How to Determine the Scale Factor of Similar Shapes

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A step-by-step guide to finding and using scale factors in geometry

Co-authored by Mario Banuelos, PhD and Aly Rusciano

Last Updated: July 30, 2025 Fact Checked

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Finding Scale Factor

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Finding Similar Figures

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Sample Problems

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The scale factor, or linear scale factor, is the ratio of two corresponding side lengths of similar figures. When two figures are similar, they have corresponding angles but are different sizes. The scale factor shows how much larger or smaller the figures are from each other. While it may seem complicated, scale factor problems are actually quite easy to solve! Read on to learn how to find the scale factor of similar figures.

How do you find the scale factor of similar figures?

Math professor Mario Banuelos, PhD explains that scale factor represents the ratio in size between similar figures. Divide the sides of Figure B by the corresponding sides of Figure A. If the result is greater than 1, Figure B is larger than Figure A. If it’s a fraction (smaller than 1), Figure B is smaller than Figure A.

Section 1 of 3:

Finding the Scale Factor of Similar Figures

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1
Verify that the figures are similar. Similar figures or shapes are ones in which the angles are congruent (the same) and the side lengths are in proportion. ^{[1]
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Research source} Similar figures are the same shape, only one figure is bigger or smaller than the other. ^{[2]
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Research source}
- The problem will likely tell you the shapes are similar, or it might show you that the angles are the same, or visually indicate that the side lengths are proportional, to scale, or that they correspond to each other.
- So, what is the scale factor? The scale factor is how much bigger or smaller one object is compared to another and is usually a fraction or ratio. Scale factor is often used when making buildings to ensure enough space and materials are provided. ^{[3]
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2
Find a corresponding side length on each figure. You may need to rotate or flip the figure so that the two shapes align and you can identify which sides have corresponding lengths. More often than not, the problem will give you the length of this side (or you may be asked to measure them). ^{[4]
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Research source} If you do not know at least one side length of each figure, you cannot find the scale factor.
- For example, you might have a triangle with a base that is 15 cm long, and a similar triangle with a base that is 10 cm long.
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3
Set up your ratio based on what you’re finding. For each pair of similar figures, there are two scale factors: one you use when scaling up, and one you use when scaling down. If you’re scaling up from a smaller figure to a larger one , use the ratio ${\text{Scale Factor}}={\frac {largerlength}{smallerlength}}$ . If you’re scaling down from a larger figure to a smaller one, use the ratio ${\text{Scale Factor}}={\frac {smallerlength}{largerlength}}$ . ^{[5]
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- For example, say you’re scaling down from a triangle with a 15 cm base to one with a 10 cm base, you’d use the ratio ${\text{Scale Factor}}={\frac {smallerlength}{largerlength}}$ .
  Filling in the appropriate values, it becomes ${\text{Scale Factor}}={\frac {10}{15}}$ .
4
Simplify the ratio to find the scale factor. As professor of mathematics Mario Banuelos, PhD, explains, “A scale factor represents the ratio.” ^{[6]
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Expert Source

Mario Banuelos, PhD
Associate Professor of Mathematics

Expert Interview} The simplified ratio or fraction of what you found in the previous step will give you the scale factor. If you’re scaling down, your scale factor will be a proper fraction. If you’re scaling up, it will be a whole number or improper fraction , which you can convert to a decimal . ^{[7]
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- For example, the ratio ${\frac {10}{15}}$ simplifies to ${\frac {2}{3}}$ . So the scale factor of two triangles, one with a base of 15 cm and one with a base of 10 cm, is ${\frac {2}{3}}$ .

