Lesson 11.1 Skills Practice
Name
Date
Whirlygigs for Sale!
Rotating Two-Dimensional Figures through Space
Vocabulary
Describe the term in your own words.
1. disc
Problem Set
Write the name of the solid figure that would result from rotating the plane figure shown around the
axis shown.
1.
2.
4.
11
sphere
© Carnegie Learning
3.
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Lesson 11.1 Skills Practice
5.
page 2
6.
Relate the dimensions of the plane figure to the solid figure that results from its rotation
around the given axis.
7.
8.
11
The base of the rectangle is equal to the radius of the cylinder’s base.
10.
© Carnegie Learning
9.
824 Chapter 11 Skills Practice
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Lesson 11.1 Skills Practice
page 3
Name
11.
Date
12.
© Carnegie Learning
11
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© Carnegie Learning
11
826 Chapter 11 Skills Practice
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Lesson 11.2 Skills Practice
Name
Date
Cakes and Pancakes
Translating and Stacking Two-Dimensional Figures
Vocabulary
© Carnegie Learning
Match each definition to its corresponding term.
1. oblique triangular prism
A. dotted paper used to show three-dimensional diagrams
2. oblique rectangular prism
B. a prism with rectangles as bases whose lateral
faces are not perpendicular to those bases
3. oblique cylinder
C. a 3-dimensional object with two parallel,
congruent, circular bases, and a lateral face not
perpendicular to those bases
4. isometric paper
D. a prism with triangles as bases whose lateral
faces are not perpendicular to those bases
5. right triangular prism
E. a 3-dimensional object with two parallel, congruent,
circular bases, and a lateral face perpendicular to those
bases
6. right rectangular prism
F. a prism with triangles as bases whose lateral faces are
perpendicular to those bases
7. right cylinder
G. a prism with rectangles as bases whose lateral faces
are perpendicular to those bases
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Lesson 11.2 Skills Practice
page 2
Problem Set
Connect the corresponding vertices of the figure and the translated figure. Name the shape that was
translated and name the resulting solid figure.
1.
2.
3.
4.
5.
6.
right triangle; right triangular prism
© Carnegie Learning
11
828 Chapter 11 Skills Practice
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Lesson 11.2 Skills Practice
page 3
Name
Date
Name the solid formed by stacking 1000 of the congruent shapes shown.
7.
8.
10.
cylinder
9.
12.
© Carnegie Learning
11.
11
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Lesson 11.2 Skills Practice
page 4
Name the solid formed by stacking similar shapes so that each layer of the stack is composed of a slightly
smaller shape than the previous layer.
13.
14.
16.
cone
15.
11
18.
© Carnegie Learning
17.
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Lesson 11.2 Skills Practice
page 5
Name
Date
Relate the dimensions of the given plane shape to the related solid figure. Tell whether the shape was
made by stacking congruent or similar shapes.
© Carnegie Learning
19. The lengths of the sides of the
triangle are the same as the lengths
of the sides of the base of the
triangular prism. The triangular
prism was made by stacking
congruent triangles.
20.
21.
11
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Lesson 11.2 Skills Practice
22.
page 6
23.
11
© Carnegie Learning
24.
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Lesson 11.3 Skills Practice
Name
Date
Cavalieri’s Principles
Application of Cavalieri’s Principles
Vocabulary
Describe the term in your own words.
1. Cavalieri’s principle
Problem Set
Estimate the approximate area or volume each irregular or oblique figure. Round your answers to the
nearest tenth, if necessary.
11
1. The height of each rectangle is 10 yards and the base of each rectangle is 1.5 yards.
© Carnegie Learning
I determined that the area is approximately 300 square yards.
area 5 base 3 height
5 (1.5 3 20)(10)
5 (30)(10)
5 300
I determined the sum of the areas of the rectangles to estimate the area of the figure because
Cavalieri’s principle says that the area of the irregular figure is equal to the sum of the areas of the
multiple rectangles when the base and height of all the rectangles are equal.
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Lesson 11.3 Skills Practice
page 2
2. The height of each rectangle is 0.6 inch and the base of each rectangle is 2 inches.
11
1 cm
4.
10 cm
12 ft
3.
© Carnegie Learning
4 ft
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Lesson 11.4 Skills Practice
Name
Date
Spin to Win
Volume of Cones and Pyramids
Problem Set
Calculate the volume of each cone. Use 3.14 for p.
1.
5 cm
4 cm
__ 
1  ​ p​r​ ​h
5 ​ __
3
5 __
​ 1 ​  p​(4)​ ​(5)
< 83.73 cubic centimeters
volume 5 ​ 1 ​ Bh
3
11
2
3
2
7 cm
© Carnegie Learning
2 cm
2.
