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Scale Factor of Similar Polygons

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Scale Factor of Similar Polygons

Have you ever wondered about the relationship between side lengths of a figure?

Jessie looked at a drawing with two quadrilaterals in it. The side lengths of the quadrilaterals were listed as follows:

MN = 3 inches QT = 6 inches

NO = 2 inches QR = 4 inches

MP = 4 inches TS = 8 inches

OP = 2 inches RS = 4 inches

Given the relationship between these sides, can you figure out the scale factor?

This Concept will teach you how to do exactly that. We'll revisit this problem at the end of the Concept.

Guidance

Similar figures are shapes that exist in proportion to each other. They have congruent angles, but their sides are different lengths. Squares, for example, are similar to each other because they always have four 90^{\circ} angles and four equal sides, even if the lengths of their sides differ. Other figures can be similar too, if their angles are equal. Let’s look at some pairs of similar figures.

Notice that in each pair the figures look the same, but one is smaller than the other. As you can see, similar figures have congruent angles but sides of different lengths.

Even though similar figures have sides of different lengths, corresponding sides still have a relationship with each other. Each pair of corresponding sides has the same relationship as every other pair of corresponding sides, so that, altogether, the pairs of sides exist in proportion to each other. For instance, if a side in one figure is twice as long as its corresponding side in a similar figure, all of the other sides will be twice as long too.

In this Concept, we are going to use these relationships to find the measures of unknown sides. This method is called indirect measurement.

Let’s take a step back to similar figures to understand how indirect measurement works.

First, let’s make sure we can recognize corresponding parts of similar figures. Similar figures have exactly the same angles. Therefore each angle in one figure corresponds to an angle in the other.

These triangles are similar because their angles have the same measures. Which corresponds to which? Angle B is 100^{\circ} . Its corresponding angle will also measure 100^{\circ} : that makes angle Q its corresponding angle. Angles A and P correspond, and angles C and R correspond.

Similar figures also have corresponding sides, even though the sides are not congruent. Corresponding sides are not always easy to spot. We can think of corresponding sides as those which are in the same place in relation to corresponding angles. For instance, side AB , between angles A and B , must correspond to side PQ , because A corresponds to P and B corresponds to Q .

Corresponding sides also have lengths that are related, even though they are not congruent. Specifically, the side lengths are proportional. In other words, each pair of corresponding sides has the same ratio as every other pair of corresponding sides. Take a look at the rectangles below.

These rectangles are similar because the sides of one are proportional to the other. We can see this if we set up proportions for each pair of corresponding sides. Let’s put the sides of the large rectangle on the top and the corresponding sides of the small rectangle on the bottom.

\frac{LM}{WX} &= \frac{8}{4}\\\frac{MN}{XY} &= \frac{6}{3}\\\frac{ON}{ZY} &= \frac{8}{4}\\\frac{LO}{WZ} &= \frac{6}{3}

Now you can clearly see each relationship. To figure out if the pairs do indeed form a proportion, we have to divide the numerator by the denominator. If the quotient is the same, then the ratios each form the same proportion and the figures are similar.

\frac{LM}{WX} &= \frac{8}{4} = 2\\\frac{MN}{XY} &= \frac{6}{3} = 2\\\frac{ON}{ZY} &= \frac{8}{4} = 2\\\frac{LO}{WZ} &= \frac{6}{3} = 2

Each quotient is the same so these ratios are proportional. The side lengths are proportional and the figures are similar.

We call this the scale factor. The scale factor is the ratio that determines the proportional relationship between the sides of similar figures. For the pairs of sides to be proportional to each other, they must have the same scale factor. In other words, similar figures have congruent angles and sides with the same scale factor. A scale factor of 2 means that each side of the larger figure is twice as long as its corresponding side is in the smaller figure.

Now it's time for you to try a few on your own. Determine each scale factor.

Example A

\frac{18}{6} and \frac{24}{8}

Solution: 3

Example B

\frac{12}{6} and \frac{8}{4}

Solution: 2

Example C

\frac{25}{5}, \frac{45}{9}, \frac{15}{3}

Solution: 5

Here is the original problem once again.

Jessie looked at a drawing with two quadrilaterals in it. The side lengths of the quadrilaterals were listed as follows:

MN = 3 inches QT = 6 inches

NO = 2 inches QR = 4 inches

MP = 4 inches TS = 8 inches

OP = 2 inches RS = 4 inches

Given the relationship between these sides, can you figure out the scale factor?

To figure out the scale factor, we can write each corresponding side as a ratio comparing side lengths.

\frac{3}{6}

\frac{2}{4}

\frac{4}{8}

\frac{2}{4}

The scale factor of these figures is \frac{1}{2} .

Vocabulary

Similar Figures
figures that have the same angle measures but not the same side lengths.
Scale Factor
the proportional relationship between two side lengths.
Proportions
two equal ratios.

Guided Practice

Here is one for you to try on your own.

What is the scale factor of the figures below?

Answer

We need to find the scale factor, so let’s set up the proportions of the sides. Let’s put all the sides from the large figure on top and the sides from the small figure on the bottom. It doesn’t matter which we put on top, as long as we keep all the sides from one figure in the same place.

\frac{QR}{HI} &= \frac{15}{5}\\\frac{TS}{KJ} &= \frac{21}{7}\\\frac{RS}{IJ} &= \frac{6}{3}\\\frac{QT}{HK} &= \frac{15}{5}

Now all we have to do is divide to find the scale factor.

\frac{QR}{HI} &= \frac{15}{5} = 3\\\frac{TS}{KJ} &= \frac{21}{7} = 3\\\frac{RS}{IJ} &= \frac{6}{3} = 3\\\frac{QT}{HK} &= \frac{15}{5} = 3

The scale factor for these similar figures is 3. This means that the first quadrilateral is exactly three times bigger than the second.

What happens if we had written the numbers the other way around?

If we had put the measurements of the smaller figure on top and the larger figure on the bottom, we would have found a scale factor of \frac{1}{3} . You can write them the other way too as long as you understand how to read the scale factor. This is another way of saying the scale factor is three. When we say that the larger figure is three times bigger than the small one, it’s the same as saying that the small figure is one-third the size of the larger one.

Video Review

- Here is a video by James Sousa on scale factor.

Practice

Directions: Find the scale factor of the pairs of similar figures below.

Directions: Use each ratio to determine scale factor.

5. \frac{3}{1}

6. \frac{8}{2}

7. \frac{2}{8}

8. \frac{10}{5}

9. \frac{12}{4}

10. \frac{16}{2}

11. \frac{15}{3}

12. \frac{24}{4}

13. \frac{4}{2}

14. \frac{6}{2}

15. \frac{3}{9}

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