How to Calculate the Scale Factor of Two Shapes
How to Find Scale Factor
Four Methods:
The scale factor, or linear scale factor, is the ratio of two corresponding side lengths of similar figures. Similar figures have the same shape but are different sizes. The scale factor is used to solve geometric problems. You can use the scale factor to find the missing side lengths of a figure. Conversely, you can use the side lengths of two similar figures to calculate the scale factor. These problems involve multiplication or require you to simplify fractions.
Steps
Finding the Scale Factor of Similar Figures

Verify that the figures are similar.Similar figures, or shapes, are ones in which the angles are congruent, and the side lengths are in proportion. Similar figures are the same shape, only one figure is bigger than the other.
 The problem should tell you that the shapes are similar, or it might show you that the angles are the same, and otherwise indicate that the side lengths are proportional, to scale, or that they correspond to each other.

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 the corresponding side lengths. You should be given the length of these two sides, or should be able to measure them.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.

Set up a ratio.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 are scaling up from a smaller figure to a larger one, use the ratioScale Factor=largerlengthsmallerlength{\displaystyle {\text{Scale Factor}}={\frac {largerlength}{smallerlength}}}. If you are scaling down from a larger figure to a smaller one, use the ratioScale Factor=smallerlengthlargerlength{\displaystyle {\text{Scale Factor}}={\frac {smallerlength}{largerlength}}}.
 For example if you are scaling down from a triangle with a 15 cm base to one with a 10 cm base, you would use the ratioScale Factor=smallerlengthlargerlength{\displaystyle {\text{Scale Factor}}={\frac {smallerlength}{largerlength}}}.
Filling in the appropriate values, it becomesScale Factor=1015{\displaystyle {\text{Scale Factor}}={\frac {10}{15}}}.
 For example if you are scaling down from a triangle with a 15 cm base to one with a 10 cm base, you would use the ratioScale Factor=smallerlengthlargerlength{\displaystyle {\text{Scale Factor}}={\frac {smallerlength}{largerlength}}}.

Simplify the ratio.The simplified ratio, or fraction, will give you your scale factor. If you are scaling down, your scale factor will be a proper fraction.If you are scaling up, it will be a whole number or improper fraction, which you can convert to a decimal.
 For example, the ratio1015{\displaystyle {\frac {10}{15}}}simplifies to23{\displaystyle {\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, is23{\displaystyle {\frac {2}{3}}}.
Finding a Similar Figure Using the Scale Factor

Find the side lengths of the figure.You should have one figure of which the side lengths are given or measurable. If you cannot determine the side lengths of the figure, you cannot make a similar figure.
 For example, you might have a right triangle with sides measuring 4 cm and 3 cm, and a hypotenuse 5 cm long.

Determine whether you are scaling up or down.If you are scaling up, your missing figure will be larger, and the scale factor will be a whole number, improper fraction, or decimal. If you are scaling down your missing figure will be smaller, and your scale factor will most likely be a proper fraction.
 For example, if the scale factor is 2, then you are scaling up, and the similar figure will be larger than the one you have.

Multiply one side length by the scale factor.The scale factor should be given to you. When you multiply the side length by the scale factor, this gives you the missing corresponding side length on the similar figure.
 For example, if the hypotenuse of a right triangle is 5 cm long, and the scale factor is 2, to find the hypotenuse of the similar triangle, you would calculate5×2=10{\displaystyle 5\times 2=10}. So the similar triangle has a hypotenuse that is 10 cm long.

Find the remaining side lengths of the figure.Continue to multiply each side length by the scale factor. This will give you the corresponding side lengths of the missing figure.
 For example, if the base of a right triangle is 3 cm long, with a scale factor of 2, you would calculate3×2=6{\displaystyle 3\times 2=6}to find the base of the similar triangle. If the height of a right triangle is 4 cm long, with a scale factor of 2 you would calculate4×2=8{\displaystyle 4\times 2=8}to find the height of the similar triangle.
Completing Sample Problems

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 isScale Factor=546{\displaystyle {\text{Scale Factor}}={\frac {54}{6}}}. Scaling down, the ratio isScale Factor=654{\displaystyle {\text{Scale Factor}}={\frac {6}{54}}}.
 Simplify the ratio. The ratio546{\displaystyle {\frac {54}{6}}}simplifies to654{\displaystyle {\frac {6}{54}}}simplifies to19{\displaystyle {\frac {1}{9}}}. So the two rectangles have a scale factor of9{\displaystyle 9}or19{\displaystyle {\frac {1}{9}}}.

Try this problem.An irregular polygon is 14 cm long at its widest point. A similar irregular polygon is 8 inches at its widest point. What is the scale factor?
 Irregular figures can be similar if all of their sides are in proportion. Thus, you can calculate a scale factor using any dimension you are given.
 Since you know the width of each polygon, you can set up a ratio comparing them. Scaling up, the ratio isScale Factor=148{\displaystyle {\text{Scale Factor}}={\frac {14}{8}}}. Scaling down, the ratio isScale Factor=814{\displaystyle {\text{Scale Factor}}={\frac {8}{14}}}.
 Simplify the ratio. The ratio148{\displaystyle {\frac {14}{8}}}simplifies to814{\displaystyle {\frac {8}{14}}}simplifies to47{\displaystyle {\frac {4}{7}}}. So the two irregular polygons have a scale factor of1.75{\displaystyle 1.75}or47{\displaystyle {\frac {4}{7}}}.

