9A Two-Dimensional Geometry

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Why Learn This?
396 Chapter 9
Geometry and
Measurement
You can find the surface areas of
composite shapes in architecture,
including many of Florida’s
famous forts.
Castillo de San Marcos
St. Augustine
MA.8.G.2.2
MA.8.G.2.2
MA.8.G.2.3
MA.8.G.2.3
Rev MA.7.G.2.1
Rev MA.7.G.2.1
Rev MA.7.G.2.1
Rev MA.7.G.2.1
MA.8.G.5.1
MA.8.G.5.1
9A Two-Dimensional
Geometry
9-1 Angle Relationships
9-2 Parallel and Perpendicular
Lines
9-3 Triangles
9-4 Polygons
9B Three-Dimensional
Geometry
9-5 Volume of Prisms and
Cylinders
9-6 Volume of Pyramids and
Cones
9-7 Surface Area of Prisms
and Cylinders
9-8 Surface Area of Pyramids
and Cones
9-9 Scaling Three-Dimensional
Figures
9-10 Measurement in Three-
Dimensional Figures
9A Two-Dimensional
C H A P T E R
9
H A
H
H
C
C A
A
A P T E R
R
R
R
FLORIDA
• Find measures of angles
formed by lines and in
polygons.
• Convert areas and volumes
between measurement systems.
Go to thinkcentral.com

Geometry and Measurement 397
Are You Ready?
Are You Ready?
Vocabulary
Choose the best term from the list to complete each sentence.
1. A(n) __?__ is a number that represents a part of a whole.
2. A(n) __?__ is another way of writing a fraction.
3. To multiply 7 by the fraction ᎏ
2
3
ᎏ, multiply 7 by the __?__ of the
fraction and then divide the result by the __?__ of the fraction.
4. To round 7.836 to the nearest tenth, look at the digit in
the __?__ place.
Complete these exercises to review skills you will need for this chapter.
Square and Cube Numbers
Evaluate.
5. 162
6. 93
7. (4.1)2
8. (0.5)3
9. (1
__
4)2
10. (2
__
5)2
11. (1
__
2)3
12. (2
__
3)3
Multiply with Fractions
Multiply.
13. 1
__
2
(8)(10) 14. 1
__
2
(3)(5) 15. 1
__
3
(9)(12) 16. 1
__
3
(4)(11)
17. 1
__
2
(82)16 18. 1
__
2
(52)24 19. 1
__
2
(6)(3 ⴙ 9) 20. 1
__
2
(5)(7 ⴙ 4)
Multiply with Decimals
Multiply.Write each answer to the nearest tenth.
21. 2(3.14)(12) 22. 3.14(52) 23. 3.14(42)(7) 24. 3.14(2.32)(5)
Multiply with Fractions and Decimals
Multiply.Write each answer to the nearest tenth.
25. 1
__
3
(3.14)(52)(7) 26. 1
__
3
(3.14)(53)
27. 1
__
3
(3.14)(3.2)2(2) 28. 4
__
3
(3.14)(2.7)3
29. 1
__
5(22
__
7 )(42
)(5) 30. 4
__
11(22
__
7 )(3.23)
31. 1
__
2(22
__
7 )(1.7)2(4) 32. 7
__
11(22
__
7 )(9.5)3
V
Choos
decimal
denominator
fraction
tenths
hundredths
numerator
Are You Ready?

Why?
Now
Before
398 Chapter 9
Study
Guide:
Preview
Study Guide: Preview
Key
Vocabulary/Vocabulario
equilateral triangle triángulo equilátero
lateral surface superficie lateral
parallel lines rectas paralelas
perpendicular lines rectas perpendiculares
polygon polígono
surface area área total
transversal transversal
volume vólumen
Vocabulary Connections
To become familiar with some of the
vocabulary terms in the chapter, consider the
following. You may refer to the chapter, the
glossary, or a dictionary if you like.
1. The word equilateral contains the roots
equi, which means “equal,” and lateral,
which means “of the side.” What do you
suppose an equilateral triangle is?
2. The Greek prefix poly means “many,” and
the root gon means “angle.” What do you
suppose a polygon is?
3. The word lateral is from the Latin lateralis,
meaning “of the side.” A lateral in football
is a pass to the side. What do you think
the lateral surface of a cone or cylinder
might be?
Previously, you
• recognized geometric concepts
and properties in fields such as
art and architecture.
• studied the concept of ␲ and
applied it to real-world situations.
• found perimeters and areas of
plane figures and composite
plane figures.
B
B
FLORIDA
C H A P T E R
H A
A P
P
P T E
9
You will study
• finding angle measures by using
relationships within figures.
• describing the effects on
perimeter, area, and volume
when the dimensions of a figure
change proportionally.
• describing the effect on
measurements when the
dimensions are converted
between measurement systems.
You can use the skills
learned in this chapter
• to find angle measures by using
relationships within figures.
• to determine whether parts of a
structure are parallel.
• to compare customary and
metric measurements of area
and volume.

January 27
I’m having trouble with Lesson 6-5. I can find what
percent one number is of another number, but I get
confused about finding percents of increase and
decrease. My teacher helped me think it through:
Find the percent of increase or decrease from 20 to 25.
• First figure out if it is a percent of increase or
decrease. It goes from a smaller to a larger number, so
it is a percent of increase because the number is getting
larger, or increasing.
• Then find the amount of increase, or the difference,
between the two numbers. 25 ⫺ 20 ⫽ 5
• Now find what percent the amount of increase, or
difference, is of the original number.
amount of increase
_____________
original number
→
5
__
20
⫽ 0.25 ⫽ 25%
So it is a 25% increase.
Try This
Geometry and Measurement 399
Reading
and
Writing
Math
FLORIDA
C H A P T E R
H A
A P T
T E R
R
R
R
R
R
R
R
9
Writing Strategy: Keep a Math Journal
By keeping a math journal, you can improve your writing
and thinking skills. Use your journal to summarize key ideas
and vocabulary from each lesson and to analyze any questions
you may have about a concept or your homework.
Journal Entry: Read the entry a student made in her journal.
Begin a math journal.Write in it each day this week, using these
ideas as starters. Be sure to date and number each page.
■ In this lesson, I already know . ..
■ In this lesson, I am unsure about . ..
■ The skills I need to complete this lesson are ...
■ The challenges I encountered were ...
■ I handled these challenges by . ..
■ In this lesson, I enjoyed/did not enjoy . ..
LA.8.2.2.3 The student
will organize information
to show understanding
or relationships among facts and
ideas... (e.g., representing key points
within text through… paraphrasing,
summarizing...).

Y
1
Z
X
65
25
o
o
65 115
o o
The measure of an
acute angle is greater
than 0° and less than 90°.
The measure of a
straight angle is 180°.
The measure of a
right angle is 90°.
Complementary angles
are two angles whose
measures add to 90°.
The measure of an
obtuse angle is greater
than 90° and less than 180°.
Supplementary angles
are two angles whose
measures add to 180°.
A
C
30°
90°
60°
D
E B
400 Chapter 9 Geometry and Measurement Lesson Tutorial Videos @ thinkcentral.com
Vocabulary
angle
right angle
acute angle
obtuse angle
straight angle
complementary
angles
supplementary
angles
adjacent angles
congruent angles
vertical angles
An angle (⬔) is formed by two rays, or sides, with a common
endpoint called the vertex. You can name an angle several
ways: by its vertex, by its vertex and a point on each ray, or
by a number. When three points are used, the middle
point must be the vertex.
Angles are usually measured in degrees (°). Since there are
360° in a circle, one degree is ᎏ
3
1
60
ᎏ of a circle.
Classifying Angles
Use the diagram to name each figure.
two acute angles
⬔AED, ⬔CEB m⬔AED ⴝ 60°;
m⬔CEB ⴝ 30°
two obtuse angles
⬔AEC, ⬔DEB m⬔AEC ⴝ 150°; m⬔DEB ⴝ 120°
a pair of complementary angles
⬔AED, ⬔CEB m⬔AED ⴙ m⬔CEB ⴝ 60° ⴙ 30° ⴝ 90°
two pairs of supplementary angles
⬔AED, ⬔DEB m⬔AED ⴙ m⬔DEB ⴝ 60° ⴙ 120° ⴝ 180°
⬔AEC, ⬔CEB m⬔AEC ⴙ m⬔CEB ⴝ 150° ⴙ 30° ⴝ 180°
E X A M P L E 1
A
B
C
D
Angle Relationships
An angl
9-1
MA.8.G.2.2
Classify and
determine the
measure of angles,....
m⬔AEC is read as
“the measure of
angle AEC.”
⬔Y, ⬔XYZ,
⬔ZYX, or ⬔1

1
4
2
3
1
4
2
3
2 4
3
1
9-1 Angle Relationships 401
Lesson Tutorial Videos @ thinkcentral.com
You can use what you know about complementary and
supplementary angles to find missing angle measurements.
Finding Angle Measures
Use the diagram to find each angle measure.
If m⬔2 ⴝ 75ⴗ, find m⬔3.
m⬔2 ⴙ m⬔3 ⴝ 180° ⬔2 and ⬔3 are supplementary.
75° ⴙ m⬔3 ⴝ 180° Substitute 75° for m⬔2.
ⴚ75° ⴚ75° Subtract 75° from both sides.
m⬔3 ⴝ 105°
Find m⬔4.
m⬔3 ⴙ m⬔4 ⴝ 180° ⬔3 and ⬔4 are supplementary.
105° ⴙ m⬔4 ⴝ 180° Substitute 105° for m⬔3.
ⴚ105° ⴚ105° Subtract 105° from both sides.
m⬔4 ⴝ 75°
Check
Use a protractor to measure ⬔3 and ⬔4.
The measurements from the protractors provide support that
m⬔3 ⴝ 105° and m⬔4 ⴝ 75°.
The angles in Example 2 are examples of adjacent angles and vertical
angles. These angles have special relationships because of their positions.
• Adjacent angles have a common vertex and a common side, but
no common interior points. Angles 1 and 2 in the diagram above
are adjacent angles.
• Congruent angles have the same measure. In Example 2, you
found that ⬔2 ⬵ ⬔4 since both angles measure 75°.
• Vertical angles are the nonadjacent angles formed by two
intersecting lines. Angles 2 and 4 are vertical angles. Vertical
angles are congruent.
E X A M P L E 2
A
B
The symbol for
congruence is ⬵,
which is read as “is
congruent to.”
Diagrams are not
always drawn to
scale. When solving
problems about
angles, it is best to
find your answers
mathematically rather
than by measuring.

A
D
F
B
C
E
32ⴗ
E
A
B C
D
50°
40°
90°
1
2
3
4
Exercises 1–16, 17, 25, 27
Homework Help
402 Chapter 9 Geometry and Measurement
Exercises MA.8.G.2.2
E
E
9-1
Think and Discuss
1. Draw a pair of angles that are adjacent but not supplementary.
2. Explain why vertical angles must always be congruent.
Art Application
An artist is designing a section of a
stained glass window. Based on the
diagram, what should be the measure
of ⬔ECF?
Step 1: Find m⬔DCE.
⬔DCE 艑 ⬔ACB Vertical angles are congruent.
m⬔DCE ⴝ m⬔ACB Congruent angles have the same measure.
m⬔DCE ⴝ 32° Substitute 32° for m⬔ACB.
Step 2: Find m⬔ECF.
m⬔DCE ⴙ m⬔ECF ⴝ 90° The angles are complementary.
32° ⴙ m⬔ECF ⴝ 90° Substitute 32° for m⬔DCE.
m⬔ECF ⴝ 58° Subtract 32° from both sides.
E X A M P L E 3
1. a right angle 2. two acute angles
3. an obtuse angle 4. a pair of complementary angles
5. two pairs of supplementary angles
Use the diagram to find each angle measure.
6. If m⬔3 ⴝ 114°, find m⬔2.
7. Find m⬔1.
See Example 1
See Example 2
GUIDED PRACTICE

V
W
X
Y
Z
60°
30°
A
F
B
C
D
E
62ⴗ
D
E
45⬚
A
C
B
1
2
3
4
Q
U
T
N
R
P
S
6
1
3
5
2
4
9-1 Angle Relationships 403
WORKED-OUT SOLUTIONS
on p. WS11
ⴝ
8. Engineering The diagram shows the
intersection of three metal supports on
an amusement park ride. Based on the
diagram, what should be the measure
of ⬔CBD?
9. a right angle
10. two acute angles
11. two obtuse angles
12. a pair of complementary angles
13. two pairs of supplementary angles
Use the diagram to find each angle
measure.
14. If m⬔2 ⴝ 126ⴗ, find m⬔3.
15. Find m⬔4.
16. Sewing The diagram shows the intersection
of three seams on a quilt. Based on the diagram,
what should be the measure of ⬔ABC?
Use the figure for Exercises 17–23. Write true or false. If a statement is false,
rewrite it so it is true.
17. ⬔QUR is an obtuse angle.
18. ⬔4 and ⬔2 are supplementary.
19. ⬔1 and ⬔6 are supplementary.
20. ⬔3 and ⬔1 are complementary.
21. If m⬔1 ⴝ 35°, then m⬔6 ⴝ 40°.
22. If m⬔SUN ⴝ 150°, then m⬔SUR ⴝ 150°.
23. If m⬔1 ⴝ x°, then m⬔PUQ ⴝ 180° ⴚ x°.
24. Make a Conjecture ⬔A and ⬔B are both supplementary to the same
angle. Describe the relationship between ⬔A and ⬔B. Justify your answer.
25. Critical Thinking The measures of two complementary angles have a
ratio of 1:2. What is the measure of each angle?
See Example 3
INDEPENDENT PRACTICE
See Example 1
See Example 2
See Example 3
PRACTICE AND PROBLEM SOLVING

