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In this lesson, similar figures will be compared in terms of their surface areas and volumes.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## Using Scale Factor of Similar Pyramids

On a trip to Egypt, Emily is visiting the Great Egyptian Museum. She sees two models of Pyramids, Khafra and Menkaure, on display. She decides to buy the model of Menkaure, but now as she exits the gift shop, she has become even more curious about the volume of Khafra.

Suppose that the models of the pyramids are similar. If the scale factor of the corresponding side lengths is and the volume of the smaller pyramid is cubic centimeters, what is the volume of the larger model?

Discussion

## Area Scale Factor

If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Let and be similar figures, and and be their respective areas. The length scale factor between corresponding side lengths is Here, the following conditional statement holds true.

### Proof

The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.

The area of a rectangle is the product of its length and its width.

Area of Area of
By the definition of similar polygons, the corresponding side lengths are proportional and equal to the scale factor
The next step is to substitute the expressions for and into the formula for which represents the area of
Simplify right-hand side
Notice that the expression on the right-hand side is times the area of or This proof has shown that the ratio of the areas of the similar rectangles is equal to the square of the ratio of their corresponding side lengths. This ratio is also called the area scale factor.
The length scale factor of two similar figures can be used to find the area of one of the two figures when the area of one of the figures is known.
Example

## Using Area Scale Factor to Determine an Unknown Area

Dominika is making a flyer for a concert and wants to compare the areas of the two pieces of paper. The widths of and papers are and centimeters, respectively.

If these two pieces of paper are similar and the area of paper is about square centimeters, find the area of the paper. Round the answer to the nearest integer.

### Hint

If the scale factor of two similar figures is then the ratio of their areas is

### Solution

The two pieces of paper are similar and two corresponding sides measure centimeters and centimeters.

Therefore, the scale factor is the ratio of these corresponding sides.
Using this information, the ratio of the areas, or area scale factor, can be calculated by the theorem about the areas of similar figures.
Now, let be the area of the piece of paper. Then, a proportion can be written using the area scale factor and the area of the paper, which is square centimeters.
Solve for
The area of the paper is square centimeters. Dominika figures that is just the right amount of area to promote the rock band Twenty-One Scale Breakers.
Example

## Using Area Scale Factor to Determine an Unknown Side

It was just learned that if the length scale factor of two similar figures and one of the areas of the figures are known, then the unknown area can be found. Next, consider if both areas but only the side length of one figure are known. It will be possible to solve for the other similar figure's corresponding side length.

The diagram shows two similar figures. Figure has an area of square inches, and Figure has an area of square inches.

If a side length of Figure is inches, find the length of the corresponding side in the other shape.

### Hint

If the scale factor of two similar figures is then the ratio of their areas is

### Solution

Recall that the ratio of areas of two similar figures is equal to the square of the ratio of their corresponding side lengths.
Since the areas are given, the following proportion can be written.
Now, take square roots of both sides of the equation to find the value of the scale factor. Keep in mind that only the principal roots will be considered because only positive numbers make sense in this situation.
Solve for
The scale factor of the figures is Finally, with the scale factor and knowing that the side length of Figure is inches, the length of the corresponding side in Figure represented by can be found.
Solve for
The corresponding length in Figure is inches.
Pop Quiz

## Practice Finding Linear Scale Factor Given Areas

Determine the linear scale factor of the shape on the right to the shape on the left.

Discussion

## Volume Scale Factor

For similar three-dimensional figures, the volume scale factor and the length scale factor are also related.

If two figures are similar, then the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths.

Let Solid and Solid be similar solids and and be their respective volumes. The length scale factor between corresponding linear measures is Given these characteristics, the following conditional statement holds true.

### Proof

The statement will be proven for similar rectangular prisms, but this proof can be adapted to prove other similar solids. As shown in the diagram, let and be the dimensions of Solid and and be the dimensions of Solid

The volume of a rectangular prism is the product of its base area and its height.

Volume of Solid Volume of Solid
By the definition of similar solids, the side lengths are proportional and equal to the scale factor
The next step is to substitute the expressions for and into the formula for the volume of Solid
Simplify right-hand side
Notice that the expression on the right-hand side is times the volume of Solid As shown, the ratio of the volumes of the similar prisms is equal to the cube of the ratio of their corresponding linear measures. This ratio is also called the volume scale factor.
Example

## Using Volume Scale Factor to Determine an Unknown Volume

The scale factor of two similar figures can be used to find the volume of one of the figures when the volume of the other figure is known.

The suitcase company Case-O-La produces snazzy suitcases of various sizes. When a large-sized suitcase is bought, the company offers its cabin-sized version at a discounted price to the same customer.

The large-sized suitcase has a height of inches and a volume of liters. If the cabin-sized suitcase has a height of inches, determine its volume. Round the answer to one decimal place.

### Hint

If the scale factor of two similar figures is then the ratio of their volumes is

### Solution

The suitcases can be considered as two similar rectangular prisms with heights and inches.

Similar solids have the same shape and all of their corresponding sides are proportional. The ratio of the corresponding linear dimensions of the similar solids is the scale factor.
If the scale factor of two similar solids is then the ratio of their corresponding volumes is Now, raise the scale factor to the third power to obtain the ratio of the volumes.
The ratio of the volumes is Now, let be the volume of the small suitcase. Since the volume of the big suitcase is liters, the ratio of to is
Solve for
The volume of the small suitcase is about liters.
Example

## Using Volume Scale Factor to Determine an Unknown Side

After reading a physics magazine, Mark feels confident in estimating the radius of the Sun. To do so, he will use the volumes of the Sun and Earth, which are and cubic kilometers, respectively.

If the radius of the Earth is about kilometers, help Mark find the radius of the Sun.

### Hint

Sun and Earth can be regarded as two similar spheres. Therefore, the volume scale factor can be used to find the radius of the Sun.

### Solution

The Sun and Earth are two similar spheres. Consequently, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear measures, which in this case is the ratio of their radii. Let and be the radii of the Earth and Sun, respectively.
Given that the volumes are known, the volume scale factor can be used to find the scale factor of the Earth to the Sun.
Next, take the cube roots of both sides of the equation to find the value of the length scale factor.
Solve for