In this explainer, we will learn how to use the properties of similar polygons to find unknown angles, side lengths, scale factors, and perimeters.
Let us begin by reviewing the definition of a polygon.
Definition: Polygon
A polygon is a closed shape with straight sides.
Examples of shapes that are and are not polygons can be seen in the table.
Given two polygons, we say that they are similar if they have the same “shape.” For now, we will use this term loosely in order to motivate this concept, and we will establish a more specific and quantifiable characterization of similar polygons below. The size of similar polygons can differ, but their shape must be the same. For instance, consider two equilateral triangles of different size.
We can see that these two triangles have different sizes, but they are both equilateral triangles. This means that they have the same shape, hence they are similar. In fact, we know that all equilateral triangles have the same shape, while the lengths of their sides may differ. We would like to extend this concept of similarity to general polygons.
For general polygons, their shape is determined by the proportions of the sides and the interior angles. This leads to the definition of similarity that can apply to general polygons.
Definition: Similarity of Polygons
Two polygons are similar if the proportions of the corresponding sides are constant and corresponding angles are equal.
An example of two shapes that are similar is the rectangles shown below.
Here, as both shapes are rectangles, they contain the same angles. However, for them to be similar, we also need to check that the sides of the two rectangles are in proportion. If we divide the corresponding lengths of the two rectangles, we get and . The scale factor between the two sides is constant, and the two rectangles are, therefore, similar. In fact, the two rectangles in this example are polygons and are, therefore, an example of similar polygons.
We can further distinguish polygons into simple or self-intersecting polygons, where simple polygons only have one enclosed region. In this explainer, we will only consider simple polygons.
When we are given that two polygons are similar, it tells us a lot of information concerning the lengths of sides of the polygons as well as the measures of the interior angles. In our first example, we will use the fact that the proportions of the corresponding sides are constant to find the missing length of a side in a similar rectangle.
Example 1: Finding the Length of a Rectangle’s Side given Its Corresponding One in a Similar Rectangle
Given that the rectangles shown are similar, what is ?
Answer
In this example, we are given that the two rectangles are similar. Recall that two polygons are similar if the proportions of the corresponding sides are constant and the corresponding angles are equal. Since all interior angles of a rectangle are right angles, the corresponding angles will not produce any new information. We can use that fact that the proportions of the corresponding sides are constant, but we first need to identify the corresponding sides in the two rectangles.
In both rectangles, we can see that one side is longer than the adjacent one. In the smaller rectangle, this is more evident since we know the exact lengths of the sides. Here, the side with length 29 cm is longer than the one with length 26 cm. In the larger rectangle, we can see from the diagram that the side with length 58 cm is longer than the side with the unknown length. Hence, we can establish the corresponding sides in the two similar rectangles as highlighted in the same color below.
The proportion of the two longer sides is given by
Because the rectangles are similar, the proportion of the two shorter sides must also equal 2. This leads to the equation
Multiplying both sides of this equation by 26 cm gives us .
In the previous example, we found an unknown length in a rectangle using the fact that it was similar to another polygon. While it was not difficult to identify the corresponding sides in the similar rectangles, this task can be difficult to manage if we only rely on the visuals. This leads to a need for mathematical notations for similar polygons, which we will now discuss.
Recall that we can refer to a polygon by first labeling the vertices and then listing the adjacent vertices successively until we return to the initial vertex. For instance, we can label the vertices of the two rectangles below as follows.
The rectangle on the left can be referred to as , and the rectangle on the right can be referred to as . However, these are not the only possible labels for these rectangles. In fact, we can begin with any vertex in each rectangle and move either clockwise or counterclockwise around each rectangle, although we commonly move in alphabetical order where possible. Other possible labels for the two rectangles, for instance, are and respectively.
We can now define the mathematical notation used to indicate similar polygons using these labels.
Definition: Notation for Similar Polygons
The similarity of a number of polygons is denoted by the symbol . When denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon.
To understand this rule in the notation of similar polygons, consider the two similar rectangles above. To denote that the rectangle is similar to the second rectangle, we need to identify each vertex in the second rectangle that corresponds to the vertices , , , and .