Physical Science
MA.8.G.2.2, MA.8.A.6.3, MA.8.A.6.4
Florida Spiral Review
30. Multiple Choice When two angles are complementary, what is the sum
of their measures?
A. 90° B. 180° C. 270° D. 360°
31. Gridded Response ⬔1 and ⬔3 are supplementary angles. If m⬔1 ⴝ 63°,
find m⬔3.
Multiply. Write the product as one power. (Lesson 4-4)
32. m3
ⴢ m2
33. w ⴢ w6
34. 78
ⴢ 73
35. 116
ⴢ 119
Find each percent of increase or decrease to the nearest percent. (Lesson 6-5)
36. from 50 to 47 37. from 150 to 147
38. from 50 to 24 39. from 24 to 50
The archerfish can spit a stream of water up to 3 meters in
the air to knock its prey into the water. This job is made
more difficult by refraction, the bending of light waves as
they pass from one substance to another. When you look
at an object through water, the light between you and the
object is refracted. Refraction makes the object appear to
be in a different location. Despite refraction, the archerfish
still catches its prey.
26. Suppose that the measure of the angle between the bug’s
actual location and the bug’s apparent location is 35°.
a. Refer to the diagram. Along the fish’s line of vision,
what is the measure of the angle between the fish and
the bug’s apparent location?
b. What is the relationship of the angles in the diagram?
27. In the image, the underwater part of the net appears to be 40° to the right
of where it actually is. What is the measure of the angle formed by the
image of the underwater part of the net and the part of the net
above the water?
28. Write About It The handle of the net in the diagram is
perpendicular to the water’s surface. Explain how to find the measure
of the acute angle that the underwater part of the net appears to make
with the water’s surface.
29. Challenge ⬔1 is supplementary to ⬔2, and ⬔2 is complementary
to ⬔3. Classify each angle as acute, right, or obtuse, and justify your
answer.
Physical Science

The sides of the windows are transversals to the top and bottom.
The top and
bottom of the
windows
are parallel.
1 2
3 4
5 6
7 8
1 2
3 4
5 6
7 8
9-2 Parallel and Perpendicular Lines 405
Parallel lines are lines in a
plane that never meet, such
as the lines that mark the
lanes in a sprinting race.
The top and bottom sides
of a window are also
parallel. However, the left
or right side and the bottom
of a window are like
perpendicular lines; that is,
they intersect at 90° angles.
A transversal is a line that intersects two or more lines that lie in the same
plane. Transversals to parallel lines form angles with special properties.
Identifying Congruent Angles Formed by a Transversal
Copy and measure the angles
formed by the transversal and
the parallel lines. Which angles
seem to be congruent?
⬔1, ⬔4, ⬔5, and ⬔8 all
measure 60°.
⬔2, ⬔3, ⬔6, and ⬔7 all
measure 120°.
Angles marked in blue appear
congruent to each other, and
angles marked in red appear
congruent to each other.
⬔1 ⬵ ⬔4 ⬵ ⬔5 ⬵ ⬔8
⬔2 ⬵ ⬔3 ⬵ ⬔6 ⬵ ⬔7
E X A M P L E 1
Vocabulary
parallel lines
perpendicular lines
transversal
MA.8.G.2.2
Classify and
determine the
measure of angles, including
angles created when parallel
lines are cut by transversals.
Parallel and Perpendicular
Lines
P
MA.8.G.2.2
Cl if d
9-2
You cannot tell if
angles are congruent
by measuring because
measurement is not
exact.

Alternate interior Alternate exterior Corresponding
1
2
3
4
5
6
a
b
c
7
74°
Math
@ thinkcentral.com
PROPERTIES OF TRANSVERSALS TO PARALLEL LINES
Some pairs of the eight angles formed by two parallel lines and a
transversal have special names.
Finding Measures of Angles Formed by Transversals
In the figure, line a
line b. Find the measure
of each angle. Justify your answer.
⬔4
m⬔4 ⴝ 74ⴗ The 74ⴗ angle and ⬔4 are
corresponding angles, so
they are congruent.
⬔3
m⬔3 ⴙ 74ⴗ ⴝ 180ⴗ ⬔3 is supplementary to the 74ⴗ angle.
ⴚ 74ⴗ ⴚ 74ⴗ Subtract 74ⴗ from both sides.
ᎏ
ᎏ
ᎏ
ᎏ
ᎏ ᎏᎏ
ᎏ
m⬔3 ⴝ 106ⴗ Simplify.
⬔5
m⬔5 ⴝ 106ⴗ ⬔3 and ⬔5 are alternate interior angles, so
they are congruent.
E X A M P L E 2
A
B
C
Think and Discuss
1. Tell how many different angles would be formed by a transversal
intersecting three parallel lines. How many different angle
measures would there be?
2. Explain how a transversal could intersect two other lines so that
corresponding angles are not congruent.
If two parallel lines are intersected by a transversal,
• corresponding angles are congruent,
• alternate interior angles are congruent,
• and alternate exterior angles are congruent.
If the transversal is perpendicular to the parallel lines, all of the
angles formed are congruent 90ⴗ angles.
The symbol for
parallel is

. The
symbol for
perpendicular is ⊥.

1 2 5 6
3 4 7 8
1
2
3
n
4
7
6
5
m
t
118°
1
2
3
4
5 8
6
7
1
4 2
3
7
6
8
5
t
r
s
90º
3
2
45º
1
a b
c
1 4
2
3
7
6
5
p q
110°
Exercises X–XX, XX
Homework Help
Exercises 1–10, 11, 15, 17
Homework Help
9-2 Parallel and Perpendicular Lines 407
Exercises
E
E
9-2
MA.8.G.2.2
on p. WS11
ⴝ
1. Measure the angles formed by the
transversal and the parallel lines.
Which angles seem to be
congruent?
In the figure, line m
line n. Find the
measure of each angle. Justify your answer.
2. ⬔1 3. ⬔4
4. ⬔6 5. ⬔7
6. Measure the angles formed by the transversal and
the parallel lines. Which angles seem to be
congruent?
In the figure, line p
line q. Find the measure of each angle.
Justify your answer.
7. ⬔1
8. ⬔4
9. ⬔6
10. ⬔7
In the figure, line t
line s.
11. Name all angles congruent to ⬔1.
12. Name all angles congruent to ⬔2.
13. Name three pairs of supplementary angles.
14. Which line is the transversal?
15. If m⬔4 is 51ⴗ, what is m⬔2?
18. Art The picture shows a painting by Theo van
Doesburg titled Counter-Composition XIII.
Line b is parallel to line a and perpendicular to
line c. Find the measures of ⬔1, ⬔2, and ⬔3.
h l f
GUIDED PRACTICE
See Example 1
See Example 2
h l f d b
See Example 1
See Example 2
h f l 
l

MA.8.G.2.2, MA.8.A.4.1
1
2
3
4
5
6 8
7
r
s t
A
B
Physical Science
25. Multiple Choice Two parallel lines are intersected by a transversal.
The measures of two corresponding angles that are formed are each 54°.
What are the measures of each of the angles supplementary to the
corresponding angles?
A. 36° B. 72° C. 108° D. 126°
26. Extended Response Suppose a transversal intersects two parallel lines.
One angle that is formed is a right angle. What are the measures of the
remaining angles? What is the relationship between the transversal and
the parallel lines?
Solve each equation for the given variable. (Lesson 7-3)
27. s ⴝ a ⴙ b ⴙ c
________
2
for b 28. V ⴝ 4
__
3
␲abc for a
⬔1 and ⬔3 are vertical angles, and ⬔2 and ⬔4 are vertical angles. ⬔1 and ⬔2 are
supplementary angles. (Lesson 9-1)
29. If m⬔1 ⴝ 25°, find m⬔3°. 30. Find m⬔2.
⬔1 ⬵ ⬔2
⬔3 ⬵ ⬔4
Draw a diagram to illustrate each of the following.
19. line m
line n and transversal h with congruent angles ⬔1 and ⬔3
20. line h
line j and transversal k with eight congruent angles
21. Make a Conjecture Two parallel lines are cut by a transversal.
Can you determine the measures of all the angles formed if given
only one angle measure? Explain.
22. Physical Science A periscope
contains two parallel mirrors that
face each other. With a periscope, a
person in a submerged submarine
can see above the surface of the
water.
a. Name the transversal in
the diagram.
b. If m⬔1 ⴝ 45°, find m⬔2,
m⬔3, and m⬔4.
23. Write About It Choose an example of abstract art or architecture with
parallel lines. Explain how parallel lines, transversals, or perpendicular lines
are used in the composition.
24. Challenge In the figure, ⬔1, ⬔4, ⬔6,
and ⬔7 are all congruent, and ⬔2, ⬔3, ⬔5,
and ⬔8 are all congruent. Does this mean
that line s is parallel to line t? Explain.
This hat from
the 1937 British
Industries Fair is
equipped with
a pair of parallel
mirrors to enable
the wearer to see
above crowds.

Math
@ thinkcentral.com
37°
y°
63°
42°
x°
Lesson Tutorial Videos @ thinkcentral.com 9-3 Triangles 409
If you tear off two corners of a
triangle and place them next to the
third corner, the three angles seem
to form a straight angle.
Draw a triangle and extend one side.
Then draw a line parallel to the
extended side, as shown.
The three angles in the triangle can
be arranged to form a straight angle, or 180°.
The sides of the triangle are
transversals to the parallel lines. The
alternate interior angles are congruent.
An acute triangle has 3 acute angles. A right triangle has 1 right angle.
An obtuse triangle has 1 obtuse angle.
Finding Angles in Acute, Right, or Obtuse Triangles
Find x° in the acute triangle.
63° ⴙ 42° ⴙ x° ⴝ 180° Triangle Sum Theorem
105° ⴙ x° ⴝ 180°
ⴚ 105° ⴚ 105° Subtract 105° from
x° ⴝ 75° both sides.
Find y° in the right triangle.
37° ⴙ 90° ⴙ y° ⴝ 180° Triangle Sum Theorem
127° ⴙ y° ⴝ 180°
ⴚ 127° ⴚ 127° Subtract 127°
y° ⴝ 53° from both sides.
E X A M P L E 1
A
B
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
MA.8.G.2.3
Demonstrate that
the sum of the
angles in a triangle is
180 degrees and apply
this fact to find unknown
measure of angles....
Triangles
If
MA.8.G.2.3
Demonstrate that
9-3
This torn triangle demonstrates an important
geometry theorem called the Triangle Sum Theorem.
TRIANGLE SUM THEOREM
Words Numbers Algebra
58°
79°
43° s°
t°
r°
The angle measures
of a triangle add to
180°.
43° ⴙ 58° ⴙ 79° ⴝ 180° r° ⴙ s° ⴙ t° ⴝ 180°

m°
m° m°
4p°
3p° 2p°
45°
45°
Think and Discuss
1. Explain whether a right triangle can be equilateral. Can it be
isosceles? scalene?
2. Explain whether a triangle can have 2 right or obtuse angles.
An equilateral triangle has 3 congruent sides and 3 congruent angles. An
isosceles triangle has at least 2 congruent sides and 2 congruent angles. A
scalene triangle has no congruent sides and no congruent angles.
Finding Angles in Equilateral, Isosceles, or Scalene Triangles
Find the angle measures in
the equilateral triangle.
3m° ⴝ 180° Triangle Sum Theorem
3m°
____
3
ⴝ 180°
____
3
Divide both sides by 3.
m° ⴝ 60°
All three angles measure 60°.
Find the angle measures in the
scalene triangle.
2p° ⴙ 3p° ⴙ 4p° ⴝ 180°
9p°ⴝ 180° Simplify.
9p°
___
9
ⴝ 180°
____
9
Divide both sides by 9.
p° ⴝ 20°
The angle labeled 2p° measures 2(20°) ⴝ 40°, the angle
labeled 3p° measures 3(20°) ⴝ 60°, and the angle labeled
4p° measures 4(20°) ⴝ 80°.
Finding Angles in a Triangle That Meets Given Conditions
The second angle in a triangle is twice as large as the first.
The third angle is half as large as the second. Find the angle
measures and draw a possible figure.
Let x° ⴝ first angle measure. Then 2x° ⴝ second angle measure,
and 1
_
2
(2x)° ⴝ x° ⴝ third angle measure.
x° ⴙ 2x° ⴙ x° ⴝ 180° Triangle Sum Theorem
4x°
___
4
ⴝ 180°
____
4
Simplify, then divide both
sides by 4.
x° ⴝ 45°
Two angles measure 45° and one angle measures 90°. The triangle
has two congruent angles. The triangle is an isosceles right triangle.
E X A M P L E 2
A
B
E X A M P L E 3

c° c°
68°
31° r°
70° 33°
q°
5d°
d°
4d°
a°
a°
a°
44°
x°
79°
y°
34° 34°
29°
121°
w°
w°
w°
w°
32°
s°
36°
m° m°
25° 40°
t°
7g°
2g°
9g°
Exercises X–XX, XX
Homework Help
Exercises 1–12, 13, 15, 23, 27
Homework Help
9-3 Triangles 411
Exercises
on p. WS11
ⴝ
MA.8.G.2.3
E
E
9-3
1. Find q° in the acute triangle.
2. Find r° in the right triangle.
3. Find the angle measures in
the isosceles triangle.
the scalene triangle.
6. The second angle in a triangle is half as large as the first. The third
angle is three times as large as the second. Find the angle measures
and draw a possible figure.
7. Find s° in the right triangle.
8. Find t° in the obtuse triangle.
the isosceles triangle.
the scalene triangle.
12. The second angle in a triangle is five times as large as the first. The third
angle is two-thirds as large as the first. Find the angle measures and draw
a possible figure.
Find the value of each variable.
13. 14. 15.
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 1
See Example 2
See Example 3

