The scale factor can be calculated when the new dimensions and the original dimensions are given. The basic formula to find the scale factor of a figure is: Scale factor = Dimension of the new shape ÷ Dimension of the original shape. For example, if the side length of a square is 6 units and if the size of the square has been increased such that the side length of the square becomes 18 units, let us find the scale factor. We will use the formula, Scale factor = Dimension of the new shape ÷ Dimension of the original shape. After substituting the given values we get, Scale factor = 18 ÷ 6 = 3. Therefore, the scale factor that is used to increase the size of the square is 3.

What Happens if the Scale Factor is Less Than 1?

If the scale factor is less than 1 (0< k <1), then the new image that is formed will be contracted or scaled-down. In other words, the new figure will have smaller dimensions as compared to the original figure after it is resized using the scale factor which is less than 1.

Where do we Use a Scale Factor?

Scale factor is a number by which the size of any geometrical figure or shape can be changed with respect to its original size. When things are too large, we use scale factors to calculate smaller, proportional measurements. It is used to compare two similar geometric shapes and also in other fields like cooking, where the ingredients can be reduced or increased according to the given situation. Scale factor can also be used to find any missing dimensions in similar figures.

What is the Formula For Scale Factor?

The basic formula that is used for calculating the scale factor is, Scale factor = Dimension of the new shape ÷ Dimension of the original shape. In case, if the original figure is scaled up, the formula is written as, Scale factor = Larger figure dimensions ÷ Smaller figure dimensions. When the original figure is scaled down, the formula is expressed as, Scale factor = Smaller figure dimensions ÷ Larger figure dimensions.

What Does a Negative Scale Factor Mean?

A negative scale factor makes the dilation rotate 180° and it creates an image on the other side of the centre of the enlargement.

What Scale Factor makes a Figure Smaller?

A scale factor which is less than 1 makes the original figure smaller. For example, let us use a scale factor of 1/3 to change the size of a figure with a given dimension of 36. We will place the given values in the formula: Scale factor = Dimension of the new shape ÷ Dimension of the original shape. Substituting the values, we get, 1/3 = Dimension of the new shape ÷ 36. After solving this, the dimension of the new shape is = 12. Since 12 is smaller than 36, it means the original figure has been reduced in size. Thus, it can be seen that the scale factor which is less than 1 makes a figure smaller.

How is a Scale Different from a Scale Factor?

• Scale is a ratio that is used to define the relation of the actual figure or object with its model. It is commonly used in maps to represent the actual figures in smaller units. For example, a scale of 1:3 means 1 on the map represents the size of 3 in the real world.
• Scale factor is a conversion factor - a number which is used to increase or decrease the size of a figure. For example, if a circle needs to be increased in size using a scale factor of 4, and the circumference of the circle is 7 units. What will be the circumference of the new enlarged circle? We will use the formula, Scale factor = Dimension of the new shape ÷ Dimension of the original shape. This can be written as, Dimension of the new shape = Scale factor × Dimension of the original shape. After substituting the given values, we will get, Dimension of the new shape = 7. After solving this, we get, Dimension of the new shape = 28 units.