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Extending to Three Dimensions

Area and Volume Scale Factors

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.

Models of Khafra and Menkauere Pyramids

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?

Theory

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.

Similar Quadrilaterals

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

Proof

The theorem will be proven for similar rectangles. The proof can be adapted to other similar figures.

Similar rectangles

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

Area of Area of
By definition of similar polygons, the corresponding side lengths are proportional and equal to the scale factor The next step will be to substitute the found expression 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
This way it has been derived that the ratio of the areas of the similar rectangles is equal to the square of the ratio of their corresponding side lengths.

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.

A4 and A2 paper

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.

A4 and A2 paper
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.

Figure A and Figure B
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 linear measures.

Two similar solids

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

Proof

The theorem will be proven for similar rectangular prisms. Take into consideration that the proof can be adapted to prove other similar solids as well. 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 definition of similar solids, the side lengths are proportional and equal to the scale factor The next step will be 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
Therefore, the ratio of the volumes of the similar prisms is equal to the cube of the ratio of their corresponding linear measures.

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.

Large-sized and cabin-sized suitcases
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.

Large-sized and cabin-sized suitcases and their heights
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.

Sun and Earth

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

The radius of Sun is about or kilometers.

Pop Quiz

Practice Finding Linear Scale Factor Given Volumes

The applet shows the volumes of two similar solids. Determine the scale factor of the solid on the right to the solid on the left.

Example

Finding Area Given Volume of Similar Solids

The corresponding faces of two similar three-dimensional figures are also similar. Subsequently, the ratio of the areas of the corresponding faces is proportional to the square of the length scale factor of the figures.

Dylan has a golden retriever and a chihuahua. He buys two similar doghouses, whose corresponding side lengths are proportional. When he is about to finish painting the doghouses, he realizes that there is not enough paint for the front face of the small doghouse!

Two similar doghouses

Dylan knows the volumes of each doghouse. They are about and cubic inches. He also knows that the front face of the big doghouse has an area of square inches. Help Dylan determine the area of the front face of the small doghouse. This will help him determine how much more paint to buy. Round the answer to the nearest integer.

Hint

Start by finding the length scale factor. Then, use the ratio for the areas of similar figures.

Solution

The length scale factor will be found first. To do so, the cube root of the volume scale factor will be calculated. Recall that the ratio of volumes of two similar solids is equal to the cube of the ratio of their corresponding side lengths. A proportion can be written using the ratio of the volumes. Now, take cube roots of both sides of the equation to find the value of the length scale factor.
Solve for
The area scale factor can be found by squaring the scale factor for length. Finally, knowing that the larger doghouse's front face area is about square inches, the corresponding area in the other one can be calculated. Let be that area. Then, should be equal to the ratio of to
Solve for
The smaller doghouse has a front face with an area about square inches. He can now go shopping for more paint to appease his cool chihuahua.
Since the corresponding faces of two similar three-dimensional figures are similar, it can be concluded that the ratio of the surface areas is the square of the scale factor.

Example

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.

Composite 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.

Two similar silos
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.

Two similar silos
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.

Closure

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

Models of Khafra and Menkauere Pyramids
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

Answer

To find the volume of the larger model, first, find the volume scale factor. Note that the volume scale factor is equal to the cube of the length scale factor. The length scale factor is given as or An equation can now be written using the known volume scale factor and the ratio of the model's volumes. Recall that it is given that the volume of the smaller model is cubic centimeters. Substitute that value into the equation to solve for the volume of the larger model.
The volume of the larger model is cubic centimeters.
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