MA.8.G.2.3, MA.8.A.6.2, MA.8.G.2.2
y˚
w˚
m˚
z˚
15˚
15˚
x˚
1
7
8 2
6 4
3
5
x y
d
28. Multiple Choice Which type of triangle can be constructed with a 50°
angle between two 8-inch sides?
A. Equilateral B. Isosceles C. Scalene D. Obtuse
29. Short Response Two angles of a triangle are 45° and 30°. What is the
measure of the third angle? Is the triangle acute, right, or obtuse?
Each square root is between two integers. Name the integers. (Lesson 4-6)
30.兹
___
42 31.兹
___
71 32.兹
___
35 33.兹
____
296
In the figure, line x
line y. (Lesson 9-2)
34. If m∠1 ⴝ 34°, what is m∠7?
35. If m∠6 ⴝ 125°, what is m∠5?
36. If m∠1 ⴝ 34°, what is m∠4?
37. If m∠5 ⴝ 34°, what is m∠2?
Sketch a triangle to fit each description, if possible.
16. acute scalene 17. obtuse equilateral 18. right scalene
19. right equilateral 20. obtuse scalene 21. acute isosceles
22. Make a Conjecture Can an acute isosceles triangle have two angles
that measure 40°? Justify your answer.
23. Critical Thinking Triangle LMN is an obtuse triangle and m⬔L ⴝ 25°.
⬔M is the obtuse angle, and its measure in degrees is a whole number. What is
the largest m⬔N can be to the nearest whole degree?
24. Social Studies American Samoa is a
territory of the United States made up of a
group of islands in the Pacific Ocean, about
halfway between Hawaii and New Zealand.
The flag of American Samoa is shown.
a. Find the measure of each angle.
b. Classify the triangles in the flag by their sides and angles.
25. Choose a Strategy Which of the following sets of angle measures can be
used to create an isosceles triangle?
A. 45°, 45°, 95° B. 49°, 51°, 80° C. 27°, 27°, 126° D. 35°, 55°, 100°
26. Write About It Explain how to cut a square or an equilateral triangle in
half to form two identical triangles. What are the angle measures in the
resulting triangles in each case?
27. Challenge Construct an equilateral triangle using only a straightedge and a
compass. Describe the steps you used.
on p. WS11
ⴝ
2

Polygons
Quadrilateral
Hexagon
Pentagon
A
C
D
E B
9-4 Polygons 413
MA.8.G.2.3
Demonstrate that
the sum of the
angles in a triangle is 180-
degrees and apply this fact to
find ... the sum of angles in
polygons.
Vocabulary
polygon
regular polygon
trapezoid
parallelogram
rectangle
rhombus
square
Many flat kites are in the shape
of polygons. A polygon is a closed
plane figure formed by three or
more segments. A polygon is
named by the number of its sides.
A diagonal of a polygon is a
segment connecting two
non-adjacent vertices. Segment
BD is a diagonal of the pentagon.
Finding Sums of the Angle Measures in Polygons
Find the sum of the angle measures in each figure.
Find the sum of the angle measures in a quadrilateral.
Draw a diagonal to divide the figure into triangles.
2 ⴢ 180° ⴝ 360° 2 triangles
Find the sum of the angle measures in a pentagon.
Draw diagonals to divide the figure into triangles.
3 ⴢ 180° ⴝ 540° 3 triangles
Look for a pattern between the number of sides and the number of triangles.
E X A M P L E 1
A
B
MA.8.G.2.3
Demonstrate that
M
9-4
Polygon Number of Sides
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
n-gon n
Hexagon:
6 sides
4 triangles
Heptagon:
7 sides
5 triangles

Quadrilaterals
Parallelograms
2 pairs of parallel sides
Trapezoids
exactly 1 pair of
parallel sides
Rhombuses
4 congruent sides
Rectangles
4 right angles
Squares
4 congruent sides
4 right angles
x°
x° x°
x° x°
y° y°
y° y°
y° y°
The pattern is that the number of triangles is always 2 less than the
number of sides. So an n-gon can be divided into n ⴚ 2 triangles. The
sum of the angle measures of any n-gon is 180°(n ⴚ 2).
All the sides and angles of a regular polygon have equal measures.
Finding the Measure of Each Angle in a Regular Polygon
Find the angle measures in each regular polygon.
5 congruent angles 6 congruent angles
5x° ⴝ 180°(5 ⴚ 2) 6y° ⴝ 180°(6 ⴚ 2)
5x° ⴝ 180°(3) 6y ° ⴝ 180°(4)
5x° ⴝ 540° 6y° ⴝ 720°
5x°
___
5
ⴝ 540°
____
5
6y°
___
6
ⴝ 720°
____
6
x° ⴝ 108° y ° ⴝ 120°
Quadrilaterals with certain properties are given additional names. A
trapezoid has exactly 1 pair of parallel sides. Aparallelogram has 2 pairs
of parallel sides. Arectangle has 4 right angles. Arhombus has 4
congruent sides. Asquare has 4 congruent sides and 4 right angles.
E X A M P L E 2
A B

E F
H G
EF || GH
W X
Z Y
t° t°
t° t°
v°
v°
v°
v°
v°
v°
v°
b° b°
b° b°
b° b°
Exercises 1–18, 21, 23, 25, 27, 29, 31, 33
Homework Help
9-4 Polygons 415
Exercises
1. 2. 3.
4. 5. 6.
GUIDED PRACTICE
See Example 1
See Example 2
MA.8.G.2.3
E
E
9-4
Think and Discuss
1. Choose which is larger, an angle in a regular heptagon or an angle
in a regular octagon. Justify your answer.
2. Explain why all rectangles are parallelograms and why all squares
are rectangles.
3. Explain why it is not possible to draw a diagonal of a triangle.
Classifying Quadrilaterals
Give all of the names that apply to each figure.
quadrilateral Four-sided polygon
trapezoid 1 pair of parallel sides
quadrilateral Four-sided polygon
parallelogram 2 pairs of parallel sides
rectangle 4 right angles
E X A M P L E 3
A
B

4 cm
4 cm
4 cm
4 cm
m° m°
m° m°
m° m°
m° m°
h°
h°
h°
h°
h°
h° h° h°
h°
h°
h°
h° v°
v°
v°
v°
v°
v°
v°
v°
v° v°
8 in.
8 in.
8 in.
8 in.
A
C
D
B
AB || CD
AD || BC
P Q
R S
PQ || RS
x°
110° 45°
75°
y°
60°
25°
w°
55°
113°
110°
125°
z°
123°
133°
150°
116°
101°
x° x°
40° 40°
m°
4m°
50°
110°
on p. WS11
ⴝ
7. 8. 9.
See Example 3
10. 11. 12.
13. 14. 15.
16. 17. 18.
See Example 1
See Example 2
See Example 3
Find the sum of the angle measures in each regular polygon. Then find the
measure of each angle.
19. 20-gon 20. 13-gon 21. 60-gon 22. pentagon 23. 16-gon
Find the value of each variable.
24. 25. 26.
27. 28. 29.

y˚
y˚
x˚
35˚
41˚ 41˚
35˚
Crown angle
Pavilion main
angle
MA.8.G.2.3, MA.8.A.1.1
Earth Science
9-4 Polygons 417
43. Multiple Choice What is the measure of each angle of a regular 15-sided polygon?
A. 146° B. 148° C. 150° D. 156°
44. Short Response The sum of the angle measures of a regular polygon is
720°. Name the regular polygon. What is the measure of each angle?
Find the slope of the line that passes through each pair of points. (Lesson 8-2)
45. (1, 9) and (ⴚ3, 13) 46. (5, ⴚ5) and (5, 7) 47. (ⴚ2, ⴚ12) and (ⴚ7, 16)
48. The second angle in a triangle is 3
_
4
as large as the first angle. The third angle in
a triangle is 2
_
3
as large as the second angle. Find the angle measures and draw a
possible figure. (Lesson 9-3)
The sum of the angle measures of a polygon is given. Name the polygon.
30. 1080° 31. 540° 32. 360° 33. 1620°
Find a counterexample to disprove each conjecture.
34. All rhombuses are squares.
35. All quadrilaterals have at least one pair of parallel sides.
Sketch a quadrilateral to fit each description. If no quadrilateral can be
drawn, write not possible.
36. a parallelogram that is not a rhombus
37. a square that is not a rectangle
38. Earth Science Precious stones are often
cut in a brilliant cut to maximize the light
they reflect. The best angles for a diamond
are shown.
a. If the pavilion main angle is 41°, find x.
b. If the crown angle is 35°, find y.
39. Architecture Fernando is designing a house. He wants one room to be in the
shape of an irregular heptagon with two corners that form right angles. What
angle measures could the remaining five corners have?
40. What’s the Error? A student said that all squares are rectangles, but
not all squares are rhombuses. What was the error?
41. Write About It Why is it possible to find the sum of the angle measures
of an n-gon using the formula (180n ⴚ 360)°?
42. Challenge Construct a square using only a straightedge and a compass.
Describe the steps you used. (Hint: The diagonals of a square are
perpendicular and congruent.)
The hope diamond
is a blue diamond
weighing 45.52
carats. By compari-
son, a diamond in a
typical engagement
ring weighs only
about 0.3 carat.

C
B
D
A
E
F
35°
15°
75°
55°
m
n
125°
3 4
1 2
83°
x°
60°
74°
x° x°
4x
3x
x
10x
4x ฀ 3
x ฀3
3 in.
3 in.
3 in.
P
N
O
M
3 in.
A B
D C
AB || CD
Ready
to
Go
On?
Ready To Go On?
Ready To Go On?
SECTION 9A
Quiz for Lessons 9-1 Through 9-4
Angle Relationships
1. two pairs of complementary angles
2. three pairs of supplementary angles
3. two right angles
Parallel and Perpendicular Lines
line n. Find the measure of each angle.
Justify your answer.
4. ⬔1 5. ⬔2 6. ⬔3 7. ⬔4
Triangles
Find x° in each triangle.
8. 9.
Find the angle measures in each triangle.
10. 11.
Polygons
12. 13.
Find the angle measures in each polygon.
14. regular 18-gon 15. regular 36-gon
A
h
9-1
P
9-2
T
i d
i i
9-3
P
i ll
Gi
G
9-4
Keyword MT11 RTGO9A
Ready To Go On?

m
n
p
3
4
5
1
2
55°
35°
A
C
B
Focus on Problem Solving 419
LA.8.2.2.3 The student will
organize information to show
understanding or relationships
among facts and ideas... (e.g., …
paraphrasing…).
Understand the
Problem
• Restate the problem in your own words
If you write a problem in your own words, you may understand it
better. Before writing a problem in your own words, you may need
to read it over several times—perhaps aloud, so you can hear
yourself say the words.
Once you have written the problem in your own words, you may
want to make sure you included all of the necessary information to
solve the problem.
Write each problem in your own words. Check to make sure you
have included all of the information needed to solve the problem.
1 In the figure, ⬔1 and ⬔2 are
complementary, and ⬔1 and ⬔5
are supplementary. If m⬔1 ⴝ 60°,
find m⬔3 ⴙ m⬔4.
2 In triangle ABC, m⬔A ⴝ 35° and
m⬔B ⴝ 55°. Use the Triangle Sum
Theorem to determine whether
triangle ABC is a right triangle.
3 The second angle in a quadrilateral is eight
times as large as the first angle. The third
angle is half as large as the second. The
fourth angle is as large as the first angle and
the second angle combined. Find the angle
measures in the quadrilateral.
4 Parallel lines m and n are intersected by a
transversal, line p. The acute angles
formed by line m and line p measure 45°.
Find the measure of the obtuse angles
formed by the intersection of line n and
line p.

Triangular prism
Base Base Base
Height Height
Rectangular prism Cylinder
Height
2
5
3
2
6
Math
@ thinkcentral.com
The largest drum ever built measures
4.8 meters in diameter and is 4.95 meters
deep. It was built by Asano Taiko Company
in Japan. You can use these measurements
to find the volume of the cylindrical drum.
Volume is the number of cubic units in a
three-dimensional figure.
Recall that a cylinder is a three-dimensional
figure that has two congruent circular bases,
and a prism is a three-dimensional figure
named for the shape of its bases. The two
bases are congruent polygons. All of the
other faces are parallelograms.
Finding the Volume of Prisms and Cylinders
Find the volume of each figure to the nearest tenth. Use 3.14 for π.
A rectangular prism with base 2 m by 5 m and height 7 m.
B ⴝ 2 ⴢ 5 ⴝ 10 m2
Area of base
V ⴝ Bh Volume of prism
ⴝ 10 ⴢ 7 ⴝ 70 m3
E X A M P L E 1
A
Volume of Prisms and
Cylinders
The larg
h l
4 8
9-5
Prep for
MA.8.G.5.1
... Convert units of
measure between different
measurement systems... and
dimensions including...
volume, and derived
units to solve problems.
Rev MA.7.G.2.1, Rev
MA.7.G.2.2
The circumference of the Taiko drum
pictured is about half that of the largest
drum ever made.
VOLUME OF PRISMS AND CYLINDERS
Words Numbers Formula
Prism: The volume V B ⴝ 2(5)
of a prism is the area ⴝ 10 units2
V ⴝ Bh
of the base B times V ⴝ (10)(3)
the height h. ⴝ 30 units3
Cylinder: The volume B ⴝ π (22
)
of a cylinder is the ⴝ 4π units2
V ⴝ Bh
area of the base B V ⴝ (4π)(6) ⴝ 24π ⴝ (πr2
)h
times the height h. ⬇ 75.4 units3
Area is measured in
square units. Volume
is measured in cubic
units.
Vocabulary
volume

