Area Formulas
with Applications
Objective To review and use formulas for perimeter,
circumference, and area.
c
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Using Nets to Find Surface Area
• Convert between fractions, decimals,
and percents. Math Journal 2, pp. 347A and 347B
Students use drawings of nets to find
the surface areas of 3-dimensional
solids.
• Measure lengths to the nearest millimeter. [Measurement and Reference Frames Goal 1]
• Use formulas to calculate the
circumference of circles, as well as
perimeter and area of figures. [Measurement and Reference Frames Goal 2]
• Apply perimeter, circumference, and area
formulas to solve for missing lengths
and areas. [Measurement and Reference Frames Goal 2]
Math Boxes 9 8
Math Journal 2, p. 348
straightedge
Students practice and maintain skills
through Math Box problems.
Study Link 9 8
Math Masters, p. 306
Students practice and maintain skills
through Study Link activities.
Key Activities
Students use formulas to calculate area. They
apply formulas for perimeter, circumference,
and area to solve for missing lengths.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 345. [Measurement and Reference Frames
Goal 2]
Materials
Math Journal 2, pp. 345–347
Student Reference Book, pp. 215–218
and 377
Study Link 97
metric ruler calculator
ENRICHMENT
Solving Perimeter Problems
Sir Cumference and the First Round Table
Students apply perimeter formulas to solve
a problem from a story.
ENRICHMENT
Using the Distributive Property
to Find Dimensions
Math Masters, pp. 306A and 306B
Students use their knowledge of the
distributive property to find the dimensions
of objects with given areas.
EXTRA PRACTICE
Exploring Areas of Parallelograms
and Triangles
Math Masters, pp. 307–309
scissors transparent tape
Students derive area formulas for
parallelograms and triangles.
ENRICHMENT
Informally Relating Circumference
and Area
Students give an informal derivation of the
relationship between the circumference and
area of a circle.
ELL SUPPORT
Relating Figures and Formulas
Students illustrate relationships between
geometric shapes and formulas.
Advance Preparation
For the optional Enrichment activity in Part 3, obtain a copy of the book Sir Cumference and the First
Round Table by Cindy Neuschwander and Wayne Geehan (Charlesbridge, 2002).
Teacher’s Reference Manual, Grades 4–6 pp. 219, 221, 222
828
Unit 9
More about Variables, Formulas, and Graphs
Interactive
Teacher’s
Lesson Guide
Differentiation Options
Ongoing Learning & Practice
Key Concepts and Skills
[Number and Numeration Goal 5]
Curriculum
Focal Points
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP8
Content Standards
Getting Started
6.RP.3a, 6.NS.4, 6.EE.1, 6.EE.2c, 6.EE.9, 6.G.1, 6.G.4
Mental Math and Reflexes
Math Message
Students convert from exponential notation to
standard notation without using a calculator.
Complete the problems on journal page 345.
Suggestions:
33 27
24 16
303 27,000
204 160,000
0.033 0.000027
0.024 0.00000016
Study Link 9 7 Follow-Up
Discuss how the spreadsheet in Problem 1 was created. Ask: What formula is stored in cell B4? = 2 ∗ π ∗ A4
What formula is stored in cell C5? = π ∗ A5 ∗ A5
Compare the two graphs in Problem 2. The first graph shows that as the radius increases, the circumference also
increases at a uniform rate. The second graph shows that area increases at a faster rate than the radius.
Use the second graph to solve Problem 3. Find the point on the graph that has a y-value of 23 on the vertical axis.
3
1
From that point, move straight down to find the x-value on the horizontal axis. The radius is between 2_
and 2_
feet.
2
4
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 345; Student Reference Book, pp. 215–218)
Algebraic Thinking Review students’ strategies and solutions.
Show how they can draw the heights in Problems 5 and 6 outside
the figures. Remind students that they must draw the heights at
right angles to the bases as shown below.