To find the corresponding vertex, we can use the following rules:
- The measurements of the interior (corresponding) angles of similar polygons at the corresponding vertices are equal.
- The ratio of the lengths of the (corresponding) sides between two pairs of corresponding vertices is constant.
In the case of our rectangles, the criterion of equal interior angles does not help us to identify the corresponding vertices since all interior angles in a rectangle are equal to . Instead, we need to use the criterion involving the proportion of sides.
We have noted that
Because the opposite sides of a rectangle have equal lengths, we also know that
We can highlight, using the same color, each of the two sides involved in these ratios.
Since we have labeled the first rectangle , the order of the sides of the rectangle following this order is blue, purple, green, and yellow. If we follow the same ordering in the second rectangle, we obtain the label . Hence, we can denote this similarity by
In the next example, we will identify a corresponding angle in a similar polygon using this notation.
Example 2: Understanding the Similarity between Polygons
Consider two similar polygons and . Which angle in corresponds to ?
Answer
In this example, we need to find the corresponding angle in a similar polygon. Recall that, when we label a pair of similar polygons using vertices, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. In particular, corresponding vertices in similar polygons mean that the corresponding interior angles of the polygons at the vertices have equal measures.
We are looking for the interior angle in quadrilateral that corresponds to in quadrilateral . Since vertex appears second in , the corresponding vertex in must be the second vertex in this label, which is .
Hence, the angle in corresponding to is .
In the next example, we will identify the corresponding side in a similar polygon from a given notation.
Example 3: Understanding the Similarity between Polygons
Complete the sentence: If , then .
Answer
In this example, we are given that quadrilateral is similar to quadrilateral , which is indicated by the symbol . Recall that, when denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Hence, we can match the corresponding pairs of vertices from the given notation as follows:
We also recall that the proportion of sides between two corresponding vertices in similar polygons are constant. The left-hand side of the given equation contains the fraction , which is the ratio of two sides of the quadrilateral . If we replace each of the vertices appearing here with the corresponding vertex from the quadrilateral , the ratio should remain the same. This means
Hence, the missing side length on the numerator of the fraction in the right-hand side of this equation is .
Now that we have become more familiar with the notation used to define similar polygons, let us consider how to show that two polygons are similar.
How To: Proving the Similarity of Polygons
To prove that two polygons are similar, we need to
- identify the corresponding vertices,
- show that the corresponding interior angles have equal measures,
- and show that the corresponding sides have equal ratio of lengths.
In the next example, we will determine whether or not two given polygons are similar.
Example 4: Verifying Whether the Two Given Polygons Are Similar
Is polygon similar to polygon ?
Answer
In this example, we need to determine whether the given polygons are similar. Recall that two polygons are similar if the proportions of the corresponding sides are constant and the corresponding angles are equal. Let us begin by verifying whether the corresponding angles in the two polygons are equal.
The first thing to note here is that the two polygons are parallelograms, which allows us to calculate the unknown sides and angle measures of each shape. If we consider , the properties of a parallelogram inform us that and . We also know that is supplementary to and, therefore, . Also, opposite angles in a parallelogram have equal measures, so and .
We can apply a similar method for the parallelogram to find all its interior angles. Adding these angles, we have the following diagram.
Recall that, when denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Since we need to determine whether and are similar, the corresponding vertices can be identified as follows:
The corresponding angles in the two parallelograms are the interior angles at the corresponding vertices. We can see from the diagram above that
This tells us that the corresponding angles of the two polygons have equal measurements. Next, let us consider the proportions of the corresponding sides of the two polygons. We know that the opposite sides of a parallelogram have equal lengths, so we can find the missing lengths from the diagram as follows.
We can now compute the ratio of the corresponding sides to make sure that they are equal:
We can see that the ratio of the corresponding sides in the two polygons are equal. This tells us that the two polygons are similar.
Hence, the answer is yes.
We know that, when two polygons are similar, the ratio of the lengths of the corresponding sides must be constant. This leads to the following definition.
Definition: Scale Factor in Similar Polygons
Given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides.