Section 2 of 3:

Finding a Similar Figure Using the Scale Factor

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1
Find the side lengths of the figure. If you’re asked to find a similar factor using the scale factor, you may be a bit stumped with only one figure given. But, solving this problem is easier than it looks! The problem should give you one figure or shape with side lengths. ^{[8]
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- For example, you might have a right triangle with sides measuring 4 cm and 3 cm and a hypotenuse (the triangle’s longest side) 5 cm long.
- Keep in mind that if you cannot determine the side lengths of a figure, you can’t make a similar figure.
2
Determine whether you’re scaling up or down. For these types of problems, you’re typically given the scale factor of the shapes or asked to use the scale factor found in a corresponding problem. Knowing whether you’re scaling up or down will determine if your missing figure is smaller or larger. If you’re scaling up, your missing figure will be larger, and the scale factor will be a whole number, an improper fraction, or a decimal. If you’re scaling down, your missing figure will be smaller, and your scale factor will likely be a proper fraction. ^{[9]
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- For example, if the scale factor is 2, then you are scaling up, and a similar figure will be larger than the original.
3
Multiply the side lengths by the scale factor. When you multiply the original figure’s side lengths by the scale factor, you get the missing corresponding side lengths for the missing figure. Once you know the missing side lengths, you’ve found your missing similar figure! ^{[10]
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- For our example problem:
  - The hypotenuse of the missing figure would be $5*2=10$ .
  - The base of the missing figure would be $3*2=6$ .
  - The height of the missing figure would be $4*2=8$ .

Section 3 of 3:

Sample Problems

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1
Find the scale factor of these similar figures: a rectangle with a height of 6 cm, and a rectangle with a height of 54 cm.
- Create a ratio comparing the two heights. Scaling up, the ratio is ${\text{Scale Factor}}={\frac {54}{6}}$ . Scaling down, the ratio is ${\text{Scale Factor}}={\frac {6}{54}}$ .
- Simplify the ratio. The ratio ${\frac {54}{6}}$ simplifies to ${\frac {9}{1}}=9$ . The ratio ${\frac {6}{54}}$ simplifies to ${\frac {1}{9}}$ . So, the two rectangles have a scale factor of $9$ or ${\frac {1}{9}}$ .
2
An irregular polygon is 14 cm long at its widest point. A similar irregular polygon is 8 cm at its widest point. What is the scale factor?
- Irregular figures can be similar if all of their sides are in proportion. So, you can calculate a scale factor using any dimension given. ^{[11]
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- Since you know the width of each polygon, set up a ratio comparing them. Scaling up, the ratio is ${\text{Scale Factor}}={\frac {14}{8}}$ . Scaling down, the ratio is ${\text{Scale Factor}}={\frac {8}{14}}$ .
- Simplify the ratio. The ratio ${\frac {14}{8}}$ simplifies to ${\frac {7}{4}}=1{\frac {3}{4}}=1.75$ . The ratio ${\frac {8}{14}}$ simplifies to ${\frac {4}{7}}$ . So, the two irregular polygons have a scale factor of $1.75$ or ${\frac {4}{7}}$ .
3
Rectangle ABCD is 8cm x 3cm. Rectangle EFGH is a larger, similar rectangle. Using a scale factor of 2.5, what is the area of Rectangle EFGH?
- Multiply the height of Rectangle ABCD by the scale factor. This gives you the height of Rectangle EFGH: $3\times 2.5=7.5$ .
- Multiply the width of Rectangle ABCD by the scale factor. This gives you the width of Rectangle EFGH: $8\times 2.5=20$ .
- Multiply the height and width of Rectangle EFGH to find the area: $7.5\times 20=150$ . So, the area of Rectangle EFGH is 150 square centimeters.

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Once I have found the scale factor how do I enlarge by the scale factor?

Community Answer

Enlarge the figure by multiplying each side by the scale factor.

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How do you find the linear scale factor of an irregular shape?

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You can find the scale factor of an irregular shape just as you would find the scale factor of a regular shape. As long as you know that the two shapes are similar, you can use one dimension on both figures to calculate the scale factor. For example, if you know the width of the shape, divide one width by the other to find the scale factor.

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Are scale factors always fractions?

Donagan

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Yes, although the fraction could be either less than or greater than 1.

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Mario Banuelos, PhD
Associate Professor of Mathematics

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How to Determine the Scale Factor of Similar Shapes

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