Name———————————————————————— Lesson
11.9
Date —————————————
Study Guide
For use with the lesson “Explore Similar Solids”
goal
Use properties of similar solids.
Vocabulary
Similar solids are two solids of the same type with equal ratios of
corresponding linear measures, such as heights or radii.
Theorem 13 Similar Solids Theorem: If two similar solids have a
scale factor of a : b, then corresponding areas have a ratio of a2 : b2, and
corresponding volumes have a ratio of a3 : b3.
Identify similar solids
Tell whether the given right rectangular prism
is similar to the right rectangular prism shown
at the right.
3
3
6
a.
b.
3
5
4
8
5
10
Solution
6
3
3
3
3
3
a. Lengths: ​ }8 ​5 }
​ 4 ​
Widths: ​ }4 ​
Heights: ​ }3 ​5 1
The prisms are not similar because the ratios of corresponding linear measures are
not all equal.
6
3
b. Lengths: ​ }
  ​ 5 }
​ 5 ​
Widths: ​ }5 ​
Heights: ​ }5 ​
10
The prisms are similar because the ratios of corresponding linear measures are all
equal. The scale factor is 3 : 5.
Exercises for Example 1
Tell whether the pair of right solids is similar. Explain your reasoning.
1.
9
2.
3
9
8
5
18
5
5
8
8
11-126
Geometry
Chapter Resource Book
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Lesson 11.9
example 1
Name———————————————————————— Lesson
11.9
Date —————————————
Study Guide continued
For use with the lesson “Explore Similar Solids”
example 2
Use the scale factor of similar solids
The cans shown are similar with a scale factor of 67 : 100. Find the surface area
and volume of the larger can.
II
I
Solution
Use Theorem 13 to write and solve two proportions.
Surface area of I
Surface area of II
a2
b
Volume of I
Volume of II
a3
b
}}
  
   ​5 }
​  2 ​​ } 
 ​ 
5}
​  3 ​
​ 
40.73
672
17.16
673
}
​ }}
 
   ​5 }
​  2  ​​ 
 
   ​ 
5}
​  3  ​ 
Surface area of II
Volume of II
100
100
VI 5 17.16 in.3
Volume of II ø 57.05
The surface area of the larger can is about 90.73 square inches, and the volume of the
larger can is about 57.05 cubic inches.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
example 3
Lesson 11.9
Surface area of II ø 90.73
SI 5 40.73 in.2
Find the scale factor
The pyramids are similar. Pyramid P has
has a volume of of 125 cubic inches and
Pyramid Q has a volume of 64 cubic
inches. Find the scale factor of
Pyramid P to Pyramid Q.
P
Solution
Use Theorem 13 to find the ratio of the two volumes.
a3
b
125
64
}
​    ​
​  3 ​5 }
a
b
5
​ } ​5 }
​ 4 ​
Write ratio of volumes.
Find cube roots.
The scale factor of Pyramid P to Pyramid Q is 5 : 4.
Exercise for Examples 2 and 3
3. Cube C has a surface area of 36 square units and Cube D has a surface area of
144 square units. Find the scale factor of C to D. Find the edge length of C, and
use the scale factor to find the volume of D.
Geometry
Chapter Resource Book
11-127
Lesson 11.9 Explore Similar
Solids, continued
Sample answer:
8. 243 ft2; 182.25 ft3 9. S 5 90 cm2, V 5 50 cm3;
S 5 810 cm2, V 5 1350 cm3 10. S ø 141.4 in.2,
3
4
5
4. Check student’s drawings:
Sample answer:
2
4
Study Guide
Practice Level A
1
1
3
9
1. Not similar; radius }
​ 9 ​5 ​ }3 ​, height }
​ 18  ​ 5 }
​ 2 ​
1. not similar 2. yes; 3 : 2 3. yes; 7 : 4
4. yes; 3 : 2 5. 3 : 1 6. 3 : 1 7. 3 : 1 8. 9 : 1
2
3
9. 27 : 1 10. D 11. S 5 832 m , V 5 1536 m
12. S 5 2464 ft2, V 5 4096 ft3
2
3
13. S 5 202.5π cm , V 5 337.5π cm
14. 1 : 2
2. Similar; The scale factor is 5 : 8. 3. 1 : 2;
}
}
Edge length of C is Ï
​ 6 ​ units; 48​Ï 6 ​ cubic units
Problem Solving Workshop:
Mixed Problem Solving
15. 7 : 2 16. 5 : 6 17. 1 : 4 18. false 19. false
1. a. 10,800 in.3 b. 72 in.3 c. 150
20. false 21. false 22. 4 : 1 23. 3750π yd3
2. a. 576 in.2 b. 2250 in.3 c. 250
24. 13,536 in.2
3. Answers will vary. 4. a. 48.11 in.3 b. yes;
20. 66π in.2 21. 576π ft3 22. 72π in.3
The juice has a volume of about 36.1 in.3, which is
smaller than the volume of the cup.
5. a. The volume doubles. b. The volume
quadruples.
c. You can substitute 2h for h in the formula for
the volume of a right cone and substitute 2r for
r in the same formula and rewrite the formula.
6. 25 in.3 7. a. 25.78 in.2; 12.31 in.3
b. 45.84 in.2; 29.18 in.3 c. 45.84 in.2; 29.18 in.3;
yes
Practice Level C
Challenge Practice
Practice Level B
1. similar; 2 : 1 2. similar; 3 : 2 3. not similar
4. similar; 1 : 0.6 5. D 6. 168π ft2; 288π ft3
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
V ø 109.5 in.3; S ø 2262.4 in.2, V ø 7008 in.3
11. S ø 201.1 ft2, V ø 268.1 ft3; S ø 2463.0 ft2,
V ø 11,494.0 ft3 12. 6: 5 13. 3 : 22 14. 450
15. 686 16. 7 17. 3, 5 18. 728π m2
19. 1920 yd3 20. smaller cylinder: r 5 4 cm,
h 5 8 cm; larger cylinder: r 5 5 cm, h 5 10 cm;
21. 5.8 ft high, 5.5 ft wide, 12.8 ft long
22. 1.56 ft3
7. 864π m2; 2592π m3 8. 170.1 cm2;
121.5 cm3 9. 144π ft2 10. B 11. 2 : 1
12. 2 : 3 13. 3: 4 14. 3 : 2 15. about 385.6 m2;
about 330.98 m3 16. 975π mm2; 4500π mm3
17. 4 : 9 18. 1 : 24 19. r 5 3 in.; h 5 8 in.
1. yes; 1 : 2 2. yes; 1 : 4 3. not similar
4. yes; 5 : 3 5. 256 in.2; 224 in.3
6. 2432π in.2, 15,360π in.3 7. 324 cm3; 1 : 3
7
1. x 5 7 2. x 5 9 3. x 5 4 4. x 5 }
​ 2 ​
11
5. x 5 1 6. x 5 }
​ 2  ​
Geometry
Chapter Resource Book
A67