6 m
15 m
7 ft
4 ft
11 ft
9-5 Volume of Prisms and Cylinders 421
B ⴝ π(62
) ⴝ 36π m2
Area of base
V ⴝ Bh Volume of a cylinder
ⴝ 36π ⴢ 15
ⴝ 540π ⬇ 1695.6 m3
B ⴝ 1
__
2 ⴢ 4 ⴢ 7 ⴝ 14 ft2
Area of base
V ⴝ Bh Volume of a prism
ⴝ 14 ⴢ 11
ⴝ 154 ft3
The formula for volume of a rectangular prism can be written as
V ⴝ ᐉwh, where ᐉ is the length, w is the width, and h is the height.
Exploring the Effects of Changing Dimensions
A cereal box measures 6 in. by 2 in. by 9 in. Explain whether
doubling the length, width, or height of the box would double
the amount of cereal the box holds.
The original box has a volume of 108 in3
. You could double the
volume to 216 in3
by doubling any one of the dimensions. So
doubling the length, width, or height would double the amount
of cereal the box holds.
A can of corn has a radius of 2.5 in. and a height of 4 in. Explain
whether doubling the height of the can would have the same
effect on the volume as doubling the radius.
By doubling the height, you would double the volume. By
doubling the radius, you would increase the volume four times
the original.
B
C
E X A M P L E 2
A
B
Original Double the Double the Double the
Dimensions Length Width Height
V ⴝ ᐉwh V ⴝ (2ᐉ)wh V ⴝ ᐉ(2w)h V ⴝ ᐉw(2h)
ⴝ 6 ⴢ 2 ⴢ 9 ⴝ 12 ⴢ 2 ⴢ 9 ⴝ 6 ⴢ 4 ⴢ 9 ⴝ 6 ⴢ 2 ⴢ 18
ⴝ 108 in3
ⴝ 216 in3
ⴝ 216 in3
ⴝ 216 in3
Original Double the Double the
Dimensions Height Radius
V ⴝ πr2
h V ⴝ πr2
(2h) V ⴝ π(2r)2
h
ⴝ 2.52
π ⴢ 4 ⴝ 2.52
π ⴢ 8 ⴝ 52
π ⴢ 4
ⴝ 25π in3
ⴝ 50π in3
ⴝ 100π in3

6 cm
3 cm
5 cm
9 cm
Think and Discuss
1. Use models to show that two rectangular prisms can have
different heights but the same volume.
2. Apply your results from Example 2 to make a conjecture about
changing dimensions in a triangular prism.
3. Use a model to describe what happens to the volume of a
cylinder when the diameter of the base is tripled.
Music Application
The Asano Taiko Company of Japan built the world’s largest
drum in 2000. The drum’s diameter is 4.8 meters, and its height
is 4.95 meters. Estimate the volume of the drum.
d ⴝ 4.8 ⬇ 5
h ⴝ 4.95 ⬇ 5
r ⴝ d
__
2
ⴝ 5
__
2
ⴝ 2.5
V ⴝ (πr2
)h Volume of a cylinder.
ⴝ (3)(2.5)2
ⴢ 5 Use 3 for π.
ⴝ (3)(6.25)(5)
ⴝ 18.75 ⴢ 5
ⴝ 93.75
⬇ 94
The volume of the drum is approximately 94 m3
.
To find the volume of a composite three-dimensional figure, find the
volume of each part and add the volumes together.
Finding the Volume of Composite Figures
Find the volume of the figure.
ⴝ ⴙ
V ⴝ (6)(9)(5 ) ⴙ
ⴝ 270 ⴙ 81
ⴝ 351 cm3
The volume is 351 cm3
.
E X A M P L E 3
E X A M P L E 4
1
__
2
(6)(3)(9)
Volume
of
figure
Volume of
rectangular
prism
Volume of
triangular
prism

21 cm
6.3 cm
7 cm
3 in.
8 in.
4 in.
5 m
16 m
18 ft 15 ft
10 ft
25 ft 20 ft
10 in.
5 in.
2 in.
11 cm
1.5 cm
13 m
9 m
6 m
ÓÊvÌ
ÈÊvÌ
ÈÊvÌ
{ÊvÌ
Exercises 1–12, 15, 21
Homework Help
9-5 Volume of Prisms and Cylinders 423
Exercises
Prep MA.8.G.5.1
E
E
9-5
1. 2. 3.
4. A can of juice has a radius 3 in. and a height 6 in. Explain whether tripling
the radius would triple the volume of the can.
5. Grain is stored in cylindrical
structures called silos.
Estimate the volume of a silo
with diameter 11.1 feet and
height 20 feet.
6. Find the volume of
the barn.
7. 8. 9.
10. A jewelry box measures 7 in. by 5 in. by 8 in. Explain whether increasing the
height 4 times, from 8 in. to 32 in., would increase the volume 4 times.
11. A toy box is 5.1 cm by 3.2 cm by 4.2 cm.
Estimate the volume of the toy box.
12. Find the volume of the treehouse.
13. While Karim was at camp, his father sent him a care package. The box
measured 10.2 in. by 19.9 in. by 4.2 in.
a. Estimate the volume of the box.
b. What might be the measurements of a box with twice its volume?
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 4
See Example 1
See Example 2
See Example 3
See Example 4

MA.8.G.5.1, MA.8.G.2.3
Life Science
27 ft 7 in.
13 ft 7 in. ?
1 in.
1 in.
8 in.
12 in.
5 in.
20. Multiple Choice Cylinder A has radius 6 centimeters and height
14 centimeters. Cylinder B has radius half as long as cylinder A. What is
the volume of cylinder B? Use 3.14 for π and round to the nearest tenth.
A. 393.5 cm3
B. 395.6 cm3
C. 422.3 cm3
D. 791.3 cm3
21. Multiple Choice A tractor trailer has dimensions of 13 feet by 53 feet by
8 feet. What is the volume of the trailer?
F. 424 ft3
G. 689 ft3
H. 2756 ft3
I. 5512 ft3
22. Find the height of a rectangle with perimeter 60 inches and length
12 inches. What is the area of the rectangle in square feet? (Lesson 5-6)
The measures of two angles of a triangle are given. Find the measure of the
third angle. (Lesson 9-3)
23. 105°, 42° 24. 12.5°, 77.5° 25. x°, 3x° 26. 94°, 0.5x°
14. Social Studies The tablet held by the Statue
of Liberty is approximately a rectangular
prism with volume 1,107,096 in3
. Estimate
the thickness of the tablet.
15. Life Science The cylindrical Giant Ocean
Tank at the New England Aquarium in Boston
has a volume of 200,000 gallons.
a. One gallon of water equals 231 cubic
inches. How many cubic inches of water
are in the Giant Ocean Tank?
b. Use your answer from part a as the volume. The tank is 24 ft deep.
Find the radius in feet of the Giant Ocean Tank.
16. Life Science As many as 60,000 bees can live in 3 cubic feet of space.
There are about 360,000 bees in a rectangular observation beehive that
is 2 ft long by 3 ft high. What is the minimum possible width of the
observation hive?
17. What’s the Error? A student read this statement in a book: “The
volume of a triangular prism with height 15 in. and base area 20 in.
is 300 in3
.” Correct the error in the statement.
18. Write About It Explain why 1 cubic yard equals
27 cubic feet.
19. Challenge A 5-inch section of a hollow brick
measures 12 inches tall and 8 inches wide on the
outside. The brick is 1 inch thick. Find the volume
of the brick, not the hollow interior.
Through the 52
large windows of
the Giant Ocean
Tank, visitors can
see 3000 corals
and sponges as
well as large
sharks, sea turtles,
barracudas,
moray eels, and
hundreds of
tropical fishes.
on p. WS12
ⴝ

Base
Height
Rectangular
pyramid
Triangular
pyramid
Height
Base
Cone
Base
Height
3
3
4
3
2
VOLUME OF PYRAMIDS AND CONES
9 cm
9 cm
4 cm
9-6 Volume of Pyramids and Cones 425
Prep for
MA.8.G.5.1
Compare, contrast,
and convert units of measure
between different measurement
systems... and dimensions
including... volume, and
derived units to solve
problems.
Rev MA.7.G.2.1
Prep for
MA 8 G 5 1
9-6
Part of the Rock and Roll
Hall of Fame building in
Cleveland, Ohio, is a glass
pyramid. The entire building
was designed by architect I. M.
Pei and has approximately
150,000 ft2
of floor space.
Finding the Volume of Pyramids and Cones
Find the volume of each figure. Use 3.14 for π.
B ⴝ 1
__
2
(4 ⴢ 9) ⴝ 18 cm2
V ⴝ 1
__
3 ⴢ 18 ⴢ 9 V ⴝ 1
__
3
Bh
V ⴝ 54 cm3
E X A M P L E 1
A
Volume of Pyramids
and Cones
Pyramid: The volume
V of a pyramid is
one-third of the
area of the base B
times the height h.
Cone: The volume
of a cone is one-
third of the area of
the circular base B
times the height h.
B ⴝ 3(3 )
ⴝ 9 units2
V ⴝ 1
__
3
(9)(4 )
ⴝ 12 units3
B ⴝ π (22 )
ⴝ 4π units2
V ⴝ 1
__
3
(4π )(3)
ⴝ 4π
⬇ 12.6 units3
V ⴝ 1
__
3
Bh
V ⴝ 1
__
3
Bh
or
V ⴝ 1
__
3
πr2
h

6 in.
2 in.
7 ft
9 ft
8 ft
7 mm
8 mm
Find the volume of each figure. Use 3.14 for π.
B ⴝ π(22 ) ⴝ 4π in2
V ⴝ 1
__
3 ⴢ 4π ⴢ 6 V ⴝ 1
__
3
Bh
V ⴝ 8π ⬇ 25.1 in3
Use 3.14 for π.
B ⴝ 9 ⴢ 7 ⴝ 63 ft2
V ⴝ 1
__
3 ⴢ 63 ⴢ 8 V ⴝ 1
__
3
Bh
V ⴝ 168 ft3
B ⴝ π(72 ) ⴝ 49π mm2
V ⴝ 1
__
3 ⴢ 49π ⴢ 8 V ⴝ 1
__
3
Bh
V ⴝ 392
___
3
π ⬇ 410.3 mm2
Use 3.14 for π.
A cone has radius 3 m and height 10 m. Explain whether doubling
the height would have the same effect on the volume of the cone
as doubling the radius.
When the height of the cone is doubled, the volume is doubled. When
the radius is doubled, the volume becomes 4 times the original volume.
Social Studies Application
The Great Pyramid of Giza in Egypt is
a square pyramid. Its height is 481 ft,
and its base has 756 ft sides. Find the
volume of the pyramid.
B ⴝ 7562
ⴝ 571,536 ft2
A ⴝ bh
V ⴝ 1
__
3
(571,536)(481) V ⴝ 1
__
3
Bh
V ⴝ 91,636,272 ft3
B
C
D
E X A M P L E 2
E X A M P L E 3
Original Dimensions Double the Height Double the Radius
V ⴝ 1
__
3
πr2
h V ⴝ 1
__
3
πr2(2h ) V ⴝ 1
__
3
π(2r )2
h
ⴝ 1
__
3
π(32 )(10) ⴝ 1
__
3
π(32 )(2 ⴢ 10) ⴝ 1
__
3
π(2 ⴢ 3 )2(10)
⬇ 94.2 m3
⬇ 188.4 m3
⬇ 376.8 m3
A lowercase b is used
to represent the
length of the base of
a two-dimensional
figure. A capital B is
used to represent the
area of the base of a
solid figure.

ENTER
ⴛ
2nd 2
3 cm
5 cm
4 cm
8 in.
12 in.
6 in.
3.2 ft
9.3 ft
12 yd
17 yd
23 yd
1.9 cm
2.4 cm
27
13
27
Exercises 1–18, 19, 21, 23, 25
Homework Help
Exercises
1. 2. 3.
4. 5. 6.
7. A square pyramid has height 6 m and a base that measures 2 m on each
side. Explain whether doubling the height would double the volume of
the pyramid.
GUIDED PRACTICE
See Example 1
See Example 2
E
E
9-6 Prep MA.8.G.5.1
Using a Calculator to Find Volume
Some traffic pylons are shaped like cones. Use a calculator to find
the volume of a traffic pylon to the nearest hundredth if the
radius of the base is 5 inches and the height is 24 inches.
Use the pi button on your calculator to find the area of the base.
5 B ⴝ πr2
Next, with the area of the base still displayed, find the volume of
the cone.
ⴛ 24 ⴛ 1 ⴜ 3 ENTER
V ⴝ 1
__
3
Bh
The volume of the traffic pylon is approximately 628.32 in3
.
E X A M P L E 4
Think and Discuss
1. Describe two or more ways that you can change the dimensions
of a rectangular pyramid to double its volume.
2. Use a model to compare the volume of a cube with 1 in. sides
with a pyramid that is 1 in. high and has a 1 in. square base.

