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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.

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.

Practice Finding Linear Scale Factor Given Areas

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

Volume Scale Factor

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

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.

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

Write in scientific notation

Finally, with the scale factor and knowing that the radius of the Earth is about kilometers, the radius of the Sun can be found. The ratio of to is equal to the scale factor.
Solve for

Write in scientific notation

Practice Finding Linear Scale Factor Given Volumes

The applet shows the volumes of two similar solids. Determine the scale factor of the blue solid to the orange solid.

Finding Volume of Similar Composite Solids

A three-dimensional figure is called a composite solid if it is the combination of two or more solids. Like similar solids, if the corresponding linear measures of two composite solids are proportional, the composite solids are said to be similar. Therefore, their length scale factor can be determined and used to find certain characteristics of the shapes.

The amount of material used to construct the larger silo is three times that of the smaller one. That is, the surface area of the larger silo is three times as large as the surface area of the smaller silo. These two silos can be considered to be similar solids. Furthermore, each silo is composed of a cone and a cylinder as shown.

If the volume of the larger silo is cubic meters, find the volume of the smaller silo. Round the answer to the nearest integer.

Hint

Use the area scale factor to determine the length scale factor.

Solution

The given silos are similar composite solids. Since they are similar, their corresponding linear measures are proportional. Therefore, each silo can be considered as a whole. To find the volume of the smaller silo, these steps will be followed.

• The ratio of their surface areas will be used to determine the length scale factor.
• The length scale factor will be used to determine the volume scale factor.
• Finally, the volume scale factor will be used to determine the volume of the smaller silo.

It is given that the surface area of the larger silo is three times as large as the surface area of the smaller silo.

To find the length scale factor, consider its relationship to the surface areas. Recall that for areas of similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Refer to the smaller silo's side length as and the larger silo's side length as
Since the surface area of the larger silo is three times the surface area of the smaller silo, the ratio of the surface area, small to large, is With that information, the following equation can be expressed.
Next, this equation can be simplified to solve for the length scale factor. Begin by taking the square root of both sides of the equation.
Solve for
Now, the length scale factor can be used to find the volume scale factor. To do so, the length scale factor needs to be raised to the third power.
Finally, knowing that the volume of the larger silo is about cubic meters, the volume of the smaller silo can be calculated. Similar to the areas, the ratio of the volumes should be equal to the volume scale factor.
Solve for
The capacity of the smaller silo is about cubic meters.
The similarity makes it possible to solve for certain characteristics of a wide range of shapes, like prisms, spheres, composite solids, and pyramids.

Relationship Between Length, Area, and Volume Scale Factor

In this course, the relationships between the length scale factor, area scale factor and volume scale factor have been discussed. If the scale factor between two similar figures is then the ratio for their areas and volumes can be expressed as the table shows.

Length Scale Factor Area Scale Factor Volume Scale Factor

Considering these expressions, the challenge presented at the beginning can be solved with more confidence.

Emily knows that the models in the museum are similar pyramids and the scale factor between the corresponding side lengths is

If the volume of the smaller model is cubic centimeters, find the volume of the larger model.

Hint

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