LESSON
16.1
?
Area of
Quadrilaterals
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—
6.8.B Model area
formulas for parallelograms,
trapezoids ... by decomposing
and rearranging parts of
these shapes. Also 6.8.D
How can you find the areas of parallelograms,
rhombuses, and trapezoids?
EXPLORE ACTIVITY
6.8.B
Area of a Parallelogram
Recall that a rectangle is a special type of parallelogram.
A Draw a large parallelogram on grid paper. Cut out your parallelogram.
B Cut your parallelogram on the dashed line as shown. Then move
the triangular piece to the other side of the parallelogram.
height (h)
base (b)
width (w)
length (l)
C What figure have you formed?
Does this figure have the same area as the parallelogram?
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base of parallelogram =
of rectangle
height of parallelogram =
area of parallelogram =
of rectangle
Math Talk
of rectangle
Mathematical Processes
What is the formula for the area of this figure? A =
or
D What is the formula for the area of a parallelogram? A =
How is the relationship
between the length and width
of a rectangle similar to the
relationship between the
base and height of a
parallelogram?
Area of a Parallelogram
The area A of a parallelogram is the
product of its base b and its height h.
A = bh
h
b
Lesson 16.1
447
EXPLORE ACTIVITY (cont’d)
Reflect
1. Find the area of the parallelogram.
7 cm
A=
14 cm
Finding the Area of a Trapezoid
Math On the Spot
To find the formula for the area of a trapezoid, notice that two copies of the
same trapezoid fit together to form a parallelogram. Therefore, the area of the
trapezoid is _12 the area of the parallelogram.
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h
b
A=
The height of the parallelogram is the same as the height
of the trapezoid. The base of the parallelogram is the sum
of the two bases of the trapezoid.
b
↓
·h
A = (b1 + b2) · h
Area of a Trapezoid
b1
h
b2
EXAMPLE 1
Math Talk
A section of a deck is in the shape of a trapezoid. What is the area of this
section of the deck?
Mathematical Processes
Does it matter which of
the trapezoid’s bases is
substituted for b1 and which
is substituted for b2?
Why or why not?
Animated
Math
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448
Unit 5
6.8.D
17 ft
b1 = 17
b2 = 39
h = 16
Use the formula for area of a trapezoid.
1 h(b + b )
A = __
2
2 1
1 · 16(17 + 39)
= __
2
1 · 16(56)
= __
2
16 ft
39 ft
Substitute.
Add inside the parentheses.
= 8 · 56
1 and 16.
Multiply __
2
= 448 square feet
Multiply.
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The area of a trapezoid is half its height
multiplied by the sum of the lengths
of its two bases.
1 h(b + b )
A = __
2
2 1
YOUR TURN
4.
Personal
Math Trainer
Amy needs to order a shade for a triangular-shaped window that has
a base of 6 feet and a height of 4 feet. What is the area of the shade?
Online Assessment
and Intervention
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Guided Practice
1. Show how you can use a copy of the triangle to form a rectangle. Find the
area of the triangle and the area of the rectangle. What is the relationship
between the areas? (Explore Activities 1 and 2, Example 1)
8 in.
14 in.
Triangle: A = _12 bh
Rectangle: A = bh
=
)(
)
in2
=(
)(
=
in2
2. A pennant in the shape of a triangle has a base of
12 inches and a height of 30 inches. What is the area
of the pennant? (Example 2)
A = _12 bh
= _12 (
=
?
?
)(
)
in2
ESSENTIAL QUESTION CHECK-IN
3. How do you find the area of a triangle?
456
Unit 5
#
)
'/ 3Ý
/9/ 4 %
© Houghton Mifflin Harcourt Publishing Company
= _12 (
Writing Equations Using the Area
of a Trapezoid
You can use the formula for area of a trapezoid to write an equation
to solve a problem.