0.8
0.4
1.6
4.9 m
7.8 m
5.5 m
5 in.
5 in.
6.67 ft
3.08 ft
22 m
20 m 16 m
13.5
37
33
on p. WS12
ⴝ
8. The Transamerica Pyramid in San Francisco has a base area of 22,000 ft2
and a height of 853 ft. What is the volume of the building?
9. Gretchen made a paper cone to hold a gift for a friend. The paper cone
was 17 inches high and had a diameter of 6 inches. Use a calculator to
find the volume of the paper cone to the nearest hundredth.
10. 11. 12.
13. 14. 15.
16. A triangular pyramid has a height of 12 in. The triangular base has a
height of 12 in. and a width of 12 in. Explain whether doubling the height
of the base would double the volume of the pyramid.
17. A cone-shaped building is commonly used to store sand. What would be
the volume of a cone-shaped building with diameter 50 m and height
20 m to the nearest hundredth?
18. Antonio made mini waffle cones for a birthday party. Each waffle cone
was 3 inches high and had a radius of 3
__
4
inch. Use a calculator to find the
volume of the waffle cone to the nearest hundredth.
Find the missing measure to the nearest tenth. Use 3.14 for π.
19. cone: 20. cylinder:
radius ⴝ 4 in. radius ⴝ
height ⴝ height ⴝ 2.5 m
volume ⴝ 100.5 in3
volume ⴝ 70.65 m3
21. triangular pyramid: 22. rectangular pyramid:
base height ⴝ base length ⴝ 3 ft
base width ⴝ 8 ft base width ⴝ
height ⴝ 6 ft height ⴝ 7 ft
volume ⴝ 88 ft3
volume ⴝ 42 ft3
23. Estimation Orange traffic cones come in a variety of sizes. Approximate
the volume in cubic inches of a traffic cone with height 2 feet and
diameter 10 inches by using 3 in place of π.
See Example 3
See Example 4
See Example 1
See Example 2
See Example 3
See Example 4

MA.8.A.1.2, MA.8.G.2.3
Architecture
29. Multiple Choice A pyramid has a rectangular base measuring
12 centimeters by 9 centimeters. Its height is 15 centimeters. What is
the volume of the pyramid?
A. 540 cm3
B. 405 cm3
C. 315 cm3
D. 270 cm3
30. Multiple Choice A cone has diameter 12 centimeters and height
9 centimeters. Using 3.14 for π, find the volume of the cone to the
nearest tenth.
F. 1356.5 cm3
G. 339.1 cm3
H. 118.3 cm3
I. 56.5 cm3
31. Gridded Response Suppose a cone has a volume of 104.7 cubic
centimeters and a radius of 5 centimeters. Find the height of the cone
to the nearest whole centimeter. Use 3.14 for π.
32. The top of the Tower of Pisa is about 3.9 meters from where it would be if the
building were completely vertical. The tower has a height of about 56.2 meters.
Find the slope to the nearest tenth. What does the value of the slope mean in
terms of the tower? (Lesson 8-2)
Find the sum of the angle measures in each polygon. (Lesson 9-4)
33. 15-gon 34. hexagon 35. n-gon 36. decagon
The Mona Lisa
by Leonardo Da
Vinci is perhaps
the world’s most
famous painting.
It is on display
at the Louvre
museum in Paris,
France.
24. Architecture The Pyramid of the Sun, in Teotihuacán, Mexico, is about
65 m tall and has a square base with side length 225 m.
a. What is the volume in cubic meters of the pyramid?
b. How many cubic meters are in a cubic kilometer?
c. What is the volume in cubic kilometers of the pyramid to the nearest
thousandth?
25. Architecture The pyramid at the
entrance to the Louvre in Paris has a
height of 72 feet and a square base
that is 112 feet long on each side.
What is the volume of this pyramid?
26. What’s the Error? A student says
that the formula for the volume of a
cylinder is the same as the formula
for the volume of a pyramid, 1
__
3
Bh. What
error did this student make?
27. Write About It How would a cone’s volume be affected if you doubled the
height? the radius? Use a model to help explain.
28. Challenge The diameter of a cone is x cm, the height is 18 cm, and the
volume is 96π cm3
. What is x?
2

3 2
3
3
5 5
2
2
2 3
5
6
6
5
Math
@ thinkcentral.com
2 m
8 m
An anamorphic image is a
distorted picture that becomes
recognizable when reflected onto
a cylindrical mirror.
Surface area is the sum of the
areas of all surfaces of a figure.
The lateral faces of a prism are
parallelograms that connect the
bases. The lateral surface of a
cylinder is the curved surface.
Finding Surface Area
Find the surface area of each figure to the nearest tenth.
Use 3.14 for π.
S ⴝ 2πr2
ⴙ 2πrh
ⴝ 2π(22) ⴙ 2π(2)(8)
ⴝ 40π m2
⬇ 125.6 m2
E X A M P L E 1
A
Vocabulary
surface area
lateral face
lateral surface
Surface Area of Prisms
and Cylinders
An anam
9-7
Prep for
MA.8.G.5.1
...Convert units of
dimensions including...area...
and derived units to solve
problems.
Rev MA.7.G.2.1
Prism: The surface area
S of a prism is twice
the base area B plus S ⴝ 2B ⴙ L
the lateral area L. The or
lateral area is the base S ⴝ 2B ⴙ Ph
perimeter P times the
height h.
S ⴝ 2(3 ⴢ 2) ⴙ (10)(5) ⴝ 62 units2
Cylinder: The surface
area S of a cylinder is
twice the base area B S ⴝ 2B ⴙ L
plus the lateral area L. or
The lateral area is the S ⴝ 2πr2
ⴙ 2πrh
base circumference
2πr times the height h. S ⴝ 2π(52) ⴙ 2π(5)(6) ⬇ 345.4 units2
SURFACE AREA OF PRISMS AND CYLINDERS

8 cm
7 cm
6 cm
9 cm
5.3 cm
9-7 Surface Area of Prisms and Cylinders 431
Use 3.14 for π.
S ⴝ 2B ⴙ Ph
ⴝ 2(1
__
2
ⴢ 9 ⴢ 5.3)ⴙ (23)(7)
ⴝ 208.7 cm2
A cylinder has diameter 10 in. and height 4 in. Explain whether
doubling the height would have the same effect on the surface
area as doubling the radius.
They would not have the same effect. Doubling the radius would
increase the surface area more than doubling the height.
Art Application
A Web site advertises that it can turn your
photo into an anamorphic image. To reflect
the picture, you need to cover the lateral
surface of a cylinder that is 49 mm in
diameter and 107 mm tall with reflective
material. Estimate the amount of reflective
material you would need.
The diameter of the cylinder is about 50 mm,
and the height is about 100 mm.
L ⴝ 2πrh Only the lateral surface needs to be covered.
ⴝ 2π(25)(100) diameter ⬇ 50 mm, so r ⬇ 25 mm
⬇ 15,700 mm2
B
E X A M P L E 2
E X A M P L E 3
Think and Discuss
1. Explain how finding the surface area of a cylindrical drinking
glass would be different from finding the surface area of a cylinder.
2. Compare the amount of paint needed to cover a cube with 1 ft
sides to the amount needed to cover a cube with 2 ft sides.
Original Dimensions Double the Height Double the Radius
S ⴝ 2πr2
ⴙ 2πrh S ⴝ 2πr2
ⴙ 2πrh S ⴝ 2πr2
ⴙ 2πrh
ⴝ 2π(5)2
ⴙ 2π(5)(4) ⴝ 2π(5)2
ⴙ 2π(5)(8) ⴝ 2π(10)2
ⴙ2π(10)(4)
ⴝ 90π in2
⬇ 282.6 in2
ⴝ 130π in2
⬇ 408.2 in2
ⴝ 280π in2
⬇ 879.2 in2

5 cm ? S ⫽ 120π cm2
5 m
4 m
4 m
17 mm
15 mm
26 mm
8 mm
7 m
8.4 m
?
S ⴝ 256 m2
12m
5m
14 cm
8 cm
3 cm
6 cm
15 cm
6 m
3 m
3 m
2.6 m
3 m
Exercises 1 –10,11,13,15,17
Homework Help
Exercises
Find the surface area of each figure to the nearest tenth. Use 3.14 for π.
1. 2. 3.
4. A rectangular prism is 3 ft by 4 ft by 7 ft. Explain whether doubling all of
the dimensions would double the surface area.
5. Tilly is covering a can with contact paper, not including its top and
bottom. The can measures 8 inches high and has a radius of 2 inches.
Estimate the amount of contact paper she needs.
6. 7. 8.
9. A cylinder has diameter 4 ft and height 9 ft. Explain whether halving the
diameter has the same effect on the surface area as halving the height.
10. Frank is wrapping a present. The box measures 6.2 cm by 9.9 cm by
5.1 cm. Estimate the amount of wrapping paper, not counting overlap,
that Frank needs.
Find the surface area of each figure with the given dimensions to the
nearest tenth. Use 3.14 for π.
11. cylinder: d ⴝ 30 mm, h ⴝ 49 mm
12. rectangular prism: 51
__
4
in. by 8 in. by 12 in.
Find the missing dimension in each figure with the given surface area.
13. 14.
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 1
See Example 2
See Example 3
Prep MA.8.G.5.1
E
E
9-7
on p. WS12
ⴝ

Sports
250 ft
36 ft
12 cm
5 cm
9 cm
MA.8.A.6.4, MA.8.G.2.2
9-7 Surface Area of Prisms and Cylinders 433
21. Multiple Choice Find the surface area of a cylinder with radius 5 feet and
height 3 feet. Use 3.14 for π.
A. 125.6 ft2
B. 150.72 ft2
C. 172.7 ft2
D. 251.2 ft2
22. Gridded Response A rectangular prism has dimensions 2 meters by
4 meters by 18 meters. Find the surface area, in square meters, of the prism.
Add or subtract. (Lesson 2-3)
23. ⴚ0.4 ⴙ 0.7 24. 1.35 ⴚ 5.6 25. ⴚ0.01 ⴚ 0.25 26. ⴚ0.65 ⴙ (ⴚ1.12)
27. The alternate exterior angles formed when two parallel lines are cut by a
transversal measure 3x° and 200ⴗ ⴚ xⴗ. What are their measures? What are
the measures of the alternate interior angles? (Lesson 9-2)
In 2006, Seth
Westcott won
the first Olympic
Men’s Snowboard
Cross gold medal.
15. Multi-Step Jesse makes rectangular glass aquariums with glass tops.
The aquariums measure 12 in. by 6 in. by 8 in. Glass costs $0.08 per
square inch. How much will the glass for one aquarium cost?
16. Sports In the snowboard half-pipe,
competitors ride back and forth on a
course shaped like a cylinder cut in half
lengthwise. What is the surface area of this
half-pipe course?
17. Multi-Step Olivia is painting the four sides and top of a large trunk.
The trunk measures 5 ft long by 3.5 ft deep by 3 ft high. A gallon of paint
covers approximately 300 square feet. She wants at least 15% extra
paint for waste and overage. How many quarts of paint does she need?
18. Choose a Strategy Which of the following nets
can be folded into the given three-dimensional figure?
A. B. C. D.
19. Write About It Describe the effect on the
surface area of a square prism when you
double the length of one of its sides.
20. Challenge A rectangular wood block that is
12 cm by 9 cm by 5 cm has a hole drilled
through the center with diameter 4 cm. What is
the total surface area of the wood block?
2

Regular pyramid
Slant
height
Right cone
Slant
height
12
12
8
12
12
8
2
5
5
2
5
SURFACE AREA OF PYRAMIDS AND CONES
434 Chapter 9 Geometry Measurement Lesson Tutorial Videos @ thinkcentral.com
Vocabulary
slant height
regular pyramid
right cone
The slant height of a pyramid
or cone is measured along its
lateral surface.
The base of a regular pyramid
is a regular polygon, and the
lateral faces are all congruent.
In a right cone, a line
perpendicular to the base
through the tip of the cone passes through the center of the base.
Finding Surface Area
Use 3.14 for π.
S ⴝ B ⴙ 1
__
2
Pᐉ
ⴝ (2.5 ⴢ 2.5) ⴙ 1
__
2
(10)(3)
ⴝ 21.25 in2
⬇ 21.3 in2
E X A M P L E 1
A
Surface Area of Pyramids
and Cones
The slan
9-8
3 in.
2.5 in.
2.5 in.
Pyramid: The surface
area S of a regular
pyramid is the base area S ⴝ B ⴙ L
B plus the lateral area L. or
The lateral area is one- S ⴝ B ⴙ 1
__
2
Pᐉ
half the base perimeter
P times the slant height ᐉ.
S ⴝ (12 ⴢ 12) ⴙ 1
__
2
(48)(8) ⴝ 336 units2
Cone: The surface area S
of a right cone is the
base area B plus the S ⴝ B ⴙ L
lateral area L. The lateral or
area is one-half the base S ⴝ πr2
ⴙ πrᐉ
circumference 2πr times
the slant height ᐉ. S ⴝ π(22) ⴙ π(2)(5) ⴝ 14π ⬇ 43.98 units2
Prep for
MA.8.G.5.1
...Convert units of
dimensions including...area...
and derived units to solve
problems.
Rev MA.7.G.2.1

2.5 cm
2 cm
ᐉ
9-8 Surface Area of Pyramids and Cones 435
Use 3.14 for π.
S ⴝ πr2
ⴙ πrᐉ
ⴝ π(4)2
ⴙ π(4)(7)
ⴝ 44π ⬇ 138.2 m2
A cone has diameter 6 in. and slant height 4 in. Explain whether
doubling the slant height would have the same effect on the
surface area as doubling the radius. Use 3.14 for π.
They would not have the same effect. Doubling the radius would
increase the surface area more than doubling the slant height.
Life Science Application
An ant lion pit is an inverted cone
with the dimensions shown. What is
the lateral surface area of the pit?
The slant height, radius, and depth of
the pit form a right triangle.
a2
ⴙ b2
ⴝ ᐉ2
Pythagorean
Theorem
(2.5)2
ⴙ 22
ⴝ ᐉ2
10.25 ⴝ ᐉ2
ᐉ ⬇ 3.2
L ⴝ πrᐉ Lateral surface area
ⴝ π(2.5)(3.2) ⬇ 25.1 cm2
B
E X A M P L E 2
E X A M P L E 3
Think and Discuss
1. Compare the formula for surface area of a pyramid to the formula
for surface area of a cone.
2. Explain how you would find the slant height of a square pyramid
with base edge length 6 cm and height 4 cm.
7 m 4 m
Original Dimensions Double the Slant Height Double the Radius
S ⴝ πr2
ⴙ πrᐉ S ⴝ πr2
ⴙ πr(2ᐉ) S ⴝ π(2r)2
ⴙ π(2r)ᐉ
ⴝ π(3)2
ⴙ π(3)(4) ⴝ π(3)2
ⴙ π(3)(8) ⴝ π(6)2
ⴙ π(6)(4)
ⴝ 21π in2
⬇ 65.9 in2
ⴝ 33π in2
⬇ 103.6 in2
ⴝ 60π in2
⬇ 188.4 in2

