Copyright © by The McGraw-Hill Companies, Inc.VOCABULARY area the number of square units needed to cover a region or a plane figure Lesson 2-3, p.. Copyright © by The McGraw-Hill Compani
Trang 1Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber
Trang 2Copyright © 2008 by The McGraw-Hill Companies, Inc All rights reserved Except as
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Trang 3California Math Triumphs
Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting
1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction
2A Chapter 2 Equivalence of Fractions
2B Chapter 3 Operations with Fractions
2B Chapter 4 Positive and Negative Fractions and Decimals
Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates
3A Chapter 2 Percents, Fractions, and Decimals
3B Chapter 3 Using Percents
3B Chapter 4 Rates and Proportional Reasoning
Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality
4A Chapter 2 Math Fundamentals
4B Chapter 3 Math Expressions
4B Chapter 4 Linear Equations
4B Chapter 5 Inequalities
Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships
5A Chapter 2 Graphing
5B Chapter 3 Proportional Relationships
5B Chapter 4 The Relationship Between
Graphs and Functions
Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry
Trang 4Authors and Consultants
AUTHORS
Frances Basich Whitney
Project Director, Mathematics K–12
Santa Cruz County Offi ce of Education
Capitola, California
Kathleen M Brown
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Dixie Dawson
Math Curriculum Leader Long Beach Unifi ed Long Beach, California
CONSULTANTS
Assessment
Donna M Kopenski, Ed.D.
Math Coordinator K–5
City Heights Educational Collaborative
San Diego, California
Instructional Planning and Support
Beatrice Luchin
Mathematics Consultant League City, Texas
ELL Support and Vocabulary
ReLeah Cossett Lent
Author/Educational Consultant Alford, Florida
Dinah-Might Activities, Inc.
San Antonio, Texas
Trang 5California Advisory Board
Carol Cronk
Mathematics Program Specialist
San Bernardino City Unifi ed
School District
San Bernardino, California
Audrey M Day
Classroom Teacher Rosa Parks Elementary School San Diego, California
Jill Fetters
Math Teacher Tevis Jr High School Bakersfi eld, California
Grant A Fraser, Ph.D.
Professor of Mathematics California State University, Los Angeles
Los Angeles, California
Eric Kimmel
Mathematics Department Chair
Frontier High School
Bakersfi eld, California
Donna M Kopenski, Ed.D.
Math Coordinator K–5 City Heights Educational Collaborative San Diego, California
Michael A Pease
Instructional Math Coach Aspire Public Schools Oakland, California
Chuck Podhorsky, Ph.D.
Math Director City Heights Educational Collaborative San Diego, California
Arthur K Wayman, Ph.D.
Professor Emeritus
California State University, Long
Beach
Long Beach, California
Frances Basich Whitney
Project Director, Mathematics K–12 Santa Cruz County Offi ce of Education
Capitola, CA
Mario Borrayo
Teacher Rosa Parks Elementary San Diego, California
Melissa Bray
K–8 Math Resource Teacher Modesto City Schools Modesto, California
CALIFORNIA ADVISORY BOARD
Glencoe wishes to thank the following professionals for their invaluable
feedback during the development of the program They reviewed
the table of contents, the prototype of the Student Study Guide, the
prototype of the Teacher Wraparound Edition, and the professional
Bonnie Awes
Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California
Kathleen M Brown
Math Curriculum Staff Developer Washington Middle School Long Beach, California
Trang 6California Reviewers
vi
CALIFORNIA REVIEWERS
Each California Reviewer reviewed at least two chapters of the Student
Study Guides, providing feedback and suggestions for improving the
effectiveness of the mathematics instruction
Bobbi Anne Barnowsky
Monica S Patterson
Educator Aspire Public Schools Modesto, California
Rechelle Pearlman
4th Grade Teacher Wanda Hirsch Elementary School Tracy, California
Armida Picon
5th Grade Teacher Mineral King School Visalia, California
Anthony J Solina
Lead Educator Aspire Public Schools Stockton, California
Trang 7Volume 6A Measurement
1-1 Unit Conversions: Metric Length 4
3AF1.4 Express simple unit conversions
in symbolic form (e.g., _ inches = _ feet × 12).
3MG1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes).
6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters)
7MG1.3 Use measures expressed
as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
Trang 82MG1.3 Measure the length
of an object to the nearest inch and/or centimeter.
3MG1.2 Estimate or determine the area and volume of solid fi gures by covering them with squares or by counting the number of cubes that would fi ll them.
3MG1.3 Find the perimeter of a polygon with integer sides.
4MG2.2 Understand that the length of a horizontal line segment equals
the difference of the x-coordinates.