Student Page
3 cm
Date
h
parallelogram
LESSON
9 8
Time
Calculate the area of each figure below. A summary of useful area formulas
appears on page 377 of the Student Reference Book.
trapezoid
b
Measure dimensions to the nearest tenth of a centimeter. Record the dimensions next
to each figure. You might need to draw and measure 1 or 2 line segments on a figure.
Round your answers to the nearest square centimeter.
If necessary, use Student Reference Book, pages 215–218 to
review how to calculate the areas of rectangles, parallelograms,
and triangles.
1.
2.
circle
triangle
Area formula
Area
Ongoing Assessment:
Recognizing Student Achievement
Area Formulas
Journal
Page 345
A = πr 2
10 cm2
A = _12 bh
Area formula
(unit)
3.
14 cm2
Area
(unit)
4.
rectangle
triangle
Use journal page 345 to assess students’ abilities to apply area formulas.
Students are making adequate progress if they can calculate the areas in
Problems 1–4. Some students may be able to apply the definition of height to
draw a reasonably correct height, measure it, and then use it to calculate the
area in Problem 4.
Area formula
Area
A=l∗w
10 cm2
A = _12 bh
Area formula
(unit)
5.
Area
6 cm2
Try This
3 cm
( i)
(unit)
6.
parallelogram
h
[Measurement and Reference Frames Goal 2]
trapezoid
p
b
Area formula
Area
Sample answer for formula:
A = 3h + _12 h(b - 3)
Area formula
A = bh
13 cm2
(unit)
Area
13 cm2
(unit)
345
Math Journal 2, p. 345
324_369_EMCS_S_G6_U09_576442.indd 345
2/26/11 1:19 PM
Lesson 9 8
829
Student Page
Date
LESSON
9 8
▶ Using Perimeter, Circumference,
Time
Perimeter, Circumference, and Area
212–218
and Area Formulas
Solve each problem. Explain your answers.
P
Rectangle PERK
K has a p
perimeter of 40 feet.
1.
12 ft
)
2
Area of rectangle PERK 96 ft
Sample
p answer: P = 2l + 2w
40 = 2l + 16
24 = 2l
12 = l
Length of side PE
8 ft
(unit
K
(unit)
A = lw
A = 12 ∗ 8
A = 96
(unit)
C
The area of parallelogram LMNK
K is 72 square inches.
3.
Algebraic Thinking Assign the problems on journal pages 346 and
347. The numbers are relatively “easy” in all problems, so expect
that some students may use trial-and-error strategies. When most
students have finished, bring the class together to share solutions.
A
15 m
40 m
Sample
p answer: A = _12bh
300 = (_12) ∗ b ∗ 15
600 = 15b
b
40 = b
Length of side AB
X
L
Many students are likely to quote or write the relevant formula,
substitute the given values in the formula, and then solve for the
missing length or area. Students may or may not reduce the
problem to an equation. For example, in Problem 3:
M
The length of side LX
X is 6 inches, and the length
of side KY
Y is 3 inches.
What is the length
g of LY?
K
N
Y
8 in.
Sample answer: A = bh
72 = (6 + 3)h
72 = 9h
8=h
Length of LY
Student A: The area of the parallelogram is base times height.
The base is 6 + 3, or 9 inches. Because 9 times 8 is 72, the height
must be 8 inches.
(unit)
Math Journal 2, p. 346
324_369_EMCS_S_G6_U09_576442.indd 346
2/22/11 4:37 PM
Adjusting
the Activity
KINESTHETIC
ELL
TACTILE
Student B: I used the formula A = bh. I know the area is 72 square
inches, and I figured out the base is 9 inches. If I substitute these
numbers in the formula, I get 72 = 9h. So, the height must be
8 inches.
In Problem 6, some students might figure out that the paths are
the same length without actually calculating the circumferences.
The circumference of the large circle is 40π, and the circumference
of each smaller circle is 20π, or 40π together.
Use a graphic organizer to demonstrate
the relationships between figures and area
formulas. Follow the format and notation
used in the Student Reference Book,
page 377.
AUDITORY
VISUAL
Student Page
Date
LESSON
9 8
Student Page
Time
Date
Perimeter, Circumference, and Area
4. The area of triangle ACE
E is 42 square yards.
cont.
