# Space Figures and Cross Sections

```11-1
Space Figures and
Cross Sections
11-1
1. Plan
Objectives
1
2
To recognize polyhedra and
their parts
To visualize cross sections of
space figures
Examples
1
2
3
4
5
Identifying Vertices, Edges,
and Faces
Using Euler’s Formula
Verifying Euler’s Formula
Describing a Cross Section
Drawing a Cross Section
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To recognize polyhedra and
For each exercise, make a copy of the cube
at the right. Shade the plane that contains
the indicated points. 1–7. See back of book.
their parts
• To visualize cross sections of
space figures
. . . And Why
To learn about medical
techniques, as in Exercise 44.
1. A, B, and C
2. A, B, and G
3. A, C, and G
4. A, D, and G
5. F, D, and G
6. B, D, and G
Lesson 1-3
B
C
D
A
F
7. the midpoints of AD, CD, EH, and GH
G
E
H
New Vocabulary • polyhedron • face • edge • vertex
• cross section
Math Background
The references to plane figures
are somewhat informal. Explain,
for example, that a base of a
cylinder is not technically a circle;
it is a circle together with the
circle’s interior. Similarly, a face
of a polyhedron is not actually a
polygon; it is a polygon together
with its interior.
1
Identifying Parts of a Polyhedron
A polyhedron is a three-dimensional
ﬁgure whose surfaces are polygons. Each
polygon is a face of the polyhedron.
An edge is a segment that is formed by the
intersection of two faces. A vertex is a
point where three or more edges intersect.
Vocabulary Tip
Polyhedron comes from
the Greek poly for
“many” and hedron for
“side.” A cube is a
polyhedron with six sides,
or faces, each of which is
a square.
More Math Background: p. 596C
1
EXAMPLE
Faces
Edge
Vertex
Identifying Vertices, Edges, and Faces
a. How many vertices are there in the polyhedron
at the right? List them.
Lesson Planning and
Resources
H
There are ﬁve vertices: D, E, F, G, and H.
See p. 596E for a list of the
resources that support this lesson.
There are eight edges: DE , EF, FG, GD,
DH, EH, FH, and GH.
PowerPoint
F
G
b. How many edges are there? List them.
D
E
c. How many faces are there? List them.
Bell Ringer Practice
There are ﬁve faces: #DEH, #EFH, #FGH, #GDH, and the quadrilateral
DEFG.
R
Check Skills You’ll Need
For intervention, direct students to:
Quick Check
Identifying Planes
Lesson 1-3: Example 4
Extra Skills, Word Problems, Proof
Practice, Ch. 1
1 List the vertices, edges, and faces of the polyhedron.
R, S, T, U, V; RS, RU, RT , VS, VU, VT , SU,
UT , TS; kRSU, kRUT, kRTS, kVSU,
kVUT, kVTS
S
T
U
V
598
Chapter 11 Surface Area and Volume
Special Needs
Below Level
L1
Review nets by having students cut-out various nets
and form their corresponding three-dimensional
figures. Clarify that there are many possible nets for
the same polyhedron.
598
learning style: tactile
L2
Some students may think that spheres and cylinders
are polyhedrons. Emphasize that the surfaces of
polyhedrons are polygons, whose sides must be line
segments. Have students draw examples of polygons
on the board.
learning style: visual
2. Teach
Leonhard Euler, a Swiss mathematician, discovered a relationship among the
numbers of faces, vertices, and edges of any polyhedron. The result is known as
Euler’s Formula.
Key Concepts
Formula
Guided Instruction
Euler’s Formula
The numbers of faces (F), vertices (V), and edges (E) of a polyhedron are
related by the formula F + V = E + 2.
2
EXAMPLE
1
Using Euler’s Formula
PowerPoint
Additional Examples
1 How many vertices, edges, and
faces of the polyhedron are there?
List them.
The polyhedron has 2 hexagons and 6 rectangles
for a total of 8 faces.
The 2 hexagons have a total of 12 edges.
The 6 rectangles have a total of 24 edges.
If the hexagons and rectangles are joined to form a
polyhedron, each edge is shared by two faces. Therefore, the
number of edges in the polyhedron is one half of the total of 36, or 18.
Connection
Euler’s Formula
8 + V = 18 + 2
Substitute.
V = 12
Euler’s Formula applies to the
polyhedron suggested by the
panels on a volleyball.
Quick Check
F+V=E+2
A
E
J
D
I
C
G
H
10 vertices, 15 edges, and 7
faces; A, B, C, D, E, F, G, H, I, J;
AF, BG, CH, DI, EJ, AB, BC, CD,
DE, EA, FG, GH, HI, IJ, JF; ;
pentagons ABCD and FGHIJ, and
quadrilaterals ABGF, BCHG,
CDIH, and EAFJ
2 Use Euler’s Formula to ﬁnd the number of edges on a polyhedron with
eight triangular faces. 12 edges
In two dimensions, Euler’s Formula reduces to
F+V=E+1
where F is the number of regions formed by V vertices linked by E segments.
EXAMPLE
B
F
Simplify.
Count the number of vertices in the ﬁgure to verify the result.
3
Teaching Tip
Encourage students to work
systematically as they list the
vertices, edges, and faces.
Count faces and edges. Then use Euler’s Formula to ﬁnd
the number of vertices in the polyhedron at the right.
Real-World
EXAMPLE
2 Use Euler’s Formula to find the
number of edges of a polyhedron
with 6 faces and 8 vertices.
12 edges
Verifying Euler’s Formula
Verify Euler’s Formula for a two-dimensional
net of the solid in Example 2.
3 Using the pentagonal prism in
Additional Example 1, verify
Euler’s Formula. Then draw a net
for the figure and verify Euler’s
Formula for the two-dimensional
figure. 7 ± 10 ≠ 15 ± 2
Draw a net:
Count the regions: F = 8
Count the vertices: V = 22
Count the segments: E = 29
8 + 22 = 29 + 1
Quick Check
3 The ﬁgure at the right is a trapezoidal prism.
a. Verify Euler’s formula F + V = E + 2 for
the prism. 6 + 8 = 12 + 2
b. Draw a net for the prism. See margin.
c. Verify Euler’s formula F + V = E + 1 for
your two-dimensional net.
Sample: 6 + 14 = 19 + 1
7 ± 18 ≠ 24 ± 1
Quick Check
Lesson 11-1 Space Figures and Cross Sections
Advanced Learners
599
3.
English Language Learners ELL
L4
Have students determine if values of F, V, and E that
satisfy Euler’s Formula make an existing polyhedron.
learning style: verbal
Make sure students understand the difference
between polyhedron and polygon. Show models of
different polyhedrons and cutouts of different
polygons.
learning style: visual
599
Guided Instruction
2
1
Describing Cross Sections
Error Prevention!
Students may think the plane of
a cross section must be horizontal
or vertical. Show a cross section
of an apple or orange cut along
a plane that is neither horizontal
nor vertical.
5
EXAMPLE
A cross section is the intersection of a solid
and a plane. You can think of a cross section
as a very thin slice of the solid.
Sclera
Lens
Cornea
Retina
Iris
Optic
nerve
Blood
vessels
Pupil
4
Teaching Tip
Point out that the example
assumes that the bottom face of
the cube is horizontal. Ask: If the
cube were tilted slightly, what
might the cross section look like?
Sample: parallelogram
Real-World
Connection
EXAMPLE
Describing a Cross Section
Describe each cross section.
a.
Cross sections are used to
study the anatomy of the eye.
b.
Visual Learners
The cross section is a square.
Encourage students to slice cubes
of butter, ice cream, or modeling
clay at home to investigate how
planes may intersect cubes.
Quick Check
4. Size of sketches
may vary, Samples:
PowerPoint
Additional Examples
The cross section is a triangle.
4 For the funnel shown, sketch each of the following.
a. a horizontal cross section a–b. See left.
b. a vertical cross section that contains the axis of
symmetry
a.
To draw a cross section, you can sometimes use the idea
from Postulate 1-3 that the intersection of two planes is exactly one line.
4 Describe this cross section.
b.
5
EXAMPLE
Drawing a Cross Section
Visualization Draw and describe a cross
section formed by a vertical plane intersecting
the front and right faces of the cube.
triangle
5 Draw and describe a cross
section formed by a vertical plane
intersecting the top and bottom
faces of a cube. Check students’
work; square or rectangle.
Resources
• Daily Notetaking Guide 11-1
A vertical plane cuts the vertical faces
of the cube in parallel segments.
Draw the parallel segments.
5. square
Join their endpoints. Shade the cross section.
L3
• Daily Notetaking Guide 11-1—
L1
Adapted Instruction
Closure
What is a polyhedron and how is
Euler’s Formula related to it?
A polyhedron is a threedimensional figure whose
surfaces are polygons. Euler’s
Formula relates the number of
faces (F), vertices (V), and edges
(E) of a polyhedron such that
F ± V ≠ E ± 2.
600
The cross section is a rectangle.
Quick Check
600
5 Draw and describe the cross section formed by a horizontal plane intersecting the
left and right faces of the cube. See left.
Chapter 11 Surface Area and Volume
17. rectangle
18. square
19. rectangle
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
3. Practice
Practice and Problem Solving
Assignment Guide
Practice by Example
Example 1
For each polyhedron, how many vertices, edges, and faces are there? List them.
M
1.
2.
4, 6, 4
(page 598)
P
(page 599)
U
P
D
V
F
E
O
3.
C
A
1–3. See back of
book for lists.
N
Example 2
B
Q
G
H
S
8, 12, 6
R
X
W
Y
(page 599)
Example 4
(page 600)
11.
56-60
61-68
To check students’ understanding
of key skills and concepts, go over
Exercises 8, 16, 22, 30, 32.
6. Faces: 20 12
Edges: 30
Vertices: j
Exercises 13–15 As a class,
explore how the shape of the
cross section changes with the
orientation of the plane.
Verify Euler’s Formula for each polyhedron. Then draw a net for the ﬁgure and
verify Euler’s Formula for the two-dimensional ﬁgure. 10–12. See back of book.
10.
13-19, 21-26, 39-45
C Challenge
46-55
Homework Quick Check
5. Faces: 8 12
Edges: j
Vertices: 6
Use Euler’s Formula to ﬁnd the number of vertices in each polyhedron
described below.
5
9
7. 6 square faces
8. 5 faces: 1 rectangle
9. 9 faces: 1 octagon
8
and 4 triangles
and 8 triangles
Example 3
2 A B
Test Prep
Mixed Review
T
10, 15, 7
Use Euler’s Formula to ﬁnd the missing number.
4. Faces: j
8
Edges: 15
Vertices: 9
1 A B 1-12, 20, 27-38
12.
Exercise 19 Draw a cube on the
board, using a different color
for each set of opposite edges.
Point out that opposite edges
are parallel and are not on the
same face.
Describe each cross section. 13. two concentric circles
13.
14.
15.
rectangle
GPS Guided Problem Solving
L3
L4
Enrichment
triangle
16. For the nut shown, sketch each of following.
a. a horizontal cross section a–b. See back of book.
b. a vertical cross section that contains the vertical
line of symmetry
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
L3
Date
Practice 10-1
Space Figures and Nets
1. Choose the nets that will fold to make a cube.
A.
Example 5
(page 600)
Visualization Draw and describe a cross section formed
by a vertical plane intersecting the cube as follows.
17–19. See margin.
17. The vertical plane intersects the front and left faces of the cube.
B.
C.
D.
Draw a net for each ﬁgure. Label each net with its appropriate dimensions.
2.
16 cm
7 cm
2 cm
3.
8 cm
4.
1 cm
2 cm
32 cm
1 cm
40 cm
Match each three-dimensional ﬁgure with its net.
5.
6.
7.
8.
18. The vertical plane intersects opposite faces of the cube.
A.
19. The vertical plane contains opposite edges of the cube.
B.
C.
D.
9. Choose the nets that will fold to make a pyramid with a square base.
A.
Lesson 11-1 Space Figures and Cross Sections
B.
C.
D.
601
© Pearson Education, Inc. All rights reserved.
A
Use Euler’s Formula to ﬁnd the missing number.
10. Faces: 5
Edges: 7
Vertices: 5
11. Faces: 7
Edges: 9
Vertices: 6
12. Faces: 8
Edges: 18
Vertices: 7
601
Exercise 38 Ask: What do you
know about a cube that might
help you solve this problem?
A cube has 6 square faces.
Connection to Calculus
B
Apply Your Skills
21. rectangle
Exercises 27–29 Formulas for the
volumes of more complicated
solids of revolution are developed
in calculus.
20. a. Open-Ended Sketch a polyhedron whose faces are all rectangles. Label the
lengths of its edges. a–b. See back of book.
b. Use graph paper to draw two different nets for the polyhedron.
Visualization Draw and describe a cross section formed
by a plane intersecting the cube as follows. 21–23. See left.
21. The plane is tilted and intersects the left and right faces
of the cube.
22. The plane contains opposite horizontal edges of the cube.
Connection to Astronomy
22. rectangle
23. The plane cuts off a corner of the cube.
Exercise 30 Early scientists used
Describe the cross section shown.
triangle
circle
24.
25.
Platonic solids to attempt to
explain the universe. Have
students investigate some of
these explanations.
26.
2 trapezoids
23. triangle
Visualization A plane region that revolves completely about a line sweeps out a
solid of revolution. Use the sample to help you describe the solid of revolution you
get by revolving each region about line <.
Sample: Revolve the
rectangular region about the
line / and you get a cylinder
as a solid of revolution.
27.
ᐉ
ᐉ
28.
ᐉ
ᐉ
cylinder attached to a cone
29.
ᐉ
sphere
cone
Sports Equipment Some balls are made from panels that
suggest polygons. The ball then suggests a polyhedron
to which Euler’s Formula, F ± V ≠ E ± 2, applies.
30. A soccer ball suggests a polyhedron with 20 regular
hexagons and 12 regular pentagons. How many
vertices does this polyhedron have? 60
31. Show how Euler’s Formula applies to the polyhedron
suggested by the volleyball pictured on page 599. (Hint: It has 6 sets of 3 panels.)
18 + 32 = 48 + 2
Euler’s Formula F ± V ≠ E ± 1 applies to any two-dimensional network where F
is the number of regions formed by V vertices linked by E edges (or paths). Verify
Euler’s Formula for each network shown.
46.
GPS 32.
47.
GO
48.
49.
602
50.
34.
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-1101
602
33.
6+4=9+1
4+6=9+1
35. Draw a network of your own. Verify Euler’s Formula for it.
Check students’ work.
Chapter 11 Surface Area and Volume
51.
52.
53.
54.
5+5=9+1
36. There are ﬁve regular polyhedrons. They are called regular because all their
faces are congruent regular polygons, and the same number of faces meet at
each vertex. They are also called Platonic Solids after the Greek philosopher
Plato (427–347 B.C.).
4. Assess & Reteach
PowerPoint
Lesson Quiz
1. Draw a net for the figure.
Tetrahedron
Octahedron
Hexahedron
Real-World
Connection
A ﬂuorite crystal forms as a
regular octahedron.
Icosahedron
Sample:
Dodecahedron
a. Match each net below with a Platonic Solid.
A.
B.
C.
D.
E.
A. icosahedron
B. octahedron
C. tetrahedron
D. hexahedron
36b. regular triangular pyramid, cube E. dodecahedron
b. The ﬁrst two Platonic solids have more familiar names. What are they?
c. Verify that Euler’s Formula is true for the ﬁrst three Platonic solids.
4 + 4 = 6 + 2, 6 + 8 = 12 + 2, 8 + 6 = 12 + 2
37. Multiple Choice A cube has a net with area 216 in.2. How long is an edge of
the cube? A
6 in.
15 in.
36 in.
54 in.
Use Euler’s Formula to solve.
2. A polyhedron with 12 vertices
and 30 edges has how many
faces? 20
3. A polyhedron with 2 octagonal
faces and 8 rectangular faces
has how many vertices? 16
4. Describe the cross section.
Draw each object. Then draw a horizontal and a vertical cross section.
38. a golf tee
39. a football
40. a baseball bat
41. a banana
42. a pear
43. a bagel
38–43. Check students’ work.
44. Writing Cross sections are used in medical training and research. Research and
write a paragraph on how magnetic resonance imaging (MRI) is used to study
cross sections of the brain. Check students’ work.
C
Challenge
45. Draw a solid that has the following cross sections.
45.
circle
5. Draw and describe a cross
section formed by a vertical
plane cutting the left and back
faces of a cube. Check
students’ drawings; rectangle.
Alternative Assessment
horizontal
vertical
Visualization Draw a plane intersecting a cube to get the cross section indicated.
46. scalene triangle
47. isosceles triangle
48. equilateral triangle
49. trapezoid
50. isosceles trapezoid
51. parallelogram
52. rhombus
lesson quiz, PHSchool.com, Web Code: aua-1101
53. pentagon
46–54. See margin.
Have each student bring a realworld polyhedron to class. Have
them verify Euler’s Formula and
then draw a net for the solid.
54. hexagon
Lesson 11-1 Space Figures and Cross Sections
603
603
Test Prep
Test Prep
Resources
Multiple Choice
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 657
• Test-Taking Strategies, p. 652
• Test-Taking Strategies with
Transparencies
For Exercises 55–56, you may need Euler’s Formula, F + V = E + 2.
55. A polyhedron has four vertices and six edges. How many faces does it have?
B
A. 2
B. 4
C. 5
D. 10
56. A polyhedron has three rectangular faces and two triangular faces.
How many vertices does it have? G
F. 5
G. 6
H. 10
J. 12
57. The plane is horizontal. What best describes
the shape of the cross section? D
A. rhombus
B. trapezoid
C. parallelogram
D. square
58. The plane is vertical. What best describes
the shape of the cross section? J
F. pentagon
G. square
H. rectangle
J. triangle
59. [2] a. square
Short Response
b. Answers may vary.
Sample: trapezoid
59. Draw and describe a cross section formed
by a plane intersecting a cube as follows.
a. The plane is parallel to a horizontal face of the cube.
b. The plane cuts off two corners of the cube.
a–b. See margin.
Mixed Review
[1] only 1 correct
drawing
GO for
Help
Lesson 10-8
60. Probability A shuttle bus to an airport terminal leaves every 20 min from a
remote parking lot. Draw a geometric model and ﬁnd the probability that a
traveler who arrives at a random time will have to wait at least 8 min for the
60%
bus to leave the parking lot.
0 4 8 12 16 20
61. Games A dartboard is a circle with a 12-in. radius. You throw a dart that hits
the dartboard. What is the probability that the dart lands within 6 in. of the
center of the dartboard? 25%
Lesson 10-3
Lesson 8-3
Find the area of each equilateral triangle with the given measure. Leave answers in
simplest radical form.
62. side 2 ft
63. apothem 8 cm
192"3 cm2
"3 ft2
Find the value of x to the nearest tenth.
65.
4.7
64. radius 100 in.
7500"3 in.2
66. 8.3
65⬚
x
x
6
36⬚
10
67. The lengths of the diagonals of a rhombus are 4 cm and 6 cm. Find the
measures of the angles of the rhombus to the nearest degree. 67 and 113
604
604
Chapter 11 Surface Area and Volume
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