4MG2.3 Understand that the length of a vertical line segment equals the
difference of the y-coordinates.
Alamo Square, San Francisco
Trang 94MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm 2 ), square meter (m 2 ), square kilometer (km 2 ), square inch (in 2 ), square yard (yd 2 ), or square mile (mi 2 ).
5MG1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two
of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle
on the parallelogram).
5MG1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to complete the surface area for these objects.
5MG1.3 Understand the concept
of volume and use the appropriate units
in common measuring systems (i.e., cubic centimeter [cm 3 ], cubic meter [m 3 ], cubic inch [in 3 ], cubic yard [yd 3 ]) to compute the volume of rectangular solids.
Trang 10x
Chapter
4-1 Lines 5MG2.1 54
4-2 Angles 5MG2.1 63
Progress Check 1 72
4-3 Triangles and Quadrilaterals 5MG2.1 73
4-4 Add Angles 5MG2.1, 5MG2.2, 6MG2.2 81
Progress Check 2 90
4-5 Congruent Figures 7MG3.4 91
4-6 Pythagorean Theorem 5MG2.1, 7MG3.3 99
Progress Check 3 108
4-7 Circles 6MG1.2 109
4-8 Volume of Triangular Prisms and Cylinders 117
6MG1.3 Progress Check 4 127
Assessment Study Guide 128
Chapter Test 134
Standards Practice 136
Standards Addressed
in This Chapter
5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straight edge, ruler, compass, protractor, drawing software).
5MG2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.
6MG1.2 Know common estimates
of π (3.14, _22
7 ) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid
6MG2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle
to solve problems involving an unknown angle.
7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to fi nd the length
of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical fi gures are congruent and what congruence means about the relationship between the sides and angles
of the two fi gures.
Mono Lake Tufa State Reserve
Trang 11HUNT
SCAVENGER
HUNT
Let’s Get Started
Use the Scavenger Hunt below to learn where things are
located in each chapter
Area of a Triangle
the area of the surface of a three-dimensional figure
complementary angles, supplementary angles
two lessons.
pages 126–131
found on page 3 The URL is ca.mathtriumphs.com
Trang 12Copyright © by The McGraw-Hill Companies, Inc.
Why is area important?
What if you compare your statistics in volleyball with your cousin’s, but the two
of you play on different-sized courts? It would be an unfair comparison Most school teams play on volleyball courts that are of regulation size.
Trang 13Copyright © by The McGraw-Hill Companies, Inc.
STEP 2 Preview Get ready for Chapter 3 Review these skills and compare
them with what you’ll learn in this chapter
STEP 1 Quiz Are you ready for Chapter 3? Take the Online Readiness
Quiz at www.mathtriumphs.com to find out
STEP 2 Preview Get ready for Chapter 3 Review these skills and compare
them with what you’ll learn in this chapter
STEP 1 Quiz Are you ready for Chapter 3? Take the Online Readiness
Quiz at ca.mathtriumphs.com to find out
You know how to measure the lengths
of items
JO
Area = length × width
Area = 2inches × 1 inch
Area = 2 square inches
You know that you can make a parallelogram into a rectangle
Area = base × height
Area = 3inches × 2 inches
Area = 6 square inches
Trang 14Copyright © by The McGraw-Hill Companies, Inc.
VOCABULARY
area
the number of square units needed to cover a region or a plane figure (Lesson 2-3, p 71)
w is the width of the
The area of the rectangle is 12 square centimeters
The units of area are square units.
Example 1
What is the area of the rectangle?
1 The length of the rectangle
is 5inches, and the width
is 3 inches
2 Substitute these values into
the formula Multiply
of cubes that would fill them.
4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm 2 ), square meter (m 2 ), square kilometer (km 2 ), square inch (in 2 ), square yard (yd 2 ), or square mile (mi 2 ).
Trang 15Copyright © by The McGraw-Hill Companies, Inc.
What is the area of the square?
1 The length of the
square is 4 feet, and
the width is 4 feet
What is the area of the square?
1 The length of the square is 5
kilometers, and the
kilometers
2 Substitute these values into the formula Multiply
What is the area of the rectangle?
1 The length of the rectangle is 7 yards, and the
Trang 16Copyright © by The McGraw-Hill Companies, Inc.
Guided Practice
Draw a rectangle for each given area.
AK
AK
AK
AK
3 What is the area of the rectangle?
Step 1 The length of the rectangle is 6 feet, and the width
Step by Step Practice
Find the area of each rectangle.
4 The length of the rectangle is 8 inches The width is
Trang 17Copyright © by The McGraw-Hill Companies, Inc.
