Lesson Explainer: Similarity of Polygons | Nagwa Lesson Explainer: Similarity of Polygons | Nagwa

# Lesson Explainer: Similarity of Polygons Mathematics

In this explainer, we will learn how to identify and prove the similarity of polygons, write the order of the corresponding vertices, and use the similarity to solve problems.

We can begin by recalling that polygons are two-dimensional shapes with straight sides. For example, squares, rectangles, triangles, hexagons, and octagons are all polygons. Polygons that have exactly the same shape and size are congruent, whereas similar polygons have the same shape and may have a different size.

We can define similar polygons more formally below.

### Definition: Similar Polygons

Two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion.

Let’s consider the two quadrilaterals and below.

If we are given that ( is similar to ), we have

We can also observe the corresponding sides.

These are and , and , and , and and .

Since corresponding sides are in the same proportion, we can write

The proportional relationship can also be given with all the numerators and denominators swapped in the entire statement; that is,

We should use the similarity statement to identify corresponding vertices, rather than solely using any given diagrams. For example, if we have two triangles such that , then , , and . We could also note that side would be corresponding to .

In the first example, we will use corresponding sides and angles to identify whether two polygons are similar.

### Example 1: Verifying Whether Two Given Polygons Are Similar

Are the two polygons similar?

We recall that two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion.

Inspecting the angles in the figure, we have two pairs of congruent angles:

We can calculate in quadrilateral using the property that the internal angles in a quadrilateral sum to . Hence, we have

We can use the same property of the angle measures in quadrilaterals to calculate . We have

Therefore, we have 4 pairs of corresponding angles that are congruent.

We now check whether we have a proportional relationship between the lengths of corresponding sides; that is, we check whether

We could also write the proportionality as

Substituting the given measurements, we have

Although we have two pairs of side lengths in the same proportion, we do not have all four pairs of sides in the same proportion. Hence, we can give the answer: no, the two polygons are not similar.

Similar polygons can also be considered as a dilation of each other. If the scale factor is 1, then the polygons are congruent. We can use the scale factor of this dilation to work out the measure of unknown sides. This scale factor may also be referred to as the ratio of enlargement. This may be particularly useful when the ratio of sides is clearer, or more intuitive.

Look at the figure below, where .

To find the length of , we could observe that the lengths of must be double the lengths of . This is because we can write that

The scale factor from to is 2. The scale factor in the opposite direction is found by dividing by 2. But, we must express scale factors in terms of a multiplier, and dividing by 2 is equivalent to multiplying by .

Therefore, to find the length of , we multiply the corresponding length in by . This gives us

This is an equivalent approach to finding the length of by writing a single equation involving the 2 pairs of sides:

We will now check whether another pair of polygons are similar.

### Example 2: Finding the Scale Factor between Two Similar Polygons

Are these two polygons similar? If yes, find the scale factor from to .

Two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion.

We can see from the diagram that there are pairs of angles that are marked as having equal measures:

We recall that the angles in a quadrilateral sum to ; hence, we can write the measures of and as

Using the congruent angles above, we can also write as

Thus,

Therefore, we have found that there are four congruent angles. This alone is insufficient to prove similarity, so, we must also determine whether the corresponding sides of the polygon are in proportion.

The sides that are corresponding are and , and , and , and and . The sides are in proportion if

Considering each ratio in turn, we have

As each of these proportions can be simplified to , the proportions are equal. Since the angles are also congruent, we have proved that the polygons are similar.

Although it is not required here to write a similarity statement between the polygons, we do need to take into account the letter ordering if doing so. If we write polygon with the letters in that order, then we must place the congruent vertices in the similar polygon in the same order. Hence, . Alternatively, would be another valid statement.

In order to find the scale factor from to , we can take any pair of corresponding sides and divide the length of the side in by the corresponding side length in .

We have

Alternatively, we have already calculated that the proportion of the corresponding sides is . We must note, however, that this proportion was calculated by dividing a side in by a corresponding side in . This would give us the scale factor from to , and we need the scale factor in the reverse direction, from to . To do this, we instead multiply by the reciprocal of , which is . This is equivalent to the decimal 0.8.

We can give the answer: yes, the polygons are similar, and the scale factor from to is 0.8.

It is worth noting some facts about similarity in regular polygons. Recall that a regular polygon is a polygon where all angles are congruent and all sides are congruent. Regular polygons include equilateral triangles, squares, regular pentagons, regular hexagons, and so on.