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Lesson 11.4 Skills Practice
3.
page 2
6 in.
3 in.
4.
4 in.
13 in.
11
5.
15 m
836 © Carnegie Learning
10 m
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Lesson 11.4 Skills Practice
Name
page 3
Date
5 mm
6.
14 mm
11
7.
5 cm
© Carnegie Learning
6.5 cm
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Lesson 11.4 Skills Practice
1 cm
8.
page 4
3.2 cm
9.
7 ft
11
4.5 ft
16.4 ft
838 © Carnegie Learning
7 ft
10.
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Lesson 11.4 Skills Practice
Name
page 5
Date
Calculate the volume of the square pyramid.
11.
9 in.
10 in.
10 in.
Volume 5 __
​ 1 ​  Bh
3
__
5 ​ 1  ​​s​2​h
3
__
5 ​ 1 ​​  (10)​2​(9)
3
12.
11
5 300 cubic inches
© Carnegie Learning
9 ft
12 ft
12 ft
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Lesson 11.4 Skills Practice
page 6
13.
11 cm
7 cm
7 cm
11
14.
20 m
840 25 m
© Carnegie Learning
25 m
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Lesson 11.4 Skills Practice
Name
page 7
Date
15.
22 ft
30 ft
30 ft
11
16.
28 mm
© Carnegie Learning
21 mm
21 mm
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Lesson 11.4 Skills Practice
page 8
17.
34.5 in.
42 in.
42 in.
11
75 cm
90 cm
842 90 cm
© Carnegie Learning
18.
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Lesson 11.4 Skills Practice
Name
page 9
Date
19.
125 yd
100 yd
100 yd
11
20.
© Carnegie Learning
180 ft
200 ft
200 ft
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© Carnegie Learning
11
844 Chapter 11 Skills Practice
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Lesson 11.5 Skills Practice
Name
Date
Spheres à la Archimedes
Volume of a Sphere
Vocabulary
Describe a similarity and a difference between each term.
1. radius of a sphere and diameter of a sphere
2. cross section of a sphere and great circle of a sphere
11
3. hemisphere and sphere
Describe the term in your own words.
© Carnegie Learning
4. annulus
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Lesson 11.5 Skills Practice
page 2
Problem Set
Calculate the volume of each sphere. Use 3.14 for . Round decimals to the nearest tenth, if necessary.
1. r 5 7 meters2. r 5 6 inches
r
r
Volume 5 __
​ 4 ​  p r 3
3
5 __
​ 4 ​  p (7)3
3
5 _____
​ 1372
 p
 ​  
3
¯ 1436.0 cubic meters
3. d 5 20 inches4. d 5 16 meters
d
d
11
r
846 r
© Carnegie Learning
5. r 5 2.5 centimeters6. r 5 11.25 millimeters
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Lesson 11.5 Skills Practice
Name
page 3
Date
7. The radius of the great circle of a sphere is 8 meters.
8. The radius of the great circle of a sphere is 12 feet.
11
© Carnegie Learning
9. The diameter of the great circle of a sphere is 20 centimeters.
10. The diameter of the great circle of a sphere is 15 yards.
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© Carnegie Learning
11
848 Chapter 11 Skills Practice
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Lesson 11.6 Skills Practice
Name
Date
Turn Up the . . .
Using Volume Formulas
Problem Set
Calculate the volume of each pyramid.
1.
5 m2.
6 ft
3m
3 ft
3m
10 ft
V 5 __
​ 1 ​  Bh
3
1
5 ​ __  ​ (3)(3)(5)
3
11
5 15 cubic meters
3.
10 in.
4 in.
4.
3 ft
6 ft
4 ft
6 in.
© Carnegie Learning
3 ft
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Lesson 11.6 Skills Practice
page 2
6.
5.
6 ft
5 ft
6m
5m
4m
8 ft
7m
Calculate the volume of each cylinder. Use 3.14 for . Round decimals to the nearest tenth, if necessary.
7.
8.
5.5 m
30 yd
7m
11
22 yd
V 5 p r 2h
5 p (5.5)2(7)
5 211.75p
¯ 664.9 cubic meters
10.
20 m
10 ft
5m
4.5 ft
850 © Carnegie Learning
9.
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Lesson 11.6 Skills Practice
page 3
Name
11.
Date
12.
4 mm
16 ft
5 ft
6 mm
11
13.
14.
9m
13 cm
© Carnegie Learning
12 m
3.5 cm
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06/06/13 5:57 PM
Lesson 11.6 Skills Practice
page 4
Calculate the volume of each cone. Use 3.14 for . Round decimals to the nearest tenth, if necessary.