5.5 in.
4 in.
4 in.
4 in.
6 mm
4 mm
9 m
6 m 6 m
8 m
5 m
5 m
5 ft
1.5 ft
3 in.
3 in. 3 in.
4.5 in.
Exercises 1–10,11,13,15
Homework Help
Exercises
1. 2. 3.
4. A cone has diameter 12 in. and slant height 9 in. Tell whether doubling
both dimensions would double the surface area.
5. The rooms at the Wigwam Village Motel in Cave City, Kentucky, are cones
about 20 ft high and have a diameter of about 20 ft. Estimate the lateral
surface area of a room.
6. 7. 8.
9. A regular square pyramid has a base with 12 yd sides and slant height 5 yd.
Tell whether doubling both dimensions would double the surface area.
10. In the late 1400s, Leonardo
da Vinci designed a parachute
shaped like a pyramid. His design
called for a tent-like structure
made of linen, measuring 21 feet
on each side and 12 feet high.
Estimate how much material
would be needed to make the
parachute.
Find the surface area of each figure with the given dimensions.
Use 3.14 for π.
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 1
See Example 2
See Example 3
d h f f h f h h
Prep MA.8.G.5.1
E
E
9-8
on p. WS12
ⴝ
11. regular triangular pyramid:
base area ⴝ 0.06 km2
base perimeter ⴝ 0.8 km
slant height ⴝ 0.3 km
12. cone: r ⴝ 121
__
2
mi
slant height ⴝ 441
__
4
mi

Social Studies
MA.8.A.6.4, MA.8.G.5.1
9-8 Surface Area of Pyramids and Cones 437
13. Earth Science When the Moon
is between the Sun and Earth,
it casts a conical shadow called
the umbra. If the shadow is
2140 mi in diameter and 260,955
mi along the edge, what is the
lateral surface area of the umbra?
14. Social Studies The Pyramid Arena in Memphis, Tennessee, is 321 feet
tall and has a square base with side length 200 yards. What is the lateral
surface area of the pyramid in square feet?
15. The table shows the dimensions of
three square pyramids.
a. Complete the table.
b. Which pyramid has the least
lateral surface area? What is its
lateral surface area?
c. Which pyramid has the greatest
volume? What is its volume?
16. Write a Problem An ice cream cone has a diameter of 4 in. and a slant
height of 11 in. Write and solve a problem about the ice cream cone.
17. Write About It The height and base dimensions of a cone are known.
Explain how to find the slant height.
18. Challenge The oldest pyramid is said to be the Step Pyramid of King
Zoser, built around 2650 B.C.E. in Saqqara, Egypt. The base is a rectangle
that measures 358 ft by 411 ft, and the height of the pyramid is 204 ft. Find
the lateral surface area of the pyramid.
Dimensions of Giza Pyramids (ft)
Slant Side of
Pyramid Height Height Base
Khufu 612 756
Khafre 471 588 704
Menkaure 216 346
The Pyramid
Arena is the third
largest pyramid
in the world.
19. Multiple Choice Find the surface area of a triangular pyramid with base
area 12 square meters, base perimeter 24 meters, and slant height 8 meters.
A. 72 m2
B. 108 m2
C. 204 m2
D. 2304 m2
20. Gridded Response What is the lateral surface area in square centimeters of a
cone with diameter 12 cm and slant height 6 cm? Use 3.14 for π.
Simplify. (Lesson 4-7)
21. 2兹
____
169 22. 3兹
___
18 23. 1
_
3兹
___
99 24. 3兹
___
30
Convert each measure. (Lesson 5-6)
25. 10 kilograms to pounds 26. 20 pints to liters 27. 10.8 seconds to minutes

3
3
3
2
2
2
1
1 1
6 cm
6 cm
6 cm
2 cm
2 cm
2 cm
9-9
Scaling Three-Dimensional
Figures
A packaging company sells boxes in a variety of sizes. It offers a supply
of cube boxes that measure 1 ft ⴛ 1 ft ⴛ 1 ft, 2 ft ⴛ 2 ft ⴛ 2 ft, and
3 ft ⴛ 3 ft ⴛ 3 ft. What is the volume and surface area of each of these boxes?
Edge Length 1 ft 2 ft 3 ft
Volume 1 ⴛ 1 ⴛ 1 ⴝ 1 ft3
2 ⴛ 2 ⴛ 2 ⴝ 8 ft3
3 ⴛ 3 ⴛ 3 ⴝ 27 ft3
Surface Area 6 ⴢ 1 ⴛ 1 ⴝ 6 ft2
6 ⴢ 2 ⴛ 2 ⴝ 24 ft2
6 ⴢ 3 ⴛ 3 ⴝ 54 ft2
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 23
ⴝ 8 times as large,
and its surface area is 22
ⴝ 4 times as large as the 1 ft cube’s.
Scaling Models That Are Cubes
A 6 cm cube is built from small
cubes, each 2 cm on an edge.
Compare the following values.
the edge lengths of the large and small cubes
6 cm cube
_________
2 cm cube
6 cm
____
2 cm
ⴝ 3
The edges of the large cube are 3 times as long as those of the
small cube.
the surface areas of the two cubes
6 cm cube
_________
2 cm cube
6 ⴢ (6 cm)2
_________
6 ⴢ (2 cm)2
216 cm2
_______
24 cm2
ⴝ 9
The surface area of the large cube is 32
ⴝ 9 times that of the
small cube.
the volumes of the two cubes
6 cm cube
_________
2 cm cube
(6 cm)3
______
(2 cm)3
216 cm3
_______
8 cm3
ⴝ 27
The volume of the large cube is 33
ⴝ 27 times that of the small cube.
E X A M P L E X
X
X
E X A M P L E 1
A
Ratio of
corresponding edges
B
Ratio of
corresponding areas
C
Ratio of
corresponding volumes
Vocabulary
capacity
MA.8.G.5.1
Compare, contrast,
and convert units
of measure between different
measurement systems (US
customary or metric (SI)) and
dimensions including... area,
volume, and derived units to
solve problems.
Multiplying the
linear dimensions of
a solid by n creates
n2
as much surface
area and n3
as much
volume.

2 m
60 m
30 m
65 m
93 m
9-9 Scaling Three-Dimensional Figures 439
Think and Discuss
1. Describe how the volume of a model compares to the original
object if the linear scale of the model is 1:2.
2. Explain one possible way to double the surface area of a
rectangular prism.
Scaling Models That Are Other Solid Figures
The Fuller Building in New York, also known as
the Flatiron Building, can be modeled as a
trapezoidal prism with the approximate
dimensions shown. For a 10 cm tall model of
the Flatiron Building, find the following.
What is the scale of the model?
10 cm
_____
93 m
ⴝ 10 cm
_______
9300 cm
ⴝ 1
___
930
Convert and
simplify.
The scale of the model is 1:930.
What are the other dimensions of the model?
left side: 1
___
930 ⴢ 65 m ⴝ 6500
____
930
cm ⬇ 6.99 cm
back: 1
___
930 ⴢ 30 m ⴝ 3000
____
930
cm ⬇ 3.23 cm
right side: 1
___
930 ⴢ 60 m ⴝ 6000
____
930
cm ⬇ 6.45 cm
front: 1
___
930 ⴢ 2 m ⴝ 200
___
930
cm ⬇ 0.22 cm
The trapezoidal base has side lengths 6.99 cm, 3.23 cm, 6.45 cm,
and 0.22 cm.
Business Application
A machine fills a cube box that has edge lengths of 1 ft with
shampoo samples in 3 seconds. How long does it take the
machine to fill a cube box that has edge lengths of 4 ft?
V ⴝ 4 ft ⴢ 4 ft ⴢ 4 ft ⴝ 64 ft3
Find the volume of the larger box.
3 s
____
1 ft3
ⴝ x s
_____
64 ft3
Set up a proportion and solve.
3 ⴢ 64 ⴝ x Cross multiply.
192 ⴝ x Calculate the fill time.
It takes 192 seconds to fill the larger box.
E X A M P L E 2
A
B
E X A M P L E 3

Exercises 1–10, 11, 13, 15, 17, 21
Homework Help
Exercises
on p. WS12
ⴝ
MA.8.G.5.1
E
E
9-9
An 8 in. cube is built from small cubes, each 2 in. on an edge. Compare
the following values.
1. the edge lengths of the large and small cubes
2. the surface areas of the two cubes 3. the volumes of the two cubes
4. The dimensions of a basketball arena are 500 ft long, 375 ft wide, and
125 ft high. The scale model used to build the arena is 40 in. long. Find
the width and height of the model.
5. A 3-ft-by-1-ft-by-1-ft fish tank in the shape of a rectangular prism drains
in 3 min. How long would it take a 7-ft-by-4-ft-by-4-ft fish tank to drain at
the same rate?
A 6 m cube is built from small cubes, each 3 m on an edge. Compare
the following values.
6. the edge lengths of the large and small cubes
7. the surface areas of the two cubes 8. the volumes of the two cubes
9. The Great Pyramid of Giza has a square base measuring 230 m on each
side and a height of about 147 m. Nathan is building a model of the
pyramid with a 50 cm square base. What is the height to the nearest
centimeter of Nathan’s model?
10. A pool 5 ft deep with a diameter of 40 ft is filled with water in 50 minutes.
How long will it take to fill a pool that is 6 ft deep with a diameter of 36 ft?
For each cube, a reduced scale model is built using a scale factor of ᎏ
1
2
ᎏ. Find the
length of the model and the number of 1 cm cubes used to build it.
11. a 2 cm cube 12. a 6 cm cube 13. an 18 cm cube
14. a 4 cm cube 15. a 14 cm cube 16. a 16 cm cube
17. What is the volume in cubic centimeters of a 1 m cube?
18. An insulated lunch box measures 7 in. by 9.5 in. by 5 in. A larger version of
the lunch box is available. Its dimensions are greater by a linear factor of
1.1. How many times greater is the volume of the larger lunch box than
the smaller one? What is the volume of the larger lunch box?
19. Art A sand castle requires 3 pounds of sand. How much sand would be
required to double all the dimensions of the sand castle?
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 1
See Example 2
See Example 3

MA.8.G.5.1, MA.8.A.1.2, MA.8.A.6.1, MA.8.A.6.3
9-9 Scaling Three-Dimensional Figures 441
25. Multiple Choice A 9-inch cube is created from small cubes, each 1 inch
on an edge. What is the ratio of the volume of the larger cube to the volume
of the smaller cube?
A. 1:9 B. 9:1 C. 81:1 D. 729:1
26. Extended Response A 5-inch cube is created from small cubes, each
1 inch on an edge. Compare the edge lengths, surface area, and volume of the
larger to the smaller cube.
27. The circumference of Jupiter is about 2.79 ⴛ 105
miles. The average diameter of
Earth is about 7.92 ⴛ 103
miles. How many times could Earth’s diameter fit along
Jupiter’s circumference to the nearest tenth? (Lesson 4-4)
Write in slope-intercept form, and then find the slope and y-intercept. (Lesson 8-3)
28. 3x ⴝ 12y ⴙ 12 29. 4x ⴙ 5y ⴝ 12 30. x ⴝ 1
__
2
y ⴚ 3
__
2
20. Recreation If it took 100,000 plastic bricks
to build a cylindrical monument with a 5 m
diameter, about how many would be
needed to build a monument with an 8 m
diameter and the same height?
21. A kitchen sink measures 21 in. by 16 in. by
8 in. It takes 4 minutes 30 seconds to fill with
water. A smaller kitchen sink takes 4 min
12 seconds to fill with water.
a. What is the volume of the smaller
kitchen sink?
b. About how many gallons of water does
the smaller kitchen sink hold?
(Hint: 1 gal ⴝ 231 in3
)
22. Choose a Strategy Six 1 cm cubes are used
to build a solid. How many cubes are used to
build a scale model of the solid with a linear scale
of 2 to 1? Describe the tools and techniques you used.
A. 12 cubes B. 24 cubes C. 48 cubes D. 144 cubes
23. Write About It If the linear scale of a model is ᎏ
1
5
ᎏ, what is the relationship
between the volume of the original object and the volume of the model?
24. Challenge To double the volume of a rectangular prism, what number is
multiplied by each of the prism’s linear dimensions? Give your answer to
the nearest hundredth.
It took more than 220 hours and
25,000 bricks to create the mask of
Tutankhamen.

6 m
5.4 m
3.6 m
1 ft2
12 in.
12 in.
144 in2
1 ft ฀12 in.
1 ft2
฀(12 in.)2
฀ 144 in.2
1 ft
1 ft .
U.S. basketball teams sometimes
play games overseas. This means they
must be able to convert dimensions
and areas between metric and
customary units.
In Lesson 5–6, you converted linear
units between measurement systems.
You can also convert units of area and
volume. For example, you can draw a
diagram to help you convert square
feet to square inches.
You can convert units of area by squaring the linear conversion factor.
Converting Units of Area
Find the area of the Olympic basketball free
throw lane in square feet to the nearest tenth.
The free throw lane is a trapezoid with
bases 3.6 m and 6 m with a height of 5.4 m.
Step 1: Find the area in square meters.
A ⴝ 1
__
2
h(b1
ⴙ b2
) ⴝ 1
__
2
(5.4)(3.6 ⴙ 6)
ⴝ 2.7(9.6) ⴝ 25.92 m2
Step 2: Find a conversion factor for meters
to feet.
1
______
0.3048 ⴢ 0.3048 m ⴝ 1 ft ⴢ 1
______
0.3048
1 foot ⴝ 0.3048 m.
1 m ⬇ 3.28 ft
Step 3: Convert the area.
25.92 m2
ⴢ (3.28 ft
_____
1 m )2
Square the linear conversion factor.
⬇ 25.92 m2
ⴢ (10.7584 ft2
_________
1 m2 )⬇ 278.9 ft2
The area of the Olympic basketball free throw lane is about 278.9 ft2
.
E X A M P L E 1
Measurement in Three-
Dimensional Figures
U.S. bas
S b
l
9-10
MA.8.G.5.1
Compare, contrast,
and convert units
of measure between different
measurement systems (U.S.
customary or metric (SI)) and
dimensions including... area,
volume, and derived units to
solve problems.
For common metric
or customary
conversions, see the
inside back cover.
For conversions
between the metric
and customary
measurement
systems, see p. 227.