10 BASKETBALL A high-school basketball court is 94 feet
long and 50 feet wide What is the area of the court?
Understand Read the problem Write what you know
The length of the basketball court is 94 feet,
and the width of the court is 50 feet
Plan Pick a strategy One strategy is to use a formula
Substitute values for length and width into the area formula
Check Use a calculator to check your answer
Step by Step Problem-Solving Practice
Look for a pattern.
Guess and check.
Act it out.
Solve a simpler problem.
GO ON
Trang 18Copyright © by The McGraw-Hill Companies, Inc.
11 CONSTRUCTION A construction crew is pouring cement for
sidewalk slabs Each slab is a square that has sides that measure
70 centimeters What is the area of each slab? 4,900 cm2
Check off each step
✔ Understand
12 ART Mrs Brady asked her class to use an entire sheet of paper to
finger paint Each sheet of paper measured 8.5 inches by 11 inches
What was the area of the finger painting?
93.5 in2
lengths and widths? Explain
Sample answer: Yes; A rectangle with side lengths of 1 inch and 6 inches has
an area of 6 in2 A rectangle with side lengths of 2 inches and 3 inches also
has an area of 6 in2.
Skills, Concepts, and Problem Solving
Draw a rectangle that has the given area.
Trang 19Copyright © by The McGraw-Hill Companies, Inc.
Find the area of each rectangle.
20 PHOTOS At the portrait studio, Ines ordered the
picture of her family shown What was the area of
Ines’s family portrait?
1,008 cm2
21 DOORS The screen door at Ethan’s house is 32 inches
wide and 85 inches tall What is the area of Ethan’s
screen door?
2,720 in2
Vocabulary Check Write the vocabulary word that
completes each sentence.
needed to cover the inside of a region or a plane figure
sides
parallel It is a quadrilateral with four right angles
25 Writing in Math Explain how to find the area of a rectangle
Sample answer: Identify the length and width of the rectangle Substitute values
of the length and width into the formula for the area of a rectangle Multiply
to find the area of the rectangle Express the answer in square units.
Trang 20Copyright © by The McGraw-Hill Companies, Inc.
Spiral Review
26 PACKAGES A package is 8 feet long, 5 feet wide, and 4 feet tall
What is the volume of the package?
160 cubic feet (Lesson 2-4, p 77)
27 FOOD A cereal box is 8 inches by 10 inches by 3 inches What is
maximum volume of cereal the box can hold?
Find the perimeter of each polygon (Lesson 2-2, p 63)
Trang 21Copyright © by The McGraw-Hill Companies, Inc.
parallelogram
a quadrilateral in which each pair of opposite sides is parallel and equal in length
KEY Concept
Area of a Parallelogram
parallelogram
Cut a triangle from the parallelogram along the dashed
line Move the triangle and place it on the other side, next
to the right edge of the parallelogram In the parallelogram,
b represents the base, and h represents the height.
I
C
I C
Notice that the new shape formed from the parallelogram is a
rectangle So, the formula for the area of a parallelogram is
similar to the formula for the area of a rectangle:
What is the area of the parallelogram?
1 The base of the parallelogram is 8 inches, and the height is
5MG1.1 Derive and use the formula for area of a triangle and of a
parallelogram by comparing each with the formula for the area of a rectangle.
Trang 22Copyright © by The McGraw-Hill Companies, Inc.
What is the area of the parallelogram?
1 The base of the
What is the area of the parallelogram?
1 The base of the parallelogram is
7 yards, and the
What is the area of the parallelogram?
1 The base of the parallelogram is 4 feet, and the
Trang 23Copyright © by The McGraw-Hill Companies, Inc.
Guided Practice
Draw a parallelogram that has the given area.
Find the area of the parallelogram.
The area of the parallelogram is 63 square inches.
Step by Step Practice
Trang 24Copyright © by The McGraw-Hill Companies, Inc.
Find the area of each parallelogram.
4 The base of the parallelogram is 5 inches, and the height
Trang 25Copyright © by The McGraw-Hill Companies, Inc.
Solve.
10 HOBBIES Tony bought a sail for his boat The sail is in
the shape of a parallelogram and is 21 feet wide at the
base and 42 feet tall What is the area of the new sail?
Understand Read the problem Write what you know
The base of the sail is 21 feet and
the height of the sail is 42 feet.
Plan Pick a strategy One strategy is to use
The area of the sail is 882 ft2
Check Use a calculator to check your multiplication
HOBBIES A sailboat can have sails shaped like parallelograms.
HOBBIES A sailboat can have sails shaped like parallelograms.
Step by Step Problem-Solving Practice
11 ART Part of the sculpture that sits in the middle of Jacob Park is
shaped like a parallelogram The front of this piece is 13 feet tall
and has a base of 8 feet What is the area of the front of the
12 ART Camila’s class is making cardboard ornaments Each
ornament is shaped like a parallelogram with a height of
16 centimeters and a base of 9 centimeters What is the area
Solve a simpler problem Work backward.
GO ON
Trang 26Copyright © by The McGraw-Hill Companies, Inc.
10 feet and a height of 5 feet to the area of a rectangle with a length of 25 feet and a width of 2 feet Explain
See TWE margin.
Skills, Concepts, and Problem Solving
Draw a parallelogram that has the area given.
Find the area of each parallelogram.
20 FARMING Mrs Rockwell’s cornfield is in the shape of a
parallelogram Refer to the photo caption at the right
What is the area of the cornfield?
FARMING The height of Mrs
Rockwell’s cornfi eld is 79 meters and the base is 52 meters.
FARMING The height of Mrs
Rockwell’s cornfi eld is 79 meters and the base is 52 meters.
Trang 27Copyright © by The McGraw-Hill Companies, Inc.
21 PARTIES Toby made a sandwich She cut the bread in the shape
of a parallelogram Each piece of bread is 45 millimeters at the base
and 65 millimeters tall What is the area of each piece of bread?
2,925 mm2
Vocabulary Check Write the vocabulary word that completes each
sentence.
22 Square unit is a unit for measuring area.
23 A(n) parallelogram is a quadrilateral in which each pair of
opposite sides is parallel and equal in length
24 Writing in Math Explain how to find the area of a parallelogram
with a base of 9 inches and a height of 10 inches
See TWE margin.
Spiral Review
25 FITNESS A trampoline has a mat that is 10 feet wide and
14 feet long What is the area of the trampoline mat? (Lesson 3-1, p 4)
140 ft2
26 Draw a figure that has an area
of 18 square units (Lesson 2-3, p 71)
Trang 28Copyright © by The McGraw-Hill Companies, Inc.
Chapter
3 Progress Check 1 (Lessons 3-1 and 3-2)
Draw each figure that has the given area 3MG1.2, 4MG1.1, 5MG1.1
Find the area of each rectangle 4MG1.1
7 DESIGN Lena hung a rectangular mirror on her bathroom wall
The mirror was 85 centimeters high and 67 centimeters wide What
was the area of Lena’s mirror? 5,695 cm2
8 FOOD Lamont decorated cookies shaped like parallelograms
Each cookie was 84 millimeters tall and 61 millimeters at the base
What was the area of each cookie? 5,124 mm2
Trang 29Copyright © by The McGraw-Hill Companies, Inc.
parallelogram
a quadrilateral in which each pair of opposite sides is parallel and equal in length (Lesson 3-2, p 11)
Cut the parallelogram along the dashed line Notice you now
have two triangles, each one-half the size of the parallelogram
The height is inside the triangle
The height is outside the triangle
The height is outside the triangle
A is the area of the
5MG1.1 Derive and use the formula for area of a triangle and of a
parallelogram by comparing each with the formula for the area of a rectangle.
Trang 30Copyright © by The McGraw-Hill Companies, Inc.
Example 2
What is the area of the triangle?
1 The base of the triangle has endpoints at (2, 1) and (5, 1)
3 Substitute values of the base and height
into the area of a triangle formula
1 The base of the triangle is 6 inches, and the
height of the triangle is 4 inches
2 Substitute values of the base and height
into the area of a triangle formula
1 The base of the triangle is 8 meters,
and the height of the triangle is 7
Trang 31Copyright © by The McGraw-Hill Companies, Inc.
YOUR TURN!
What is the area of the triangle?
1 The base of the triangle has endpoints at (1, 2)
and (7, 2) It is 6 units long
2 The height of the triangle has endpoints at (4, 2) and
(4, 4) Its height is 2 units long.
3 Substitute values of the base and height into the area
Draw a triangle that has the given area.
Trang 32Copyright © by The McGraw-Hill Companies, Inc.
What is the area of the triangle?
3 Step 1 The base of the triangle is 5 miles, and the height of
the triangle is 16 miles.
Step 2 Substitute values of the base and height into the area of a
Step by Step Practice
Find the area of each triangle
4 The base of the triangle is 7 meters, and the height of the
Trang 33Copyright © by The McGraw-Hill Companies, Inc.
6 The area of the triangle is 45
8 VOLUNTEERING Pam earned a club patch for her jacket
by volunteering last week The patch is shaped like a
triangle It is 66 millimeters tall and has a base of 53
millimeters What is the area of Pam’s patch?
Understand Read the problem Write what you know
The base of the patch is 53 millimeters, and the
height of the patch is 66 millimeters.
Plan Pick a strategy One strategy is to use a formula
Substitute values of the base and height into the formula for the area of a triangle
Check Use a calculator to check your multiplication
Step by Step Problem-Solving Practice
GO ON
Problem-Solving Strategies
✓
Draw a diagram.
Look for a pattern.
Guess and check.
Use a formula.
Solve a simpler problem.
Trang 34Copyright © by The McGraw-Hill Companies, Inc.
9 FLAGS The Coleman Camp flag is raised every morning The flag
is in the shape of a triangle It is 36 inches long at its base and has a
height of 49 inches What is the area of the camp flag? 882 in2
Check off each step
✔ Understand
10 COSTUMES Manuel bought a triangular-shaped bandanna for his
costume in the school play The bandanna is 62 centimeters tall and
has a base of 67 centimeters What is the area of the bandanna?
2,077 cm2
11 Compare the area of a triangle with a base of 12 feet and
a height of 6 feet to the area of a parallelogram with a base of 12 feet and a height of 6 feet
See TWE margin.
Skills, Concepts, and Problem Solving
Draw a triangle that has the given area.
Trang 35Copyright © by The McGraw-Hill Companies, Inc.
Find the area of each triangle.
14 The area of the triangle is 1,000 square
18 FOOD For a picnic, Wayne sliced cheese in the shape of triangles
Each slice was 48 millimeters at the base and 51 millimeters tall
What was the area of each slice of cheese?
1,224 mm2
19 HORSES Earl searched for a missing horse near his uncle’s
ranch He searched in an area that was shaped like a triangle
Refer to the photo caption at the right What was the area
of the piece of land that Earl searched?
306,556 yd2
Vocabulary Check Write the vocabulary word that completes
each sentence.
20 Area is the number of square units needed to cover the
inside of a region or a plane figure
21 A(n) triangle is a polygon with three sides and three angles.
HORSES The area in which Earl searched for the horse had a base of
886 yards and a height of
692 yards.
HORSES The area in which Earl searched for the horse had a base of
886 yards and a height of
692 yards.
GO ON
Trang 36Copyright © by The McGraw-Hill Companies, Inc.
22 Writing in Math Explain how the area of a triangle is related to
the area of a rectangle
Answers will vary Sample answer: A rectangle can be cut into two equal
triangles So, the area of a triangle is half the area of a rectangle.
Spiral Review
23 WEATHER Zing turned on the weather channel She saw a
region that had a tornado watch The region was shaped like a
parallelogram and measured 82 miles across at the base and
56 miles high What was the area of the tornado watch? (Lesson 3-2, p 11)
4,592 mi2
24 Find the area of the figure (Lesson 2-3, p 71)
SLGRQ
SLGRQ
The area of the rectangle is 20 square units.
Draw a line segment of each length (Lesson 2-1, p 56)
Trang 37Copyright © by The McGraw-Hill Companies, Inc.
face
the flat part of a dimensional figure that is considered one of the sides
The surface area of a rectangular prism is the sum of the
areas of all the faces of the figure Surface area is measured
in square units.
A rectangular prism has six faces
Example 1
1 Draw a net of the rectangular
prism Label the faces A, B, C,
cubes that would fill them.
4MG1.1 Measure the area of ular shapes by using appropriate units, such as square centimeter (cm 2 ),
rectang-square meter (m 2 ), square kilometer (km 2 ), square inch (in 2 ), square yard (yd 2 ), or square mile (mi 2 ).
5MG1.2 Construct a cube and rectangular box from two- dimensional patterns to compute the surface area for these objects.
Trang 38Copyright © by The McGraw-Hill Companies, Inc.
3 There are six faces on the cube Find the
sum of the areas of all six faces
Trang 39Copyright © by The McGraw-Hill Companies, Inc.
Trang 40Copyright © by The McGraw-Hill Companies, Inc.
What is the surface area of the rectangular prism?
3
Step 1 Draw a net of the rectangular prism
Label the faces A, B, C, D, E, and F
A = × w
A = 4 × 3 = 12Step 4 Find the area of faces C and E
A = × w
A = 2 × 3 = 6Step 5 Find the sum of the areas of all the faces
8 + 12 + 6 + 8 + 12 + 6 = 52
The surface area of the rectangular prism is 52 square units.
Step by Step Practice
Find the surface area of each rectangular prism.
Draw a net of the rectangular prism
Label the faces A, B, C, D, E, and F
Follow the steps at the top of page 31
to find the surface area
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