In this explainer, we will learn how to calculate areas of similar polygons given two corresponding side lengths or the scale factor between them and the area of one of the polygons.
We can begin by recapping what it means for two polygons to be similar.
Definition: Similar Polygons
Two polygons are similar if they have the same number of sides, their corresponding angles are congruent, and their corresponding sides are in the same proportion.
For example, we could consider the following rectangles.
Rectangle is similar to rectangle . Both rectangles have 4 sides, all the corresponding angles are congruent, and we can write that
The scale factor from rectangle to rectangle can be found by dividing any of the side lengths of by the corresponding side length in . For instance,
If a scale factor in one direction is given as , the scale factor in the opposite direction is .
We can find the length ratio between two similar polygons by writing the ratio of the length of one side in a polygon, with its corresponding side in the other polygon. In the rectangles above, we could write the length ratio of : as . Substituting the widths of these rectangles would give us the equivalent ratio, .
We will now explore how we can use the length ratio of similar polygons to find the area ratio.
We can consider the following triangle, , with base and height . A similar triangle, , is created by a dilation of scale factor . Therefore, the dimensions of will be and .
We might pose the following questions: How does the area of differ from the area of ? Is it also times larger?
We remember that the area of a triangle of base and perpendicular height is calculated by
Thus, the area of is calculated by
For , the area is given by
We can compare the areas of both triangles by observing that
When this polygon, triangle , was dilated with scale factor to give triangle , the areas have a scale factor of . This result is true for all polygons.
Definition: Areas of Similar Polygons given a Scale Factor
If the length scale factor between two similar polygons is , then their area scale factor is .
We will now investigate how we can use a length ratio between similar polygons to identify an area ratio.
Consider the following similar parallelograms, and .
The length ratio can be found by writing the ratios of the lengths of any corresponding sides. For example, we can write as . Simplifying this, we have
We can now consider the areas of each parallelogram. We recall that the area of a parallelogram is found by multiplying the base length by the perpendicular height.
Therefore, we calculate
We can calculate the area of as
We can write the ratio of the areas of as
Comparing the length ratio, , with the area ratio, , we notice that each of the parts of the length ratio is squared to give the corresponding part of the area ratio. That is,
This is true for all polygons. We can formalize this below.
Definition: Area Ratio of Similar Polygons
If the length ratio of two similar polygons is given as , then the ratio of their areas is .
We will now see how we can apply this in the following examples.
Example 1: Finding the Area of a Similar Rectangle given Two Corresponding Lengths and a Diagram
Given the following figure, find the area of a similar polygon in which = 6.
We can recall that two polygons are similar if they have the same number of sides, their corresponding angles are congruent, and their corresponding sides are in the same proportion. A similar polygon to would also be a rectangle with the sides in proportion.
We can find the length ratio of rectangles by writing the lengths . We are given that , and we can use the figure to establish that units. Therefore, substituting these values into the ratio, we have
We can use the length ratio to find the ratio of areas between two similar polygons. If the length ratio of two similar polygons is given as , then the ratio of their areas is . Therefore, we have
We can write that
To find the area of a rectangle, we multiply the length by the width. Using the figure, we observe that the length of is 5 units, and the width is 3 units. Thus, to find the area of , we calculate
If we define the area of as , then we have the ratio of the areas, , as
For the ratios to be equivalent, the value of must be 60, since
We may also consider that, in words, the area ratio of simply means that the area of is 4 times larger than the area of . As the smaller rectangle has an area of 15 square units, then square units.
Therefore, the area of polygon is .
We will now see an example where we write the ratio of the areas of two similar polygons, given the ratio between their lengths.
Example 2: Finding the Ratio of the Areas of Two Similar Shapes given the Ratio of Their Lengths
Rectangle is similar to rectangle with their sides having a ratio of . If the dimensions of each rectangle are doubled, find the ratio of the areas of the larger rectangles.
We can recall that two polygons are similar if their corresponding angles are congruent and their corresponding sides are in the same proportion. We can use the given length ratio of the rectangles to help us calculate the ratio of their areas.
Two similar polygons with corresponding sides in a length ratio of have an area ratio of .
We are given the length ratio of the rectangles as . Therefore, we can calculate the area ratio of as
We are given that the length of each rectangle is doubled. We could define these new rectangles as and . Thus, every length of and will be double the original lengths. If we imagined that the lengths were a fixed value, for example, if the corresponding lengths were 8 cm and 9 cm, then, when doubled, these lengths would be 16 cm and 18 cm. When written as a ratio, , this would simplify to .
Hence, if we have a pair of similar polygons with a given length ratio, and if the same scale factor is applied to both rectangles, then the length ratio remains the same. Furthermore, the area ratio will also remain the same. Thus, we can give the answer that the area ratio of these larger, doubled, rectangles is .
In the next example, we will use information about the perimeter of a square to help us work out the length and area ratios of two similar shapes.