98
212–218
C
D
B
E
What is the area of rectangle
g BCDE?
240 yd
y 2
Sample
p answer:
A = _12 bh
42 = _12 ∗ 7 ∗ h
84 = 7h
12 = h
Time
LESSON
Surface Area
Surface area is the total area of all the surfaces of a 3-dimensional object.
In Problems 1 and 2, a 3-dimensional solid is represented by a net. Use the net
and the following formulas to answer the questions.
Area of rectangle
g BCDE
A 7 yd
13 yd
A = bh
A = (13 + 7) ∗ 12
A = 240
Area of a rectangle: A = b ∗ h
1 ∗
Area of a triangle: A = _
(b ∗h)
2
A is the area of the rectangle.
b is the length of its base.
h is the height of the rectangle.
A is the area of the triangle.
b is the length of its base.
h is the height of the triangle.
1.
The net below represents a right triangular prism.
For Problems 5 and 6, use 3.14 for π.
square is covered by the area of the circle?
4 in.
6. Which path is longer: once around the figure 8—from A to B
A
20 ft
B
20 ft
in.2.
The area of each triangular face is
b.
The area of the larger rectangular face is
c.
The area of each of the smaller rectangular faces is
d.
The surface area of the triangular prism is
24
60
in.2.
15
in.2.
in.2.
Math Journal 2, p. 347A
Math Journal 2, p. 347
3/2/11 2:24 PM
347A_347B_EMCS_S_G6_MJ2_U09_576442.indd 347A
Unit 9
3
a.
C
They
y are the same distance
around. Sample
p answer:
Large circle: C = πd; C = π(40 ft); C = 125.6 ft
Smaller circle: C = πd; C = π ∗ 20 ft = 62.8 ft
Figure 8: 62.8 ft ∗ 2 = 125.6 ft
324_369_EMCS_S_G6_MJ2_U09_576442.indd 347
6 in.
20 in.
10 in.
20%
Sample
p answer:
Square:
q
A = s2
A = 202 = 400
Circle: A = πr 2
A = π((52) = 78.5
∗ 100)%
78.5 / 400 = 0.19625; (0.19625
(
) = 19.625%
Answer
to C to B and back to A—or once around the large
g circle?
1 12 in.
2 21 in.
5. To the nearest percent, about what percent of the area of the
830
PROBLEM
PRO
P
RO
R
OBL
BLE
B
L
LE
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IIN
NG
(Math Journal 2, pp. 346 and 347)
R
B
The area of triangle BAC
C is 300 meters2.
What is the length of side AB?
2.
E
INDEPENDENT
ACTIVITY
More about Variables, Formulas, and Graphs
3/9/11 11:13 AM
Student Page
Date
▶ Using Nets to Find Surface Area
Surface Area
98
2.
continued
The net below represents a square pyramid. The triangular faces are all congruent.
5.5 cm
2 Ongoing Learning & Practice
Time
LESSON
INDEPENDENT
ACTIVITY
(Math Journal 2, pp. 347A and 347B)
6.3 cm
6.3 cm
6.3 cm
6.3 cm
The area of each triangular face is
b.
The area of the square face is
c.
3.
17.325 cm .
39.69 cm .
The surface area of the square pyramid is 108.99
a.
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 348)
cm2.
Roxanne cut out the shape below from a piece of cardboard. She will fold it to
make a right rectangular prism. She plans to decorate it by completely covering the
outside with paper. How much paper does Roxanne need to decorate the prism?
10 in.
8 in.
Roxanne needs
▶ Math Boxes 9 8
2
2
6 in.
Have students look at journal page 347A. Tell them that a net is a
2-dimensional representation of a 3-dimensional object that shows
what the object would look like if it were “unfolded.” Explain that
since a net includes all of a 3-dimensional solid’s faces, the area of
a net is equal to the surface area of the 3-dimensional solid it
represents. Because the nets are 2-dimensional polygons, students
can decompose the polygons into triangles and rectangles to find
the areas. Students use this information to complete journal pages
347A and 347B.
376
in.2 of paper to decorate the prism.
Math Journal 2, p. 347B
347A_347B_EMCS_S_G6_MJ2_U09_576442.indd 347B
3/9/11 11:13 AM
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 9-6. The skills in Problems 4 and 5
preview Unit 10 content.
▶ Study Link 9 8
INDEPENDENT
ACTIVITY
(Math Masters, p. 306)
Home Connection Students solve area problems, some
of which involve variables and extraneous information.
Study Link Master
Name
Student Page
Date
STUDY LINK
Time
Area Problems
9 8
rectangle
4.
9 cm
1 ft
8 in
112 in.
2.5 ft
Area
triangle
5.
triangle
Area
6.
m
Solution
trapezoid
10 m
10 ft
45.5 mm2
m
30
.9
10
m
7m
Area
14
11 ft
mm
13 m
ft
$73.41
9m
3m + 4p + 6m + 9
b.
2k + 7n - 9n - 4k
c.
3
4
5
_
w - (-_) + _w
d.
-6x - 6y - 6x - 7y
2
5
+ 4p + 9
-2k - 2n
4w +
2
_4
5
-12x - 13y
248 249
252
3.
Seven out of 9 cards are faceup.
If 56 cards are faceup, how many cards
are there altogether?
5.
Identify whether the preimage (1) and image (2)
are related by a translation, a reflection,
or a rotation.
4.
Draw the line(s) of symmetry for each
figure below.
18 m
55 ft2
Area
Simplify each expression by combining
like terms.
a.
x = 3(19.49 + 4.98)
108 cm
30 m
2.
Number model
2
2
Use the distributive property to write a
number model for the problem. Then solve.
Niesha bought 3 sets of screwdrivers at
$19.49 each and 3 boxes of screws at
$4.98 each. What was the total cost of
Niesha’s purchase?
12 cm
20 cm
2
1.
215–217
parallelogram
30 in.
16 in.
Area
3.
m
2.
.
7 in.
parallelogram
Math Boxes
26
24 m
98
Time
LESSON
Calculate the area of each figure in Problems 1–6. Remember to include
the unit in each answer.
1.
Date
696 m2
Area
Try This
72
In Problems 7 and 8, all dimensions are given as variables. Write a true
statement in terms of the variables to express the area of each figure.
Example:
7.
8.
b
c
d
a
Area
x
b
c
1
_
ºcºd
2
Area
182
y
5
4
3
Write your answer on the line below.
m
a∗b
117–119
y
a
a
cards
Area
2
n
1
(n + m) ∗ y
rotation
1
-5 -4 -3 -2 -1
1 0
-1
2
1
2
3
4
5
x
-2
2
-3
3
Practice
9.
x ÷ 5.3 = 12
x=
63.6
10.
-3.1 = -31w
w=
-4
4
0.1
-5
5
Math Masters, p. 306
285-328_EMCS_B_G6_MM_U09_576981.indd 306
180 181
Math Journal 2, p. 348
2/26/11 3:11 PM
324_369_EMCS_S_G6_U09_576442.indd 348
2/22/11 4:37 PM
Lesson 9 8
831
Teaching Master
Name
Date
LESSON
98
Time
3 Differentiation Options
Finding Dimensions
Pete is considering buying a tent for a backpacking trip. The package says that the base
of the enclosed part of the tent is rectangular and has an area of 45 square feet, and that
there is a rectangular awning at the front of the tent that covers an area of 20 square feet.
From the picture, Pete can tell that the main tent and awning have the same width. He
draws this picture to illustrate the information on the label.
ENRICHMENT
Main tent
45 sq ft
Awning
20 sq ft
m
a
▶ Solving Perimeter Problems
w
Use what you know about area models and the distributive property to help you find
the dimensions of the tent. Assume the dimensions are in whole numbers of feet.
Explain your thinking.
Sample answer: From the area formula for a
rectangle, I know that the total area of the main
tent and awning together is (m + a) ∗ w. From
the picture, I know that the total area is 45 + 20.
To write this in the form (m + a) ∗ w, I can use
the distributive property. First, I need to find a
common factor of 45 and 20. The only common
factor is 5. 45 + 20 = 9 ∗ 5 + 4 ∗ 5 =
(9 + 4) ∗ 5. This shows that m = 9, a = 4,
and w = 5.
After reading the story, students calculate the various perimeters
for the first three configurations mentioned. Using previous
configurations and applying their knowledge of the scale factor
(1 cm : 2 ft), students should conclude that the length between
two parallel sides of the hexagon is 10 feet and that the sides are
about 4 feet long. They will need to estimate the circumference of
the final circular table. A good estimate is between 26 and 27 ft2,
1 feet tall.
assuming Lady Di is about 5_
2
Math Masters, p. 306A
3/9/11 12:11 PM
Radius
(r)
Circumference
(C )
50
63
126
75
8 in.
10 cm
20 ft
12 m
Area
(A)
A
_
201
314
1257
452
4
5
10
6
ENRICHMENT
C
▶ Using the Distributive Property
LESSON
98
Date
Finding Dimensions
To further explore area and area formulas, students use
information about the areas of rectangles and their knowledge of
the distributive property to find the dimensions of rectangles.
If students have trouble getting started, you might have them
write this form of the distributive property on their papers:
a ∗ c + b ∗ c = (a + b) ∗ c. Remind them how they used the
distributive property to factor sums in Lesson 9-2.
Time
continued
The Web site for a hotel says it has a pool with a 49-square-meter deep end and a
63-square-meter shallow end. The diagram below shows the shape of the pool.
ENRICHMENT
Shallow end
Deep end
▶ Informally Relating
Sample answer: 63 + 49 = 7 ∗ (9 + 7)
7m
a. How wide is the pool?
9m
b. How long is the shallow end?
7m
c. How long is the deep end?
2.
Apron
Sample answer: 170 + 51 = 17 ∗ (10 + 3)
3 ft
a. How far does the apron extend in front of the curtain?
10 ft
b. How far does the main stage extend behind the curtain?
17 ft
c. How wide is the stage?
Math Masters, p. 306B
Unit 9
15–30 Min
To explore the relationship between area and circumference
of a circle, draw the table in the margin on the board with only
the Radius (r) column filled in. Students complete the other
columns and describe the patterns they observe in the table. Help
students notice that the ratio of a circle’s area to its circumference
is half the measure of its radius. Round all values to the nearest
whole number.
A diagram of the stage in a school auditorium is shown below. The area of the
part behind the curtain, or the main stage, is 170 square feet. The area of the
section in front of the curtain, or the apron, is 51 square feet.
306A-306B_EMCS_B_G6_MM_U09_576981.indd 306B
INDEPENDENT
ACTIVITY
Circumference and Area
Main stage
832
15–30 Min
(Math Masters, pp. 306A and 306B)
Use the distributive property to solve the problems. Assume all dimensions are in
whole-number units. On the line below each drawing, show how you used the
distributive property to find the dimensions.
1.
INDEPENDENT
ACTIVITY
to Find Dimensions
Teaching Master
Name
15–30 Min
Literature Link To further explore circles, students read the
book Sir Cumference and the First Round Table, by Cindy
Neuschwander and Wayne Geehan (Charlesbridge, 2002), a
fantasy about how the knights of Camelot decided to use a
round table.
Pete labeled his picture with the variable m to represent the length of the main tent,
the variable a to represent the length of the awning, and the variable w to represent
the width. He wants to know the value of each of these dimensions.
306A-306B_EMCS_B_G6_MM_U09_576981.indd 306A
SMALL-GROUP
ACTIVITY
3/9/11 12:11 PM
More about Variables, Formulas, and Graphs
Teaching Master
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
▶ Exploring Areas of
15–30 Min
Name
Date
LESSON
98
䉬
1.
Do not cut out the shapes on this page. Instead cut out Parallelogram A
on Math Masters, page 309 and follow the directions there.