Use the scale factor to answer this problem.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 will give you the height of Rectangle EFGH:3×2.5=7.5{\displaystyle 3\times 2.5=7.5}.
 Multiply the width of Rectangle ABCD by the scale factor. This will give you the width of Rectangle EFGH:8×2.5=20{\displaystyle 8\times 2.5=20}.
 Multiply the height and width of Rectangle EFGH to find the area:7.5×20=150{\displaystyle 7.5\times 20=150}. So, the area of Rectangle EFGH is 150 square centimeters.
Finding the Scale Factor in Chemistry

Divide the molar mass of the compound by that of the empirical formula.When you have the empirical formula of a chemical compound and you need to find the molecular formula of that same chemical compound, you can find the scaling factor you need by dividing the molar mass of the compound by the molar mass of the empirical formula.
 For example, you might need to find the molar mass of an H2O compound with a molar mass of 54.05 g/mol.
 The molar mass of H2O is 18.0152 g/mol.
 Find the scaling factor by dividing the molar mass of the compound by the molar mass of the empirical formula:
 Scaling factor = 54.05 / 18.0152 = 3
 For example, you might need to find the molar mass of an H2O compound with a molar mass of 54.05 g/mol.

Multiply the empirical formula by the scaling factor.Multiply the subscripts of each element within the empirical formula by the scaling factor you just calculated. This will give you the molecular formula of the chemical compound sample involved in the problem.
 For example, to find the molecular formula of the compound in question, multiply the subscripts of H20 by the scaling factor of 3.
 H2O * 3 = H6O3
 For example, to find the molecular formula of the compound in question, multiply the subscripts of H20 by the scaling factor of 3.

Write the answer.With this answer, you have successfully found the answer for the empirical formula as well as the molecular formula of the chemical compound involved in the problem.
 For example, the scaling factor for the compound is 3. The molecular formula of the compound is H6O3.
Community Q&A

QuestionOnce I have found the scale factor how do I enlarge by the scale factor?wikiHow ContributorCommunity AnswerEnlarge the figure by multiplying each side by the scale factor.Thanks!

QuestionAre scale factors always fractions?Top AnswererYes, although the fraction could be either less than or greater than 1.Thanks!

QuestionHow do you find the linear scale factor of an irregular shape?wikiHow ContributorCommunity AnswerYou 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.Thanks!

QuestionWhat's the length and width of an actual room, which measures scale length 6.5" and width 5", and scale factor is 3 inches:8 feet?Top AnswererFor the length, divide 3 inches into 6.5 inches, and multiply by 8 feet. For the width, divide 3 inches into 5 inches, and multiply by 8 feet.Thanks!

QuestionThe total length of something is 4000 MM. The scale is 1:20 meters. How would I calculate this?wikiHow ContributorCommunity AnswerFirst you need to convert the millimeters to meters. There are 1,000 mm in a meter, so 4,000 mm = 4 meters. Since the scale factor is 1:20 meters, multiply the number of meters by 20 to find the similar length: 4 x 20 = 80 meters.Thanks!

QuestionA plan of a house measures 20cm by 13cm. How can I find the area of the actual house if the linear scale factor is 50?wikiHow ContributorCommunity AnswerMultiply each dimension by 50 to get the actual length and width of the house: 20 x 50 = 1,000; 13 x 50 = 650. Then, multiply the length and width to find the area: 1,000 x 650 = 650,000 square cm.Thanks!

QuestionWhat is the ratio scale if it's only known that the scale factor is 4?Top AnswererThat means the scale factor is 1:4, or ¼.Thanks!

QuestionI have to draw a site map for a home's location on a lot measuring 70' x 150'. The city has given me a 3.5" X 1.75" area to draw in. What would be a good scale?Top AnswererTry 1" = 50'.Thanks!

QuestionHow would I find the scale factor of a rectangle where the length is 3.8 and the larger version is 31.75 in length?Top Answerer31.75 ÷ 3.8 = 8.355, or roughly 8 1/3. The exact scale is 1 : 8.355, or about 1 : 8 1/3 (or 3 : 25). That's an unusual scale. You could call it 1 : 8, but that wouldn't be exactly accurate.Thanks!

QuestionTriangle ABC is enlarged 200%. What is the scale factor for the enlargement?Top Answerer3:1. (A 100% increase means a doubling in size. A 200% increase is a tripling in size.)Thanks!
Quick Summary
To find scale factor, start by finding the length of a corresponding side on each figure. If you're scaling up from a smaller figure to a larger one, plug the lengths into the equation scale factor = larger length over smaller length. If you're scaling down from a larger figure to a smaller one, use the equation scale factor = smaller length over larger length. Plug in the lengths and simplify the fraction to find the scale factor.
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Video: How to Find Scale Factor with Similar Figures
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