Example 3: Finding the Area of a Similar Polygon given a Length Scale Factor and Perimeter
Square is an enlargement of square by a scale factor of . If the perimeter of square equals 56 cm, what is the area of square ? Give your answer to the nearest hundredths.
We are given that square is an enlargement of square by a scale factor of . All squares are similar; that is, they have all corresponding angles equal and all pairs of corresponding sides in proportion. As the scale factor is , all four sides of will be the length of those of .
We are not given the information about the side lengths of or , but we can calculate the lengths of square using the information about its perimeter. We recall that the perimeter is the distance around the outside of a shape. If we define the side length of as , then as there are 4 sides of equal length, we can write that
We substitute the given value of 56 cm for the perimeter to give
Now, we have that the side length of square is 14 cm. Next, to find the area, we recall that the area of a square of side length is given by Therefore, to find the area of square , we substitute the length, , which gives
Given that the length scale factor from square to square is , we can calculate the area scale factor.
We can recall that if the length scale factor between two similar polygons is , then their area scale factor is . Since the length ratio from to can be given as , then their area ratio can be written as
We can then write that
Substituting our value for the area of , 196 cm2, we have
Hence, the area of square is 441 cm2.
We will now look at another example.
Example 4: Calculating the Perimeter of a Similar Shape given the Two Areas
The areas of two similar polygons are 361 cm2 and 81 cm2. Given that the perimeter of the first is 38 cm, find the perimeter of the second.
In this question, as the polygons are similar, we know that they have the same number of sides, their corresponding angles are congruent, and their corresponding sides are in the same proportion. We can use the given areas to write an area ratio and then establish the length ratio between the two polygons.
We can write the area ratio of as
Two similar polygons with corresponding sides in a length ratio of have an area ratio of .
Therefore, to calculate the length ratio of , we take the positive value of the square root of each of the ratio terms. This gives
We can use this length ratio to calculate the perimeter of the second polygon, polygon 2. In this case, it does not matter what specific shape the polygon is, for example, triangle, square, or hexagon. As the perimeter is a measure of length, the length ratio will still apply.
We can define the perimeter of polygon 2 as and compare the length ratio with the ratio of perimeters as follows:
Since 38 is double the value of 19, both values in the length ratio must be doubled to give those of the perimeters. Hence,
Thus, we can give the answer: the perimeter of the second polygon is 18 cm.
We will now see an example of how we can prove that two rectangles are similar in order to help us solve a real-world problem.
Example 5: Solving a Real-World Problem Involving Area
It cost 3 799 pounds to fit wooden flooring in a class with dimensions 28 m and 10 m. How much would it cost to fit wooden flooring in a similar room with dimensions 84 m and 30 m.
In this question, we can assume that since we are given two dimensions, of differing sizes, then both classrooms are rectangular in shape. We can define the first rectangle as and the second as . One way to solve this problem is to establish if these rectangular classrooms are, in fact, similar. Two polygons are similar if they have the same number of sides, their corresponding angles are congruent, and their corresponding sides are in the same proportion.
We know that the rectangles will have the same number of sides, and the corresponding angles will all be of equal measure, as they are all . We need to check if the sides are in proportion. Sketching a diagram can be useful.
We can write the ratio of widths, , as
This ratio simplifies to
The ratio of lengths can be written as which simplifies to
In order to check if two polygons have corresponding sides in proportion, we need to check all sides. However, as this is a rectangle, we know that there are two pairs of congruent sides. We have shown, therefore, that both the lengths and widths simplify to the same ratio, , and so all sides are in the same proportion. Hence, the two classroom rectangles are similar.
The length ratio of is equal to . We can then use the property that similar polygons with corresponding sides in a length ratio of have an area ratio of . We can calculate the area ratio of as
Alternatively, we could calculate the area ratio by working out the area of each rectangle. The area of a rectangle with length and width is given by
The area of rectangle , with length 28 m and width 10 m, is
The area of rectangle can be calculated as
Simplifying the ratio of these areas gives
Either method produces the same area ratio, .
In order to find the cost of the flooring, we must use the area ratio, rather than the length ratio. This is because the cost of the flooring changes directly according to the area of the room, rather than one of its dimensions.
We are given the cost of the flooring for the rectangle with dimensions 28 m and 10 m, rectangle . We could define the cost of the flooring for rectangle as . We compare the ratios for the area and cost as
Each value in the cost ratio must be 3 799 times larger than those in the area ratio. Therefore,
We can give the answer that the cost to fit flooring in the classroom with dimensions 84 m and 30 m is .
We can now summarize the key points.
- Two polygons are similar if they have the same number of sides, their corresponding angles are congruent, and their corresponding sides are in the same proportion.
- If the length scale factor between two similar polygons is , then their area scale factor is .
- If the length ratio of two similar polygons is given as , then ratio of their areas is .
- As perimeter is a length, we can also say that the ratio of the areas of two similar polygons is equal to the square of the ratio of their perimeters.