15.
h = 6 mm
16.
r = 5 mm
12 m
20 m
V 5 __
​ 1 ​  p r 2h
3
5 __
​ 1 ​  p (5)2(6)
3
5 50p
¯ 157 cubic millimeters
11
17.
18.
10 ft
7 ft
11 yd
© Carnegie Learning
4 yd
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Lesson 11.6 Skills Practice
page 5
Name
Date
19.
20.
6 m
[
3 ft
]
3 ft
[
8m
]
11
21.
8 in.
22.
2 mm
5 mm
© Carnegie Learning
3 in.
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Lesson 11.6 Skills Practice
page 6
Calculate the volume of each sphere. Use 3.14 for p. Round decimals to the nearest tenth, if necessary.
23.
9 in.
__ 
5 __
​ 4 ​ p​
  (9)​​
V 5 ​ 4 ​ p​r3​ ​
3
3
3
5 972p
< 3052.08 cubic inches
11
24.
© Carnegie Learning
10 cm
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Lesson 11.6 Skills Practice
Name
page 7
Date
25.
14 mm
11
26.
© Carnegie Learning
2.5 ft
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© Carnegie Learning
11
856 Chapter 11 Skills Practice
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Lesson 11.7 Skills Practice
Name
Date
Tree Rings
Cross Sections
Problem Set
Describe the shape of each cross section shown.
1.
2.
11
The cross section is a rectangle.
4.
© Carnegie Learning
3.
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Lesson 11.7 Skills Practice
5.
7.
page 2
6.
8.
9.
858 10.
© Carnegie Learning
11
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Lesson 11.7 Skills Practice
page 3
Name
Date
Use the given information to sketch and describe the cross sections.
11. C
onsider a cylinder. Sketch and describe three different cross sections formed when a plane
intersects a cylinder.
circle
rectangle
ellipse
12. C
onsider a rectangular prism. Sketch and describe three different cross sections formed when a
plane intersects a rectangular prism.
© Carnegie Learning
11
13. C
onsider a pentagonal prism. Sketch and describe three different cross sections formed when a
plane intersects a pentagonal prism.
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Lesson 11.7 Skills Practice
page 4
14. C
onsider a triangular prism. Sketch and describe three different cross sections formed
when a plane intersects a triangular prism.
15. C
onsider a triangular pyramid. Sketch and describe three different cross sections formed
when a plane intersects a triangular pyramid.
16. C
onsider a hexagonal pyramid. Sketch and describe three different cross sections formed
when a plane intersects a hexagonal pyramid.
© Carnegie Learning
11
860 Chapter 11 Skills Practice
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Lesson 11.7 Skills Practice
Name
page 5
Date
Consider two cross sections of the given solid. One cross section is parallel to the base of the solid, and
the other cross section is perpendicular to the base of the solid. Determine the shape of each of these
cross sections.
17.
A cross section that is parallel to the base is a hexagon congruent to the hexagonal bases.
A cross section that is perpendicular to the base is a rectangle.
11
18.
© Carnegie Learning
19.
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Lesson 11.7 Skills Practice
page 6
20.
21.
11
© Carnegie Learning
22.
862 Chapter 11 Skills Practice
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Lesson 11.7 Skills Practice
Name
page 7
Date
23.
24.
11
Draw a solid that could have each cross section described.
25. cross section parallel to the base
© Carnegie Learning
The solid is a cone. (The solid could also be a cylinder.)
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Lesson 11.7 Skills Practice
page 8
26. cross section perpendicular to the base
27. cross section parallel to the base
11
© Carnegie Learning
28. cross section parallel to the base
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Lesson 11.7 Skills Practice
Name
page 9
Date
29. cross section perpendicular to the base
30. cross section parallel to the base
© Carnegie Learning
11
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© Carnegie Learning
11
866 Chapter 11 Skills Practice
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Lesson 11.8 Skills Practice
Name
Date
Two Dimensions Meet Three Dimensions
Diagonals in Three Dimensions
Problem Set
Draw all of the sides you cannot see in each rectangular solid using dotted lines. Then, draw a
three-dimensional diagonal using a solid line.
1.
2.
© Carnegie Learning
11
3.
4.
5.
6.
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Lesson 11.8 Skills Practice
page 2
Determine the length of the diagonal of each rectangular solid.
7.
10"
4"
6"
The length of the diagonal of the rectangular solid is about 12.33 inches.
The length of the first leg is 10 inches.
Length of Second Leg: Length of Diagonal:
d 2 5 62 1 42 d 5 7.212 1 102
5 36 1 16 5 51.98 1 100
___
_______
d 5 √
​ 52 ​ ¯ 7.21 d 5 ​√ 151.98 ​ 
¯ 12.33
11
8.
