Upload
hatuong
View
237
Download
0
Embed Size (px)
Citation preview
?
7 mm
15 mm12 mm
Math TalkMathematical Processes
Math On the Spot
my.hrw.com
L E S S O N
10.1Volume of Rectangular Prisms and Pyramids
ESSENTIAL QUESTIONHow do you find the volume of a rectangular prism and a rectangular pyramid?
Finding the Volume of a Rectangular PrismRemember that the volume of a rectangular prism is given by the formula
V = ℓ × w × h, or V = ℓwh.
The base of a rectangular prism is a rectangle with length ℓ and width w, so the
area of the base B is equal to ℓw. The volume formula can also be written as
V = Bh. In fact, the volume of any prism is the product of the base B and the
height h. Remember that volume is given in cubic units.
Find the volume of the rectangular prism.
Find the area of the base.
B = ℓw
B = 12 × 15
B = 180 m m 2
Find the volume.
V = Bh
V = 180 × 7
V = 1,260 m m 3
The volume of the rectangular prism is 1,260 cubic millimeters.
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Equations, expressions, and relationships—7.8.A Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas. Also 7.9.A
7.9.A
Does it matter which dimension of a rectangle
you consider its height?
Volume of a Prism
The volume V of a prism is the area of its base B times its height h.
V = Bh
Use the formula.
Substitute for ℓ and w.
Use the formula.
Substitute for B and h.
315Lesson 10.1
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
• Im
age C
redit
s: ©
Phot
odisc
/Ge
tty Im
ages
Net A Net B
Math Trainer
Online Assessment and Intervention
Personal
my.hrw.com
NetsA net is a two-dimensional pattern of shapes that can be folded into a
three-dimensional figure. The shapes in the net become the faces of the
three-dimensional figure.
Copy Net A and Net B on graph paper, and
cut them out along the blue lines.
One of these nets can be folded along the
black lines to make a cube. Which net will
not make a cube?
See if you can find another net that can be folded into a cube.
Draw a net that you think will make a cube on your graph paper, and
then cut it out. Can you fold it into a cube? Sketch your net below.
Compare your results with several of your classmates. How many
different nets for a cube did you and your classmates find?
STEP 1
STEP 2
STEP 3
EXPLORE ACTIVITY 1
Reflect1. What If? If you know the volume V and the height h of a prism,
how would you find the area of the base B?
2. Use the formula V = Bh to find the volume of a gift box
that is 3.5 inches high, 7 inches long, and 6 inches wide.
YOUR TURN
7.9.A
Unit 5316
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Height
Base
Exploring the Volume of a Rectangular PyramidA pyramid is a three-dimensional shape whose base is a polygon and whose other
faces are all triangles. Like a prism, a pyramid is named by the shape of its base.
Rectangular Pyramid Triangular Pyramid Pentagonal Pyramid
The faces of a pyramid that are not the base have a
common vertex, called the vertex of the pyramid.
The perpendicular distance from the vertex to the
base is the height of the pyramid.
Reflect3. What shapes will appear in a net for a rectangular prism that is not a
cube? How many of these shapes will there be?
How do you know that each net cannot be folded into a cube without
actually cutting and folding it?
4. 5.
6. Make a Conjecture If you draw a net for a cylinder, such as a soup can,
how many two-dimensional geometric shapes would this net have?
Name the shapes in the net for a cylinder.
Lesson 10.1 317
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Math Trainer
Online Assessment and Intervention
Personal
my.hrw.com
In this activity, you will compare the volumes of a pyramid and a prism with
congruent bases and equal heights. Remember that congruent figures have
the same shape and size.
Make three-dimensional models. Make
larger versions of the nets shown. Make
sure the bases and heights in each net are
the same size. Fold each net, and tape it
together to form a prism or a pyramid.
Fill the pyramid with beans. Make sure that the beans are level
with the opening of the pyramid. Then pour the beans into the
prism. Repeat until the prism is full. How many times did you fill
the prism from the pyramid?
Write a fraction that compares the volume of the pyramid to the
volume of the prism.
volume of pyramid ________________
volume of prism = _____
Reflect7. Draw Conclusions A rectangular pyramid has a base area
of B and a height of h. What is a formula for the volume of
the pyramid? Justify your reasoning.
8. Communicate Mathematical Ideas The prism and the pyramid in this
activity have congruent bases and equal heights. Are they congruent
three-dimensional shapes? Explain.
STEP 1
STEP 2
STEP 3
EXPLORE ACTIVITY 2
9. The volume of a rectangular prism is 4 1 _ 2 in 3 . What is the volume of a
rectangular pyramid with a congruent base and the same height? Explain
your reasoning.
YOUR TURN
Math TalkMathematical Processes
7.8.A
Describe ways in which a prism and a pyramid
are different.