1 ft ฀12 in.
1 ft3
฀(12 in.)3
฀ 1728 in3
12 in.
12 in.
12 in.
4 ft
7 ft
75 ft
8 ft
6 ft 4 ft
.
9-10 Measurement in Three-Dimensional Figures 443
Lesson Tutorial Videos
You can also convert units
of volume using a similar
method. Convert units of
volume by cubing the
linear conversion factor.
Converting Units of Volume
Find the volume of the cone in cubic meters.
Step 1: Find the volume in cubic feet.
V ⴝ 1
__
3
␲r2
h ⬇ 1
__
3
(3.14)(4)2
(7)
⬇ 117.2 ft3
Step 2: Convert the volume.
117.2 ft3
ⴢ (0.3048 m
________
1 ft )3
Cube the linear conversion factor.
⬇ 117.2 ft3
ⴢ (0.02832 m3
_________
1 ft3 )⬇ 3.3 m3
The volume of the cone is about 3.3 m3
.
Recreation Application
A lap pool is filling at a
rate of 25 liters per second.
How long will it take to
fill the pool?
1 ft3
艐 7.5 gal, 1 gal 艐 3.8 L
Step 1: Find the volume of the pool in cubic feet.
V ⴝ Bh ⴝ (1
__
2
h(b1
ⴙ b2
))h The pool is a trapezoidal prism.
ⴝ (1
__
2
(75)(4 ⴙ 6))8 ⴝ 3000 ft3
Step 2: Convert cubic feet to gallons.
1 ft3
⬇ 7.5 gal, so 3000 ft3
⬇ 3000(7.5) gal ⬇ 22,500 gal
Step 3: Convert gallons to liters.
1 gal ⬇ 3.8 L, so 22,500 gal ⬇ 22,500(3.8) L ⬇ 85,500 L
Step 4: Find the time it takes to fill the pool.
85,550 L ⴢ 1 s
____
25 L
ⴝ 3420 s
It will take about 3420 seconds, or about 57 minutes, to fill the pool.
E X A M P L E 2
E X A M P L E 3
Think and Discuss
1. Explain how to convert an area in square feet to square centimeters.
2. Draw a diagram to show the number of cubic feet in a cubic yard.
Make a reference
sheet of conversion
factors.

10 m
18 m
2.5 ft
1 ft
1.5 ft
4 ft
1
2 9 ft
1
4
7 ft
3
4
85 in.
6 in.
2.1 cm
5 cm
7 cm
4 cm
10 cm
Exercises 1–7, 9, 11, 15, 21
Homework Help
Exercises
1. A vacant lot is shaped like a right triangle as
shown. Find the area of the vacant lot in
square yards.
2. Find the volume of the aquarium in
cubic centimeters.
3. Tents are priced at $15 per cubic meter.
Find the price of the tent shown
to the nearest dollar.
4. Find the surface area of the cylinder in
square centimeters. Use 3.14 for ␲.
5. A grocer sells a wedge of cheese with the
dimensions shown. What is the volume
of the cheese wedge, in cubic inches?
6. A pyramid has a height of 18 meters, and its base covers 90 square
meters. Find the volume of the pyramid in cubic feet.
7. The cylinder is filled with crystals that cost $20 per
teaspoon. What is the value of the crystals in the
cylinder? (1 in3
⬇ 3.3 tsp)
Convert.
8. 1 mi2
ⴝ yd2
9. 1 km2
ⴝ m2
10. 468 in2
ⴝ ft2
11. 6000 cm2
⬇ ft2
12. 500 cm3
⬇ in3
13. 37 yd3
⬇ m3
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 1
See Example 2
See Example 3
MA.8.G.5.1
E
E
9-10
on p. WS12
ⴝ

MA.8.G.5.1, MA.8.A.6.1, MA.8.A.4.2
1.6 ft
4.4 ft
3.5 in.
3 in. 2 in.
Juice
Juice
9-10 Measurement in Three-Dimensional Figures 445
25. Which is the correct conversion factor for converting square feet to
square millimeters?
A. 304.8 mm2
_________
1 ft2 B. 609.6 mm2
_________
1 ft2 C.
92,903.04 mm2
____________
1 ft2 D.
28,316,846.6 mm2
______________
1 ft2
26. Short Response Tracie bought a rug that measures 9 feet by 12 feet. The cost is
$6.50 per square meter. How much did she pay for the rug? Explain.
Simplify the powers of 10. (Lesson 4-2)
27. 10ⴚ4
28. 10ⴚ1
29. 100
30. 10ⴚ5
Solve. (Lesson 7-5)
31. ⴚ3x ⬍ 9 32. 2x ⴙ 17 ⱖ 9 33. 1
__
2
⬎ x
__
4
ⴙ 1
Find the conversion factor to make each conversion.
14. square feet to square meters 15. square miles to square meters
16. cubic centimeters to cubic inches 17. cubic feet to cubic inches
18. Geography The state of Florida covers an area of 58,560 square miles.
Convert this area to square kilometers. (1 mi ⬇ 1.61 km)
19. Manufacturing The dimensions of a juice box
are shown. (1 cm3
⬇ 0.034 fl oz)
a. What is the volume in cubic inches?
b. What is the volume in cubic centimeters?
c. Is the box larger or smaller than one that holds 12 fluid ounces?
About how many fluid ounces does the box contain?
20. Baking Brandon bought a frozen pizza with an area of 1700 cm2
.
The baking time is 5 minutes per 60 in2
. How long should the pizza
bake to the nearest minute?
21. Water One quart of water weighs about 2 pounds.
The 60 lb container shown at right is filled with
water. (1 ft3
⬇ 29.9 qt)
a. What is the weight of the filled container in pounds?
b. What is the weight of the filled container in kilograms?
22. Critical Thinking Which is greater, an area of 5 in2
or 50 cm2
? a volume
of 5 in3
or 50 cm3
? Explain any difference in the units of your answers.
23. What’s the Error? A student wrote that 60 square feet equals
720 square inches because 60 ⴢ 12 ⴝ 720. Explain and correct the error.
24. Challenge One parsec is equivalent to about 3.26 light-years. One
light-year is about 5.9 ⴛ 1012
miles. About how many square miles are
in a square parsec? Give your answer in scientific notation.

5 cm
6 cm 7 cm
24 in.
4 in. 2 ft
8 ft
12 ft
7 cm
5 cm
6 cm
9 m 4 m
12 m
10.9 m
10 m
10 m
5 ft
9 ft
130 cm
130 cm
Ready
to
Go
On?
Ready To Go On?
Ready To Go On?
SECTION 9B
Quiz for Lessons 9-5 Through 9-10
Volume of Prisms and Cylinders
1. 2. 3.
Volume of Pyramids and Cones
4. 5. 6.
Surface Area of Prisms and Cylinders
Find the surface area of the indicated figure to the nearest tenth. Use 3.14 for π.
7. the prism from Exercise 1 8. the cylinder from Exercise 2
Surface Area of Pyramids and Cones
Find the surface area of the indicated figure to the nearest tenth. Use 3.14 for π.
9. the pyramid from Exercise 5
10. the cone from Exercise 6 if the slant height is 9.2 m
Scaling Three-Dimensional Figures
11. The dimensions of a skating arena are 400 ft long, 280 ft wide, and
100 ft high. The scale model used to build the arena is 20 in. long.
Find the width and height of the model.
Measurement in Three-Dimensional Figures
Find the area of the square to the nearest tenth.
12. in square meters.
13. in square feet.
Find the volume of the cone. Use 3.14 for π.
14. in cubic inches.
15. in cubic meters to the nearest tenth.
V
i d h
Fi
F
9-5
V
9-6
S
9-7
S
9-8
S
1
11 Th
11
9-9
M
i d h
Fi
F
9-10
Ready To Go On?

FLORIDA
E
C
D
A
F
x¡
x¡
40°
B
n
m
60°
50°
Real-World
Connections
Real World Connection 447
Water, Water Everywhere The Dry Tortugas are a group
of islands about 70 miles west of Key West. Visitors to Dry Tortugas
National Park can explore Fort Jefferson, an abandoned fort that
dates from the 1840s.
The figure shows an overhead view of Fort Jefferson. Use the
figure for 1–6.
1. What type of polygon is polygon
ABCDEF?
2. What is the sum of the angle
measures in polygon ABCDEF?
3. What is m⬔AFE? Why?
4. Find x° in the isosceles triangle.
5. In the figure, line m is parallel to line n.
What can you say about m⬔ADC?
Explain.
6. Wall
___
AB of Fort Jefferson is 250 ft long and wall
___
BC is 375 ft long. In
a scale model of the fort, wall
___
AB is 2 ft long.
a. What is the scale of the model?
b. What is the length of wall
___
BC in the model?
Dry Tortugas

Triple Concentration
Planes in Space
Some three-dimensional figures can be generated by plane figures.
Experiment with a circle first. Move the circle around. See if you
recognize any three-dimensional shapes.
If you spin a circle If you slide a circle up If you spin a circle
around a diameter, you along a line perpendicular around a line outside the
get a sphere. to the plane that the circle circle but in the same
is in, you get a cylinder. plane as the circle, you get
a donut shape called a
torus.
Draw or describe the three-dimensional figure generated by
each plane figure.
a square slid along a line perpendicular to the plane it is in
a rectangle spun around one of its edges
a right triangle spun around one of its legs
1
2
3
The goal of this game is to form Pythagorean
triples, which are sets of three whole numbers
a, b, and c such that a2
ⴙ b2
ⴝ c2
. A set of
cards with numbers on them are arranged
face down. A turn consists of drawing 3 cards
to try to form a Pythagorean triple. If the
cards do not form a Pythagorean
triple, they are replaced in their
original positions.
A complete set of rules and cards are available online. Go to thinkcentral.com
Games

A
B
It’s in the Bag! 449
Materials
• white paper
• scissors
• decorative paper
• stapler
• hole punch
• twine or yarn
• cardboard tube
• glue
• markers
The Tube Journal
PROJECT
Use this journal to take notes on perimeter, area,
and volume. Then roll up the journal and store it
in a tube for safekeeping!
Directions
1 Start with several sheets of paper that measure
81
_
2
inches by 11 inches. Cut an inch off the end
of each sheet so they measure 81
_
2
inches by
10 inches.
2 Stack the sheets and fold them in half
lengthwise to form a journal that is
approximately 41
_
4
inches by 10 inches. Cover
the outside of the journal with decorative
paper, trim it as needed, and staple everything
together along the edge. Figure A
3 Punch a hole through the journal in the top left
corner. Tie a 6-inch piece of twine or yarn
through the hole. Figure B
4 Use glue to cover a cardboard tube with
decorative paper. Then write the name and
number of the chapter on the tube.
Taking Note of the Math
Use your journal to take notes on perimeter, area,
and volume. Then roll up the journal and store it in
the cardboard tube. Be sure the twine hangs out of
the tube so that the journal can be pulled out easily.

1
2
3
68°
Study
Guide:
Review
Vocabulary
Study Guide: Review
MA.8.G.2.2
acute angle . . . . . . . . . . . . 400
acute triangle . . . . . . . . . . 409
adjacent angles . . . . . . . . 401
angle . . . . . . . . . . . . . . . . . . 400
capacity . . . . . . . . . . . . . . . 438
complementary angles . . 400
congruent angles . . . . . . 401
equilateral triangle . . . . . 410
isosceles triangle . . . . . . . 410
lateral face . . . . . . . . . . . . 430
lateral surface . . . . . . . . . 430
obtuse angle . . . . . . . . . . . 400
obtuse triangle . . . . . . . . . 409
parallel lines . . . . . . . . . . . 406
parallelogram . . . . . . . . . 414
perpendicular lines . . . . 406
polygon . . . . . . . . . . . . . . . . 414
rectangle . . . . . . . . . . . . . . 415
regular polygon . . . . . . . . 415
regular pyramid . . . . . . . 434
rhombus . . . . . . . . . . . . . . . 415
right angle . . . . . . . . . . . . . 400
right cone . . . . . . . . . . . . . 434
right triangle . . . . . . . . . . 409
scalene triangle . . . . . . . . 410
slant height . . . . . . . . . . . . 434
square . . . . . . . . . . . . . . . . . 415
straight angle . . . . . . . . . . 400
supplementary angles . . 400
surface area . . . . . . . . . . . 430
transversal . . . . . . . . . . . . 406
trapezoid . . . . . . . . . . . . . . 415
Triangle Sum Theorem . 409
vertical angles . . . . . . . . . 401
volume . . . . . . . . . . . . . . . . 420
Complete the sentences below with vocabulary words from the
list above.
1. Lines in the same plane that never meet are called ___?___.
Lines that intersect at 90° angles are called ___?___.
2. A quadrilateral with 4 congruent angles is called a ___?___.
A quadrilateral with 4 congruent sides is called a ___?___.
3. The nonadjacent angles formed by two intersecting lines
are called ___?___.
Angle Relationships (pp. 400–404)
■ Find the angle measure.
m⬔1
m⬔1 ⴙ 122° ⴝ 180°
ⴚ122° ⴚ122°
ᎏᎏᎏᎏ ᎏᎏ
m⬔1 ⴝ 58°
Find each angle measure.
4. m⬔1
5. m⬔2
6. m⬔3
E X A M P L E S EXERCISES
9-1
1 122°
2 3
abulary
Voca
abulary
Voca
FLORIDA
C H A P T E R
H A
A P
P
P T E R
R
R
R
R
R
R
R
9
Multi–Language
Glossary