Math On the Spot
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EXAMPLE 2
6.8.C
A garden in the shape of a trapezoid has an area of
44.4 square meters. One base is 4.3 meters and the
other base is 10.5 meters long. The height of the
trapezoid is the width of the garden. How wide is
the garden?
A = _12 h (b1 + b2)
Math Talk
Mathematical Processes
How can you check
that the answer is
reasonable?
4.3 m
10.5 m
Write the formula.
44.4 = _12 h (4.3 + 10.5)
Use the formula to write an equation.
44.4 = _12 h (14.8)
Add inside parentheses.
44.4 = 7.4 h
1 and 14.8.
Multiply __
2
44.4 ___
____
= 7.4
h
7.4
7.4
Divide both sides of the equation by 7.4.
6= h
The garden is 6 meters wide.
Reflect
Communicate Mathematical Ideas Explain why the first step after
substituting is addition.
YOUR TURN
3.
Personal
Math Trainer
Online Assessment
and Intervention
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460
Unit 5
The cross section of a water bin is shaped like a trapezoid. The bases of the
trapezoid are 18 feet and 8 feet long. It has an area of 52 square feet. What
is the height of the cross section?
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2.
LESSON
16.4
?
Solving Volume
Equations
ESSENTIAL QUESTION
Equations,
expressions, and
relationships—
6.8.C Write equations
that represent problems
related to . . . volume of right
rectangular prisms where
dimensions are positive
rational numbers. Also 6.8.D
How do you write equations to solve problems involving
volume of right rectangular prisms?
Problem Solving by Finding Volume
To find the volume of a box, which is in the shape of a rectangular prism, you
can multiply the length, the width, and the height. The volume of a threedimensional shape is always in cubic units, such as cubic meters or cubic inches.
Math On the Spot
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Volume of a Rectangular Prism
The volume V of a rectangular prism is the product
of its length ℓ, its width w, and its height h.
h
V = ℓwh
Math Talk
ℓ
Mathematical Processes
w
Why are V = ℓwh and V = Bh
both formulas for the
volume of a rectangular
prism?
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EXAMPL 1
EXAMPLE
6.8.D
A rectangular swimming pool is
25 meters long and 17 _12 meters wide.
It has an average depth of 1_12 meters.
What is the volume of the pool?
1
12 m
17 12 m
Label the rectangular prism to represent
the pool.
ℓ = 25 meters
w = 17 _12 meters
25 m
h = 1 _12 meters
Use the formula to write an equation.
V = ℓwh
V = 25 · 17 _12 · 1 _12
35 _
·3
= 25 · __
2 2
2,625
Write mixed numbers as fractions greater than 1.
= ____
4
Multiply.
= 656 _14 cubic meters
Write as a mixed number in simplest form.
Lesson 16.4
465
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
1. Miguel has a toolbox that measures 18 _12 inches by 12 _12 inches
by 4 inches. What is the volume of the toolbox?
V=
cubic inches
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Writing Equations Using the Volume
of a Rectangular Prism
Math On the Spot
You can use the formula for the volume of a rectangular prism to write an
equation. Then solve the equation to find missing measurements for a prism.
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EXAMPLE 2
6.8.C
Samuel has an ant farm with a volume of
375 cubic inches. The width of the
ant farm is 2.5 inches and the length is
15 inches. What is the height of
Samuel’s ant farm?
V = ℓwh
Write the formula.
375 = 15 · 2.5 · h
Use the formula to write
an equation.
Multiply.
375
37.5h
____
= ____
37.5
37.5
Divide both sides of the equation by 37.5.
10 = h
The height of the ant farm is 10 inches.
Reflect
2. Communicate Mathematical Ideas Explain how you would find the
solution to Example 2 using the formula V = Bh.
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
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466
Unit 5
?
3. Find the height of this shape, which has a
15
volume of __
cubic feet.
16
3
4 ft
1
2 ft
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Thom Lang/CORBIS
375 = 37.5h