Considering two squares of different side lengths, since within each square all angles are congruent, then each of these angles are also congruent to their corresponding angles in the other square. Furthermore, the proportion of corresponding side lengths will be the same for each side length. Therefore, we can say that any regular -sided polygon is similar to another regular -sided polygon, where the values of are the same. That is, all equilateral triangles are similar, all squares are similar, and so on.

Of course, since in similar polygons corresponding angles must be congruent and corresponding sides must be in the same proportion, we cannot say that squares are similar to rectangles (as the corresponding side lengths will not be in the same proportion), nor can we say that rhombuses are similar to squares (as corresponding angles are not congruent).

In the next example, we will see how we can use the information that polygons are similar to determine an unknown side length.

### Example 3: Finding the Length of a Side in a Quadrilateral given the Corresponding Sides in a Similar Quadrilateral and Their Lengths

Given that , determine the length of .

We are given that the two polygons and are similar. This means that their corresponding angles are congruent, and their corresponding sides are in proportion. We can use the proportionality of the sides to help us find the unknown side length, .

In the figure, we are given the lengths of the two sides, and . We can identify from the similarity statement that these two sides are corresponding, as . We can then use the side to help us work out the corresponding length of . We can write that

Substituting the given lengths, we have

We have determined that the length of is 105 in.

It is a common error to confuse similarity and congruence. Congruent polygons have equal corresponding pairs of angles and equal corresponding sides. Two errors commonly seen when dealing with similarity are either mistakenly writing that corresponding sides are equal, rather than in proportion, or writing that corresponding sides are in proportion and corresponding angles are in proportion. If we have two similar polygons, for example, triangles, the angle measures in both triangles must still sum to , regardless of the difference in their sizes.

In the next example, we will see how we use a similarity relationship to determine an unknown side length and an unknown angle measure. We will consider how to find the side length both by using the proportionality relationship between two corresponding pairs of sides and by calculating the scale factor.

### Example 4: Finding the Side Length and Angle Measure in Similar Quadrilaterals

Given that , find and the length of .

We are given the information that the two polygons are similar. Their corresponding angles are congruent and their corresponding sides are in proportion.

To find , we note that we do not have enough information about the angles in polygon to work out . However, because , we know that the given angle measure of is corresponding to . It must also be .

Using the property that the sum of the internal angle measures in a quadrilateral is , we have

Next, the length of can be determined by using the corresponding side, , in . The proportion of these sides will be the same proportion as that between all other pairs of corresponding sides in the polygons. We are given the lengths of another pair of corresponding sides, and .

Therefore,

Substituting the lengths, we have

Alternatively, we could have calculated the length of by finding the scale factor from to . In order to find the scale factor, we use a known pair of side lengths. Hence,

Using this scale factor, the length of must be multiplied by to give the length of . This is given by

Thus, using either method to find the side length, we have determined that and the length of is 123.1 cm.

We will now see how we can solve a problem involving similar polygons and a perimeter.

### Example 5: Finding the Side Lengths of a Polygon given Its Perimeter and the Side Lengths of a Similar Polygon

A polygon has sides of lengths 2 cm, 4 cm, 3 cm, 8 cm, and 4 cm. A second similar polygon has a perimeter of 31.5 cm. What are the lengths of its sides?

We recall that similar polygons have corresponding angles that are congruent and corresponding sides in proportion.

We are given that the side lengths of one polygon, a pentagon, are 2 cm, 4 cm, 3 cm, 8 cm, and 4 cm. We are required to determine the side lengths of a similar polygon using only the information about its perimeter, which is the distance around the edge of the polygon. As the sides of similar polygons are in proportion, then the perimeter, which is also a measure of length, will be in the same proportion.

We calculate the perimeter of the first polygon as follows:

The scale factor from the first polygon to the second polygon can be found by

In order to find the sides in the second polygon, we multiply each corresponding side length in the first polygon by the scale factor of . Hence, we have

The sides in the second polygon can be given as 3 cm, 6 cm, 4.5 cm, 12 cm, and 6 cm.

We can now summarize the key points below.

### Key Points

• Two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion.
• We can use the similarity statement to identify corresponding sides and angles, and we must ensure that the letter ordering is correct when writing a similarity relationship between polygons.
• We can calculate an unknown side by writing the proportional relationship between the side and its corresponding side, along with the proportion between another pair of corresponding sides, or by first calculating the dilation scale factor.
• The scale factor between the perimeters of two similar polygons is the same as that between corresponding side lengths.
• All regular polygons are similar to other regular polygons with the same number of sides.