13 cm
8 cm
7 cm
4.2 mm
7.5 mm
p⬚
78ⴗ
47ⴗ
128°
m°
m°
1
p
q
2
3
4
5
114°
Study Guide: Review 451
Study
Guide:
Review
MA.8.G.2.2
MA.8.G.2.3
Parallel and Perpendicular Lines (pp. 405–408)
Line j
line k. Find each angle measure.
■ m⬔1
m⬔1 ⴝ 143°
■ m⬔2
m⬔2 ⴙ 143° ⴝ 180°
ⴚ 143° ⴚ 143°
ᎏᎏᎏᎏ ᎏᎏ
m⬔2 ⴝ 37°
Line p
line q. Find each angle measure.
7. m⬔1
8. m⬔2
9. m⬔3
10. m⬔4
11. m⬔5
Triangles (pp. 409–412)
■ Find n°.
n° ⴙ 50° ⴙ 90° ⴝ 180°
n° ⴙ 140° ⴝ 180°
ⴚ 140° ⴚ 140°
ᎏᎏ
ᎏᎏᎏ ᎏᎏ
n° ⴝ 40°
12. Find m°. 13. Find p°.
Polygons (pp. 413–417)
■ Find the angle measures in a regular 12-gon.
12x° ⴝ 180°(12 ⴚ 2)
12x° ⴝ 180°(10)
12x° ⴝ 1800°
x° ⴝ 150°
Find the angle measures in each regular
polygon.
14. a regular octagon
15. a regular 11-gon
Volume of Prisms and Cylinders (pp. 420–424)
■ Find the volume to the nearest tenth.
V ⴝ Bh ⴝ (πr2)h
ⴝ π(32)(4)
ⴝ (9π)(4) ⴝ 36π cm3
⬇ 113.0 cm3
Find the volume to the nearest tenth.
16. 17.
Volume of Pyramids and Cones (pp. 425–429)
■ Find the volume.
V ⴝ 1
__
3
Bh ⴝ 1
__
3
(5)(6)(7)
ⴝ 70 m3
Find the volume of each figure.
Use 3.14 for π.
18. 19.
P
9-2
T
9-3
P
9-4
V
9-5
V
9-6
MA.8.G.2.3
Prep MA.8.G.5.1
143° 2
1
j
k
50°
n°
3 cm
4 cm
8 in.
9 in. 5 in.
12 in.
5 in.
7 m
6 m
5 m
Prep MA.8.G.5.1

14.9 mm
20 mm
12 mm
10 mm
8 mm
8 cm
6 cm
6 cm
10 in.
12 in.
Study
Guide:
Review
Measurement in Three-Dimensional Figures (pp. 442–445)
■ Find the area in square
feet to the nearest tenth.
Find the area in m2
.
A ⴝ 1
__
2
(3)(4) ⴝ 6 m2
.
Convert to ft2
. 1 m 艐 3.28 ft
A ⴝ 6 m2
ⴢ (3.28 ft
_____
1m )2
ⴝ 64.5504 ft2
艐 65.6 ft2
.
A rectangular acre of land is 264 ft ⴛ 165 ft.
28. Find the area in yd2
.
29. Find the area in m2
.
For a cylinder with r ⴝ 6 in. and h ⴝ 1 ft,
30. Find the volume in in3
.
31. How long will it take for the cylinder to
fill with liquid at 3.5 cm3
/s?
M
9-10
Scaling Three-Dimensional Figures (pp. 438–441)
■ A 4 in. cube is built from small cubes, each
2 in. on an edge. Compare the volumes
of the large cube and the small cube.
vol. of large cube
______________
vol. of small cube
ⴝ 43
in3
_____
23
in3
ⴝ 64 in3
_____
8 in3
ⴝ 8
The volume of the large cube is 8 times
that of the small cube.
A 9 ft cube is built from small cubes, each
3 ft on an edge. Compare the indicated
measures of the large cube and the
small cube.
25. edge lengths 26.surface areas
27. volumes
S
9-9
Surface Area of Pyramids and Cones (pp. 434–437)
■ Find the surface area.
S ⴝ B ⴙ 1
__
2
Pᐉ
ⴝ 16 ⴙ 1
__
2
(16)(5)
ⴝ 56 in2
Find the surface area of each figure.
22. 23.
24. Explain whether doubling the height
and diameter of the cone in Exercise 23
would double the surface area.
S
9-8
Surface Area of Prisms and Cylinders (pp. 430–433)
■ Find the surface area.
S ⴝ 2B ⴙ Ph
ⴝ 2(6) ⴙ (10)(4)
ⴝ 52 in2
Find the surface area of the figure.
20.
21. A tissue box is a rectangular prism with
two square bases 11 cm on a side and a
height of 14 cm. A pattern covers all of the
surfaces.What is the area of the pattern?
S
9-7 Prep MA.8.G.5.1
Prep MA.8.G.5.1
MA.8.G.5.1
3 in.
2 in.
4 in.
4 in.
5 in.
4 in.
MA.8.G.5.1
4 m
3 m

Chapter
Test
135°
1 2
3 4
m
n
13 in.
15 in.
24 in. 6 m
4 m
4 m
5 m
5 m
4 in.
6 in.
Chapter 9 Test 453
m
FLORIDA
C H A P T E R
H A
A P T E R
R
R
R
R
R
R
R
9
ChapterTest
ChapterTest
line n.
1. Name two pairs of supplementary angles.
2. Find m⬔1. 3. Find m⬔2.
4. Find m⬔3. 5. Find m⬔4.
6. Two angles in a triangle have measures of 44° and 57°. What is the
measure of the third angle?
7. The angles in a triangle have measures 2x°, 2x°, and 5x°. Find the angle
measures and classify the triangle.
8. Find the sum of the angle measures in a 20-gon.
9. The measure of each angle in a regular polygon is 172°. How many
sides does the polygon have?
10. a cube of edge length 8 ft 11. a cylinder of height 5 cm and radius 2 cm
12. a cone of diameter 12 in. and height 18 in.
13. a rectangular prism with base 5 m by 3 m and height 6 m
14. a pyramid with a 3 ft by 3 ft square base and height 4 ft
15. Explain whether doubling the diameter and height of the cone in Problem 12 would
double the volume of the cone.
16. 17. 18.
19. The dimensions of a history museum are 400 ft long, 200 ft wide, and
75 ft tall. The scale model used to build the museum is 40 in. long.
Find the width and height of the model.
A cone has a radius of 3 inches, a height of 4 inches, and a slant height
of 5 inches.
20. Find the surface area of the cone in square centimeters to
the nearest tenth.
21. Find the volume of the cone in cubic centimeters
to the nearest tenth.

6 cm
3 cm
Figure I
4 cm
6 cm
3 cm
Figure III
6 cm
3 cm
Figure IV
3 cm
9 cm
Figure II
Test
Tackler
Standardized Test Strategies
Standar
Test Tackler
Test Tackler
FLORIDA
C H A P T E R
H A
A P T E R
R
R
R
R
R
R
R
9
Multiple Choice: Context-Based Test Items
Sometimes a multiple-choice test item requires you to use information in the
answer choices to determine which choice fits the context of the problem.
Which statement is supported by the figure?
A. ⬔1 and ⬔4 are supplementary. C. The measure of ⬔7 is 35°.
B. ⬔3 and ⬔2 are vertical angles. D. ⬔5 and ⬔6 are congruent.
Read each answer choice to find the best answer.
Choice A: ⬔1 and ⬔4 are vertical angles and therefore, congruent.
Congruent angles are supplementary only if they are right angles.
⬔1 and ⬔4 measure 35°.
Choice B: ⬔3 and ⬔2 are vertical angles. This is the correct answer choice.
Choice C: The measure of ⬔7 cannot be 35° because ⬔7 is supplementary
to a 35° angle. Therefore, ⬔7 has a measure of 145°.
Choice D: ⬔5 and ⬔6 are supplementary angles but not right angles.
Supplementary angles are congruent only if they are both right angles.
Which two figures have the same area?
F. Figure I and Figure II H. Figure II and Figure III
G. Figure I and Figure III I. Figure I and Figure IV
Find the areas of all four figures and compare them.
Figure I: 3 ⴢ 6 ⴝ 18 cm2
Figure III: 1
__
2 ⴢ 4(3 ⴙ 6) ⴝ 18 cm2
Figure II: 3 ⴢ 9 ⴝ 27 cm2
Figure IV: 1
__
2 ⴢ 6 ⴢ 3 ⴝ 9 cm2
Figures I and III have the same area. Choice G is correct.
atement is
EXAMPLE 1
o figures
EXAMPLE 2
1
2 4
3
5 7
35º
6

8 in.
12 in.
x k 8 4
y 14 22 k
r
Test Tackler 455
Test
Tackler
Do not choose an answer until you have
read all of the answer choices.
Do n
read
D
r
HOT
HOT
Tip!
Tip!
Read each test item and answer the
questions that follow.
Item A
The area of the square is 16 cm2
.
Which of the following is NOT correct
about the circle?
A. C ⴝ 4π cm
B. A ⴝ 16π cm2
C. d ⴝ 4 cm
D. r ⴝ 2 cm
1. Since only one answer choice has
incorrect information, why can you
automatically eliminate answer
choices C and D?
2. How can you find the side length
of the square? What does the
side length tell you about the circle?
3. Use your answer to Problem 2 to
determine whether answer choice A
has correct information.
4. How can you tell that choice B is the
correct answer?
Item B
Which figure is an acute isosceles
triangle?
F. H.
G. I.
5. What is an acute triangle?
6. What is an isosceles triangle?
7. Why is choice F incorrect?
Item C
What value(s) of k results in a linear
function?
A. 6 only C. 6 and 24
B. 12 only D. 6 and 12
8. Why would it be difficult to solve
for k?
9. If k has a value of 6, is the function
linear?
10. What answer choices can you
rule out based on your answer to
question 9?
Item D
The area of the trapezoid is 30 in2
.
Which equation CANNOT be used to
find the height of the trapezoid?
F. 30 ⴝ 1
__
2
(8 ⴙ 12)h
G. 60 ⴝ (8 ⴙ 12)h
H. 30 ⴝ 1
__
2
(8 ⴚ 12)h
I. 1
__
2
(8 ⴙ 12)h ⴝ 30
11. What is the formula for the area of
a trapezoid?
12. What steps would you take to solve
the formula for h?

Silver
Gold
Precious Metals Company
Supply of Gold and Silver
Mastering
the
Standards Mastering the
Standards
Standards
Florida Test
Practice
Cumulative Assessment, Chapter 1–9
Multiple Choice
1. The surface area of an object is
measured in square feet. Which
conversion results in the largest
number of square units?
A. to square centimeters
B. to square inches
C. to square meters
D. to square miles
2. For which of the following polygons is
the sum of three of its angle measures
equal to 324°?
F. H.
G. I.
3. If
gx
__
g5
ⴝ gⴚ8
and gⴚ3
ⴢ gy
ⴝ g12
, what is
the value of x ⴙ y?
A. ⴚ9 C. 12
B. ⴚ3 D. 15
4. A jeweler buys a pearl and resells it
for $299, which is $26 more than a
300% increase over his cost. What
did the pearl cost the jeweler to the
nearest dollar?
F. $54 H. $81
G. $68 I. $91
5. 27 cubic blergs is equal to 1 cubic quil.
Which is the correct conversion factor for
converting square quil to square blergs?
A. (1 blerg
______
3 quil )
2
C. (3 quil
______
1 blerg)
2
B. ( 1 quil
_______
3 blergs)
2
D. (3 blergs
_______
1 quil )
2
6. For the equation 2
_
3
x ⴙ 1
_
6
ⴝ 1, what is
the value of x when it is rounded up
to the next whole number?
F. 0 H. 2
G. 1 I. 4
7. A relation has a domain of 1, 3, and 5
and a range of 1, 5, 7, and 10. Which
table can NOT represent the relation?
A. x 5 3 3 1 1
y 10 1 7 5 5
B. x 1 5 1 3 3
y 10 5 7 1 5
C. x 5 3 5 5 1
y 5 5 7 10 1
D. x 1 5 1 10 3
y 7 10 7 1 5
8. The value of gold is about 60 times the
value of silver. What is the closest ratio
of gold value to silver value for the
Precious Metal Company’s supply?
F. 0.86 to 1 H. 51.82 to 1
G. 6.97 to 1 I. 69.47 to 1
mulative Asses
Cum
i l Ch i
M lti
mulative A
Cum
FLORIDA
C H A P T E R
A P T E R
R
R
R
R
R
R
R
H A
A
9

68 m
48 m
36 m
60 m
32
80
q
m
n
22 in.
6 in.
7 in.
18° 62°
x
Cumulative Assessment, Chapters 1–9 457
Mastering
the
Standards
Use logic to eliminate answer choices
that are incorrect. This will help you
to make an educated guess if you are
having trouble with the question.
U
Use l
hat
o m
havin
U
t
t
h
HOT
HOT
Tip!
Tip!
9. Suzanne plans to install a fence around
the perimeter of her land. How much
fencing does she need?
A. At least 194 yd
B. At least 212 yd
C. At least 232 yd
D. At least 294 yd
10. Sheron found the area of her bedroom
using square units of centimeters,
meters, inches, and feet. The numbers
of square units she found are listed
below. What is the area of Sheron’s
bedroom in square inches?
F. 9 H. 14,400
G. 100 I. 93,025
Gridded Response
11. What is the value of x in the
expression 54
53
___
(51
)2 ⴝ 5x
?
12. Lines m and n are parallel. What is the
value of q in degrees?
13. How much larger than the average
of the measures of the angles in the
triangle, in degrees, is the measure of
the obtuse angle?
Short Response
S1. The number of diagonals in a polygon
is 1
_
2
n(n ⴚ 3). A regular polygon has
angles of 140° and side lengths of 15 cm.
a. What is the polygon?
b. What is the sum of the angle
measures of the polygon?
c. What is the perimeter of the polygon?
d. What is the number of diagonals of
the polygon?
S2. The cylinder formed by the net
shown takes 1 minute 36 seconds to
fill with sand.
a. At what rate per second does the
cylinder fill with sand to the nearest
tenth?
b. If the measurements are converted
to centimeters, what is the filling
rate per second to the nearest tenth?
Extended Response
E1. The regular octagon has a side length
of 1 m.
a. Find the area
using the 9
individual shapes
in the octagon
(the triangles are
isosceles).
b. Use 1 rectangle and 2 trapezoids to
find the area.
c. Find the area in square feet. Show
your work.
1 m
0.7 m

9A Two-Dimensional Geometry

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