Parallelogram A
Parallelograms and Triangles
Time
Areas of Parallelograms
Tape your rectangle in the space below.
Sample
answer:
(Math Masters, pp. 307–309)
To provide extra practice with formulas, have students transform
parallelograms into rectangles to derive the area formula for
parallelograms and combine two identical triangles to derive the
area formula for triangles.
4
3
base ⫽
height ⫽
length ⫽
cm
width ⫽
cm
12
Area of parallelogram ⫽
2.
cm
cm
12
Area of rectangle ⫽
cm2
Do the same with Parallelogram B on Math Masters, page 309.
Parallelogram B
Tape your rectangle in the space below.
Sample
answer:
SMALL-GROUP
ACTIVITY
ELL SUPPORT
▶ Relating Figures
5–15 Min
6
5
base ⫽
and Formulas
height ⫽
length ⫽
cm
width ⫽
cm
30
Area of parallelogram ⫽
3.
To provide language support, use a graphic organizer such as the
one below to present the area formulas in this lesson.
Square
cm2
4
3
Rectangle
s
cm2
6
5
cm
cm
30
Area of rectangle ⫽
cm2
Write a formula for the area of a parallelogram.
A⫽bⴱh
Math Masters, p. 307
Parallelogram
h
h
b
b
Area Formulas
Circles
Triangles
h
h
d
r
Teaching Master
b
1
2 bh
d )2
2
Name
LESSON
98
䉬
1.
Date
Time
Areas of Triangles
Do not cut out the triangle below. Instead cut out Triangles C and D
from Math Masters, page 309 and follow the directions there.
Triangle C
Tape your parallelogram in the space below.
C
base ⫽
height ⫽
4
4
width ⫽
cm
8
Area of triangle ⫽
2.
length ⫽
cm
cm2
4
4
cm
cm
Area of parallelogram ⫽
16
cm2
Do the same with Triangles E and F.
Triangle E
Tape your parallelogram in the space below.
E
F
b
D
b
h
base ⫽
height ⫽
6
2
Area of triangle ⫽
3.
length ⫽
cm
width ⫽
cm
6
cm2
6
2
cm
cm
Area of parallelogram ⫽
12
cm2
Write a formula for the area of a triangle.
1
ᎏᎏ
2
A ⫽ bh
Math Masters, p. 308
Lesson 9 8
833
Name
LESSON
98
Date
Time
Finding Dimensions
Pete is considering buying a tent for a backpacking trip. The package says that the base
of the enclosed part of the tent is rectangular and has an area of 45 square feet, and that
there is a rectangular awning at the front of the tent that covers an area of 20 square feet.
From the picture, Pete can tell that the main tent and awning have the same width. He
draws this picture to illustrate the information on the label.
Main tent
45 sq ft
Awning
20 sq ft
m
a
w
Pete labeled his picture with the variable m to represent the length of the main tent,
the variable a to represent the length of the awning, and the variable w to represent
the width. He wants to know the value of each of these dimensions.
Copyright © Wright Group/McGraw-Hill
Use what you know about area models and the distributive property to help you find
the dimensions of the tent. Assume the dimensions are in whole numbers of feet.
Explain your thinking.
306A
Name
Date
LESSON
98
Finding Dimensions
Time
continued
Use the distributive property to solve the problems. Assume all dimensions are in
whole-number units. On the line below each drawing, show how you used the
distributive property to find the dimensions.
1.
The Web site for a hotel says it has a pool with a 49-square-meter deep end and a
63-square-meter shallow end. The diagram below shows the shape of the pool.
Shallow end
2.
a.
How wide is the pool?
b.
How long is the shallow end?
c.
How long is the deep end?
Deep end
A diagram of the stage in a school auditorium is shown below. The area of the
part behind the curtain, or the main stage, is 170 square feet. The area of the
section in front of the curtain, or the apron, is 51 square feet.
Apron
a.
How far does the apron extend in front of the curtain?
b.
How far does the main stage extend behind the curtain?
c.
How wide is the stage?
306B
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Main stage