7m
4m
868 © Carnegie Learning
8m
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Lesson 11.8 Skills Practice
page 3
Name
Date
9.
15 cm
6 cm
10 cm
11
10.
7 yd
5 yd
© Carnegie Learning
7 yd
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Lesson 11.8 Skills Practice
page 4
11.
5"
3"
15"
11
12.
12 ft
2 ft
870 © Carnegie Learning
2 ft
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Lesson 11.8 Skills Practice
page 5
Name
Date
Diagonals are shown on the front panel, side panel, and top panel of each rectangular solid. Sketch three
triangles using the diagonals from each of the three panels and some combination of the length, width,
and height of the solid.
13.
4"
6"
H
7"
W
L
7"
H
H
11
6"
4"
W
L
W
14.
L
7m
H
5m
8m
W
© Carnegie Learning
L
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Lesson 11.8 Skills Practice
15.
page 6
4 ft
H
6 ft
5 ft
W
L
16.
11
9 cm
H
5 cm
8 cm
W
17.
8 yd
H
4 yd
10 yd
W
L
872 © Carnegie Learning
L
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Lesson 11.8 Skills Practice
page 7
Name
Date
18.
2"
H
8"
9"
W
L
11
A rectangular solid is shown. Use the diagonals across the front panel, the side panel, and the top panel
of each solid to determine the length of the three-dimensional diagonal.
19.
3"
H
6"
© Carnegie Learning
8"
W
L
SD2 5 __
​ 1 ​  (82 1 62 1 32)
2
SD2 5 __
​ 1 ​  (64 1 36 1 9)
2
​ 1 ​  (109)
SD2 5 __
2
SD2 5 54.5
_____
SD 5 √
​ 54.5 ​ ¯ 7.4
_____
The length of the three-dimensional diagonal is √
​ 54.5 ​ or approximately 7.4 inches.
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Lesson 11.8 Skills Practice
20.
page 8
9m
3m
H
10 m
W
L
11
21.
8 ft
H
10 ft
12 ft
W
© Carnegie Learning
L
874 Chapter 11 Skills Practice
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Lesson 11.8 Skills Practice
Name
page 9
Date
22.
6m
5m
6m
H
W
L
11
23.
8 yd
4 yd
10 yd
H
W
© Carnegie Learning
L
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Lesson 11.8 Skills Practice
24.
page 10
3"
13"
H
15"
W
L
11
Use a formula to answer each question. Show your work and explain your reasoning.
25. A
packing company is in the planning stages of creating a box that includes a diagonal support. The
box has a width of 5 feet, a length of 6 feet, and a height of 8 feet. How long will the diagonal support
need to be?
The diagonal support should be approximately 11.18 feet. I determined the answer by calculating
the length of the box’s diagonal.
d 2 5 52 1 62 1 82
d 2 5 25 1 36 1 64
d ¯ 11.18
876 © Carnegie Learning
d 2 5 125
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Lesson 11.8 Skills Practice
Name
page 11
Date
26. A
plumber needs to transport a 12-foot pipe to a jobsite. The interior of his van is 90 inches in length,
40 inches in width, and 40 inches in height. Will the pipe fit inside the plumber’s van?
27. Y
ou are landscaping the flower beds in your front yard. You choose to plant a tree that measures
5 feet from the root ball to the top. The interior of your car is 60 inches in length, 45 inches in width,
and 40 inches in height. Will the tree fit inside your car?
© Carnegie Learning
11
28. J
ulian is constructing a box for actors to stand on during a school play. To make the box stronger, he
decides to include diagonals on all sides of the box and a three-dimensional diagonal through the
center of the box. The diagonals across the front and back of the box are each 2 feet, the diagonals
across the sides of the box are each 3 feet, and the diagonals across the top and bottom of the box
are each 7 feet. How long is the diagonal through the center of the box?
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Lesson 11.8 Skills Practice
page 12
29. Carmen has a cardboard box. The length of the diagonal across the front of the box is 9 inches. The
length of the diagonal across the side of the box is 7 inches. The length of the diagonal across the
top of the box is 5 inches. Carmen wants to place a 10-inch stick into the box and be able to close
the lid. Will the stick fit inside the box?
11
© Carnegie Learning
30. A
technician needs to pack a television in a cardboard box. The length of the diagonal across the
front of the box is 17 inches. The length of the diagonal across the side of the box is 19 inches. The
length of the diagonal across the top of the box is 20 inches. The three-dimensional diagonal of the
television is 24 inches. Will the television fit in the box?
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