In the next example, we will find the scale factor between similar polygons.
Example 5: Finding the Ratio of Enlargement between Two Similar Quadrilaterals given Their Dimensions
Quadrilaterals and are similar. Determine the ratio of enlargement rounded to two decimal places if necessary.
Answer
In this example, we are given that quadrilaterals and are similar. We need to find the ratio of enlargement, which is the ratio of the length of the corresponding sides in the similar polygons. We begin by choosing a direction for the enlargement. It can be easier to choose a scale factor larger than 1 by placing the larger lengths on the numerator of the ratio.
Let us begin by identifying the corresponding sides. Recall that, when denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Since we are given that quadrilaterals and are similar, we can identify the following pairs of corresponding vertices as follows:
The corresponding sides are the sides between a pair of adjacent corresponding vertices. In the given diagram, we are given the length of in the first quadrilateral. Since the corresponding vertices to and in the second quadrilateral are and , respectively, the corresponding side to is .
Hence, the ratio of the lengths and will give us the ratio of enlargement. We can see that
Since is larger, we will put this length on the numerator. Hence, the ratio of enlargement is
Rounding the ratio to two decimal places, we obtain 1.06.
In the previous example, we found the scale factor, or the ratio of enlargement, in similar polygons. The scale factor can be used to find missing lengths in similar polygons, as we will see in the next example.
Example 6: Finding the Side Length and Angle Measure in Similar Quadrilaterals
Given , find and the length of .
Answer
In this example, we are given that quadrilateral is similar to . Recall that, when we label a pair of similar polygons using vertices, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Since we are given that quadrilaterals and are similar, we can identify the following pairs of corresponding vertices:
In particular, the corresponding interior angles of the polygons at these pairs of vertices have equal measures:
We want to find , which is the same as . We recall that the sum of the interior angles of a quadrilateral must equal . We know the measures of two angles in this quadrilateral, and . If we can find , we can find by using this property.
Using the similarity of the two quadrilaterals, we can see that , where is given in the diagram. Hence, . We can sum all interior angles of the second quadrilateral and set this equal to :
Rearranging this equation so that is the subject, we obtain
Let us find the length of . To find this missing length, we recall the scale factor of similarity. Recall that, given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides. Since the corresponding vertices of and are and , respectively, we know that is a corresponding side of . Hence, the scale factor is given by
This means that we can obtain the lengths of any side in the second polygon by multiplying the correspond length in the first polygon by 2. In particular, this leads to
Dividing both sides of the equation by 2, we obtain
Hence,
In our final example, we will consider a real-world problem involving similar polygons.
Example 7: Finding the Height of a Model given the Scale Factor between It and Another Model given Its Dimensions
A college professor was using a projector to give his lectures. A slide whose dimensions are 11 inches wide and 7 inches high was projected into an image that was inches wide. Find the height of the projected image.
Answer
In this example, we need to find the height of the image that is created using a projector. We know that a projector enlarges an image without changing its shape. This leads to the concept of similar polygons, which are polygons whose shapes are the same. We can assume that the slide and the projected image are similar polygons.
We are given that the slide is a rectangle that is 11 inches wide and 7 inches high. We are also given that the projected image is inches wide. Let us draw two similar rectangles with this information with vertices labeled.
Using the notations of similar polygons, we can write
We need to find the height of the projected image, which is . To find this missing length, we recall the scale factor of similarity. Recall that, given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides. Since and are corresponding sides, the scale factor is given by
This gives us the scale factor of . This means that we can obtain the length of a side in the projected image by multiplying the length of the corresponding side in the slide by . In particular, this leads to
Hence, the height of the projected image is in.
Let us finish by recapping a few important concepts from this explainer.
Key Points
- Two polygons are similar if the proportions of the corresponding sides are constant and corresponding angles are equal.
- The similarity of two polygons is denoted by the symbol . When denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon.
- To prove that two polygons are similar, we need to
- identify the corresponding vertices,
- show that the corresponding interior angles have equal measures,
- and show that the corresponding sides have equal ratio of lengths.
- Given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides.