GUIDED PRACTICE
✓
Concept Check ✓
Skill Check ✓
Vocabulary Check
3 APPLY
1. Define polyhedron in your own words. Sample answer: A polyhedron is a solid
figure bounded by faces that are polygons and enclose a single region of space.
2. Is a regular octahedron convex? Are all the Platonic solids convex? Explain.
4.
BASIC
Day 1: pp. 723–726 Exs. 10–30
even, 32–35, 43–52, 54, 55,
60–70 even
Decide whether the solid is a polyhedron. Explain.
2. Yes; yes; any 2 points on the
surface of any Platonic solid 3.
can be connected by a line
segment that lies entirely
inside or on the solid.
ASSIGNMENT GUIDE
5.
AVERAGE
3. Yes; the figure is a solid that
is bounded by polygons that
enclose a single region of
space.
Use Euler’s Theorem to find the unknown number.
4. Yes; the figure is a solid that
?㛭
?㛭
7. Faces: 5
8. Faces: 㛭㛭㛭㛭㛭㛭
8
7
is bounded by polygons that 6. Faces: 㛭㛭㛭㛭㛭㛭
?㛭
enclose a single region of
Vertices: 6
Vertices: 㛭㛭㛭㛭㛭㛭
Vertices: 10
6
space.
Edges: 12
Edges: 9
Edges: 15
Day 1: pp. 723–726 Exs. 10–30
even, 32–35, 42–52, 54, 55,
60–70 even
ADVANCED
Day 1: pp. 723–726 Exs. 10–30
even, 32–35, 42–52, 54–59,
60–70 even
9. Faces: 20
Vertices: 12
?㛭
Edges: 㛭㛭㛭㛭㛭㛭
30
5. No; it does not have faces
that are polygons.
BLOCK SCHEDULE
(with Ch. 11 Assess) pp. 723–726
Exs. 10–30 even, 32–35, 42–52,
54, 55, 60–70 even
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 825.
EXERCISE LEVELS
Level A: Easier
10–18, 25–31, 36–41, 60–62, 68–70
Level B: More Difficult
19–24, 32–35, 43–55, 63–67
Level C: Most Difficult
42, 56–69
IDENTIFYING POLYHEDRA Tell whether the solid is a polyhedron. Explain
your reasoning.
10.
11.
12.
10. No; some of its faces are
not polygons.
11. Yes; the figure is a solid
that is bounded by
polygons that enclose a
single region of space.
12. Yes; the figure is a solid
that is bounded by
polygons that enclose a
single region of space.
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 10,
14, 18, 24, 28, 34, 44, 48. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 12
Resource Book, p. 23)
•
Transparency (p. 87)
ANALYZING SOLIDS Count the number of faces, vertices, and edges of
the polyhedron.
13.
14.
15.
5, 5, 8
STUDENT HELP
HOMEWORK HELP
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
Example 6:
Exs. 10–15
Exs. 16–24
Exs. 25–35
Exs. 36–52
Ex. 53
Exs. 47–52
10, 16, 24
8, 12, 18
ANALYZING POLYHEDRA Decide whether the polyhedron is regular
and/or convex. Explain. 16–18. See margin.
16.
17.
COMMON ERROR
EXERCISE 17 Some students
may think the polyhedron is regular
because all of its faces are regular
polygons. In this case, however,
they are not all congruent, as is
required.
18.
16–18. See next page.
12.1 Exploring Solids
723
723
TEACHING TIPS
EXERCISES 25–28 To help students visualize cross sections when
a plane intersects a solid, demonstrate with actual objects that can
be cut. For example, you could use
solids made of styrofoam or clay
(cut with a very fine wire).
MATHEMATICAL REASONING
EXERCISE 26 The idea of intersecting a cone by a plane at various
angles can be used to introduce the
conic sections: the circle, ellipse,
parabola, and hyperbola. For the
hyperbola, use two cones sharing
the same vertical axis, touching
only at their vertices.
16. Regular, convex; the 6 faces of the
polyhedron are • squares; any
two points on the surface of the
polyhedron can be connected by
a line segment that lies entirely
inside or on the polyhedron.
17. Not regular, convex; the faces of
the polyhedron are not congruent
(two are hexagons and six are
squares); any two points on the
surface of the polyhedron can be
connected by a line segment that
lies entirely inside or on the polyhedron.
18. Not regular, not convex; the faces
of the polyhedron are not congruent (two are squares, eight are
trapezoids, and four are rectangles); some of the segments connecting a vertex of one square
face to a vertex of the other lie
outside the polyhedron.
33. yes;
LOGICAL REASONING Determine whether the statement is true or
false. Explain your reasoning.
19. False; see the
octahedron in part (b)
of Example 2 on page
720.
20. False; each face of a
polyhedron is a
polygon, so each face
has at least 3 sides
and therefore must
intersect with at least
3 other faces. Thus a
polyhedron must have
at least 4 faces.
21. True; the faces are £
squares.
19. Every convex polyhedron is regular.
20. A polyhedron can have exactly 3 faces.
21. A cube is a regular polyhedron.
22. A polyhedron can have exactly 4 faces.
23. A cone is a regular polyhedron.
24. A polyhedron can have exactly 5 faces.
CROSS SECTIONS Describe the cross section.
25. circle
26. circle
27. pentagon
28. rectangle
22. True; see the regular
tetrahedron on page
721.
23. False; it does not have
polygonal faces.
24. True; see the pyramid
on page 719.
COOKING Describe the shape that is formed by the cut made in the
food shown.
29. Carrot circle
30. Cheese rectangle or
square
31. Cake rectangle
CRITICAL THINKING In the diagram, the bottom face of the pyramid is
a square.
32. Name the cross section shown. pentagon
33. Can a plane intersect the pyramid at a point?
If so, sketch the intersection. See margin.
34. Describe the cross section when the pyramid
is sliced by a plane parallel to its bottom face.
square
35. Is it possible to have an isosceles trapezoid
as a cross section of this pyramid? If so,
draw the cross section. See margin.
POLYHEDRONS Name the regular polyhedron.
36.
37.
38.
35. yes;
tetrahedron
724
724
Chapter 12 Surface Area and Volume
octahedron
dodecahedron
FOCUS ON
CAREERS
CRYSTALS In Exercises 39–41, name the Platonic solid that the crystal
resembles.
39. Cobaltite dodecahedron 40. Fluorite octahedron
CAREER NOTE
EXERCISES 39–41
Additional information about
mineralogists is available at
www.mcdougallittell.com.
41. Pyrite cube
STUDENT HELP NOTES
Homework Help Students can
find help for Exs. 47–52 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
42. VISUAL THINKING Sketch a cube and describe the figure that results from
connecting the centers of adjoining faces. regular octahedron
RE
FE
L
AL I
MINERALOGY
INT
By studying the
arrangement of atoms in a
crystal, mineralogists are
able to determine the
chemical and physical
properties of the crystal.
EULER’S THEOREM In Exercises 43–45, find the number of faces, edges,
and vertices of the polyhedron and use them to verify Euler’s Theorem.
43–45. See margin.
43.
44.
45.
42.
NE
ER T
CAREER LINK
www.mcdougallittell.com
46. MAKING A TABLE Make a table of the number of faces, vertices, and edges
for the Platonic solids. Use it to show Euler’s Theorem is true for each solid.
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Exs. 47–52.
See margin.
USING EULER’S THEOREM In Exercises 47–52, calculate the number of
vertices of the solid using the given information.
47. 20 faces;
12 vertices
all triangles
48. 14 faces; 12 vertices
8 triangles and
6 squares
49. 14 faces; 24 vertices
8 hexagons and
6 squares
43. 5 faces, 6 vertices,
9 edges; 5 + 6 = 9 + 2
46. For each solid, the number of
faces, vertices, edges, and the
verification of Euler’s Theorem
are given in that order.
tetrahedron: 4, 4, 6, 4 + 4 = 6 + 2
cube: 6, 8, 12, 4 + 4 = 6 + 8 = 12 + 2
octahedron: 8, 6, 12, 8 + 6 = 12 + 2
dodecahedron: 12, 20, 30, 12 + 20 =
30 + 2
icosahedron: 20, 12, 30, 20 + 12 =
30 + 2
44. 5 faces, 5 vertices,
8 edges; 5 + 5 = 8 + 2
45. 5 faces, 6 vertices,
9 edges; 5 + 6 = 9 + 2
50. 26 faces; 18 squares
51. 8 faces; 4 hexagons
52. 12 faces;
and 8 triangles 24 vertices and 4 triangles 12 vertices all pentagons 20 vertices
ADDITIONAL PRACTICE
AND RETEACHING
53.
For Lesson 12.1:
In molecules of
cesium chloride, chloride atoms are arranged
like the vertices of cubes. In its crystal
structure, the molecules share faces to form
an array of cubes. How many cesium
chloride molecules share the faces of a given
cesium chloride molecule? 6 molecules
SCIENCE
CONNECTION
12.1 Exploring Solids
725
• Practice Levels A, B, and C
(Chapter 12 Resource Book,
p. 13)
• Reteaching with Practice
(Chapter 12 Resource Book,
p. 16)
•
See Lesson 12.1 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
725
Test
Preparation
4 ASSESS
54. MULTIPLE CHOICE A polyhedron has 18 edges and 12 vertices. How many
faces does it have? C
A
¡
DAILY HOMEWORK QUIZ
B
¡
4
C
¡
6
8
D
¡
E
¡
10
12
55. MULTIPLE CHOICE In the diagram, Q and S
Transparency Available
1. Determine whether the statement A cylinder is a convex
polyhedron is true or false.
Explain your reasoning.
are the midpoints of two edges of the cube.
Æ
What is the length of QS, if each edge of the
cube has length h? B
A
¡
False; a cylinder is not a polyhedron because its faces are
not polygons.
★ Challenge
2. Describe the cross section.
EXTRA CHALLENGE
www.mcdougallittell.com
D
¡
h
ᎏᎏ
2
B
¡
兹2苶h
E
¡
C
¡
h
ᎏ
兹2苶
R
2h
ᎏ
兹2苶
S
q
T
2h
SKETCHING CROSS SECTIONS Sketch the intersection of a cube and a
plane so that the given shape is formed. 56–59. See margin.
56. An equilateral triangle
57. A regular hexagon
58. An isosceles trapezoid
59. A rectangle
MIXED REVIEW
FINDING AREA OF QUADRILATERALS Find the area of the figure.
(Review 6.7 for 12.2)
735 m 2
280 ft2
96 in.2
60.
61.
62.
14 ft
trapezoid
3. Find the number of faces,
edges, and vertices of the
polyhedron and use them to
verify Euler’s Theorem.
15 m
8 in.
17 m
32 m
16 ft
15 m
21 ft
12 in.
FINDING AREA OF REGULAR POLYGONS Find the area of the regular
polygon described. Round your answer to two decimal places.
(Review 11.2 for 12.2)
7, 15, 10; 7 + 10 = 15 + 2
63. An equilateral triangle with a perimeter of 48 meters and an apothem of
4.6 meters. 110.40 m2
4. A solid has 14 faces:
6 octagons and 8 triangles.
How many vertices does it
have? 24
64. A regular octagon with a perimeter of 28 feet and an apothem of 4.22 feet.
59.08 ft2
65. An equilateral triangle whose sides measure 8 centimeters. 27.71 cm2
66. A regular hexagon whose sides measure 4 feet. 41.57 ft2
67. A regular dodecagon whose sides measure 16 inches. 2866.22 in.2
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 12.1 are available in
blackline format in the Chapter 12
Resource Book, p. 20 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Use Euler’s Theorem
to explain why there can be no
polyhedron with six vertices and
seven edges.
FINDING AREA Find the area of the shaded region. Round your answer
to two decimal places. (Review 11.5)
68.
69.
115⬚
7 cm
49.17 cm2
726
Chapter 12 Surface Area and Volume
See sample answer at right.
56–59. See Additional Answers
beginning on page AA1.
726
Additional Test Preparation Sample answer:
1. By Euler’s Theorem, F + V = E + 2, so F + 6 = 7 + 2,
which gives F = 3. Since you cannot enclose any
region in three-dimensional space with only three
polygons, the polyhedron cannot exist.
70.
140ⴗ
43 ft
5808.80 ft2
32 in.
357.44 in.2