1. Scaling Three-Dimensional Figures
Course 3
8-10
Scaling Three-Dimensional Figures
Warm Up
Find the surface area of each rectangular prism.
1. length 14 cm, width 7 cm, height 7 cm
2. length 30 in., width 6 in., height 21 in
3. length 3 mm, width 6 mm, height 4 mm
4. length 37 in., width 9 in., height 18 in.
490 cm2
1872 in2
108 mm2
2322 in2

2. Scaling Three-Dimensional Figures
Course 3
8-10
Scaling Three-Dimensional Figures
Problem of the Day
A model of a solid-steel machine tool is built to a scale of 1 cm = 10 cm. The real object will weigh 2500 grams. How much does the model, also made of solid steel, weigh?
2.5 g

3. Scaling Three-Dimensional Figures
Course 3
8-10
Scaling Three-Dimensional Figures
TB P
Learn to make scale models of solid figures.

4. Insert Lesson Title Here
Course 3
8-10
Scaling Three-Dimensional Figures
Insert Lesson Title Here
Vocabulary
capacity

5. Scaling Three-Dimensional Figures
Course 3
8-10
Scaling Three-Dimensional Figures

6. Insert Lesson Title Here
Course 3
8-10
Scaling Three-Dimensional Figures
Insert Lesson Title Here
Corresponding edge lengths of any two cubes are in proportion to each other because the cubes are similar. However, volumes and surface areas do not have the same scale factor as edge lengths.
Each edge of the 2 ft cube is 2 times as long as each edge of the 1 ft cube. However, the cube’s volume, or capacity, is 8 times as large, and its surface area is 4 times as large as the 1 ft cube’s.

7. Insert Lesson Title Here
Course 3
8-10
Scaling Three-Dimensional Figures
Insert Lesson Title Here
Multiplying the linear dimensions of a solid by n creates n2 as much surface area and n3 as much volume.
Helpful Hint

8. Additional Example 1A: Scaling Models That Are Cubes
Course 3
8-10
Scaling Three-Dimensional Figures
Additional Example 1A: Scaling Models That Are Cubes
A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.
the edge lengths of the two cubes
3 cm cube
1 cm cube
3 cm
1 cm
Ratio of corresponding edges
= 3
The length of the edges of the larger cube is 3 times the length of the edges of the smaller cube.

9. Additional Example 1B: Scaling Models That Are Cubes
Course 3
8-10
Scaling Three-Dimensional Figures
Additional Example 1B: Scaling Models That Are Cubes
A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.
the surface areas of the two cubes
3 cm cube
1 cm cube
54 cm2
6 cm2
Ratio of corresponding areas
= 9
The surface area of the large cube is 9 times that of the small cube.

10. Additional Example 1C: Scaling Models That Are Cubes
Course 3
8-10
Scaling Three-Dimensional Figures
Additional Example 1C: Scaling Models That Are Cubes
A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.
the volumes of the two cubes
3 cm cube
1 cm cube
27 cm3
1 cm3
Ratio of corresponding volumes
= 27
The volume of the large cube is 27 times that of the smaller cube.

11. Additional Example 2A: Scaling Models That Are Other Solid Figures
Course 3
8-10
Scaling Three-Dimensional Figures
Additional Example 2A: Scaling Models That Are Other Solid Figures
A box is in the shape of a rectangular prism. The box is 4 ft tall, and its base has a length of 3 ft and a width of 2 ft. For a 6 in. tall model of the box, find the following.
What is the scale factor of the model?
6 in.
4 ft =
6 in.
48 in. = 1 8
Convert and simplify. 1 8
The scale factor of the model is .

12. Additional Example 2B: Scaling Models That Are Other Solid Figures
Course 3
8-10
Scaling Three-Dimensional Figures
Additional Example 2B: Scaling Models That Are Other Solid Figures
A box is in the shape of a rectangular prism. The box is 4 ft tall, and its base has a length of 3 ft and a width of 2 ft. For a 6 in. tall model of the box, find the following.
What are the length and the width of the model?
Length:  3 ft = in. = 4 in. 1 8 36 2
Width:  2 ft = in. = 3 in. 1 8 24
The length of the model is 4 in., and the width is 3 in. 1 2

13. Additional Example 3: Business Application
Course 3
8-10
Scaling Three-Dimensional Figures
Additional Example 3: Business Application
It takes 30 seconds for a pump to fill a cubic container whose edge measures 1 ft. How long does it take for the pump to fill a cubic container whose edge measures 2 ft?
Find the volume of the 2 ft cubic container.
V = 2 ft  2 ft  2 ft = 8 ft3
Set up a proportion and solve.
30 s
1 ft3 x
8 ft3 =
Cancel units.
30  8 = x
Multiply.
240 = x
Calculate the fill time.
It takes 240 seconds, or 4 minutes, to fill the larger container.

14. Scaling Three-Dimensional Figures Insert Lesson Title Here
Course 3
8-10
Scaling Three-Dimensional Figures
Insert Lesson Title Here
Lesson Quiz: Part I
A 10 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.
1. the edge lengths of the two cubes
2. the surface areas of the two cubes
3. the volumes of the two cubes
10:1
100:1
1000:1

15. Scaling Three-Dimensional Figures Insert Lesson Title Here
Course 3
8-10
Scaling Three-Dimensional Figures
Insert Lesson Title Here
Lesson Quiz: Part II
4. A pyramid has a square base measuring 185 m on each side and a height of 115 m. A model of it has a base 37 cm on each side. What is the height of the model?
5. A cement truck is pouring cement for a new 4 in. thick driveway. The driveway is 90 ft long and 20 ft wide. How long will it take the truck to pour the cement if it releases 10 ft3 of cement per minute?
23 cm
60 min