Unit 5318
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Math Trainer
Online Assessment and Intervention
Personal
my.hrw.com
Math On the Spot
my.hrw.com
Kyle needs to build a crate in the shape of a rectangular prism. The
crate must have a volume of 38 1_2
cubic feet, and a base area of 15 2_5
square feet. Find the height of the crate.
V = Bh
38 1 _ 2
= 15 2 _ 5
· h
77 __ 2
= 77 __ 5
h
5 __ 77
· 77 __ 2
= 77 __ 5
h · 5 __ 77
5 _ 2
= h
The height of the crate must be 5 _ 2
, or 2 1 _ 2
, feet.
A glass paperweight in the shape of a rectangular pyramid has a base
that is 4 inches by 3 inches and a height of 5 inches. Find the volume
of the paperweight.
V = 1_3
Bh
V = 1_3
· 12 · 5
V = 20
The paperweight has a volume of 20 cubic inches.
EXAMPLEXAMPLE 2
A
B
Solving Volume ProblemsYou can use the formulas for the volume of a rectangular prism and the volume
of a rectangular pyramid to solve problems.
10. A rectangular prism has a volume of 160 cubic centimeters
and a height of 4 centimeters. What is the area of its base?
11. A square pyramid has a base edge of 5.5 yards and a height of
3.25 yards. Find the volume of the pyramid to the nearest tenth.
YOUR TURN
7.9.A
Volume of a Rectangular Pyramid
The volume V of a pyramid is one-third the area of its base B times
its height h.
V = 1 _ 3
Bh
Use the formula.
Substitute for V and B.
Change the mixed numbers to fractions.
To divide both sides by 77 ___
5 , multiply both
sides by the reciprocal.
Use the formula.
Think: B = ℓw = 4 · 3 = 12
319Lesson 10.1
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
9 ft
6 ft4 ft12
Guided Practice
1. Find the volume of the rectangular prism. (Example 1)
V = Bh
V = ( × _____ ) ( )
V = ft 3
Identify the three-dimensional shape that can be formed from each net.
(Explore Activity 1 and Explore Activity 2)
2. 3. 4.
5. The volume of a rectangular prism is
161.2 m 3 . The prism has a base that is
5.2 m by 3.1 m. Find the height of the
prism. (Example 2)
6. Explain how to use models to show the relationship between the volume of a rectangular prism
and a rectangular pyramid with congruent bases and heights.
ESSENTIAL QUESTION CHECK-IN??
V = Bh
=
( ×
) × h
=
h
= h
The height of the prism is .
Unit 5320
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
48 in.16 in.
20 cm
12 cm10 cm
11 in.
4 in.
8 in.
9 in.
Personal Math Trainer
Online Assessment and
Interventionmy.hrw.com
Name Class Date
8. Explain the Error A student found the
volume of a rectangular pyramid with a
base area of 92 square meters and a height
of 54 meters to be 4,968 cubic meters.
Explain and correct the error.
9. A block of marble is in the shape of a
rectangular prism. The block is 3 feet long,
2 feet wide, and 18 inches high. What is the
volume of the block?
10. Multistep Curtis builds a doghouse with
base shaped like a cube and a roof shaped
like a pyramid. The cube has an edge
length of 3 1 _ 2 feet. The height of the pyramid
is 5 feet. Find the volume of the doghouse
rounded to the nearest tenth.
11. Miguel has an aquarium in the shape of a
rectangular prism. The base is 30.25 inches
long and 12.5 inches wide. The aquarium
is 12.75 inches high. What is the volume of
the aquarium to the nearest cubic inch?
12. After a snowfall, Sheree built a snow
pyramid. The pyramid had a square base
with side lengths of 32 inches and a height
of 28 inches. What was the volume of the
pyramid to the nearest cubic inch?
13. A storage chest has the shape of a
rectangular prism with the dimensions
shown. The volume of the storage chest is
18,432 cubic inches. What is its height?
14. Draw Conclusions A shipping company
ships certain boxes at a special rate. The
boxes must not have a volume greater
than 2,500 cm 3 . Can the box shown be
shipped at the special rate? Explain.
15. Communicate Mathematical Ideas Is the
figure shown a prism or a pyramid? Justify
your answer.
Independent Practice10.17.8.A, 7.9.A
321Lesson 10.1
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Work Area
16. A cereal box can hold 144 cubic inches of cereal. Suppose the box is
8 inches long and 1.5 inches wide. How tall is the box?
17. A shed in the shape of a rectangular prism has a volume of 1,080 cubic
feet. The height of the shed is 8 feet, and the width of its base is 9 feet.
What is the length of the shed?
18. Draw Conclusions Sue has a plastic paperweight shaped like a
rectangular pyramid. The volume is 120 cubic inches, the height is
6 inches, and the length is 10 inches. She has a gift box that is a
rectangular prism with a base that is 6 inches by 10 inches. How tall
must the box be for it to hold the pyramid?
19. Represent Real-World Problems A public swimming pool is in the
shape of a rectangular prism. The pool is 20 meters long and 16 meters
wide. The pool is filled to a depth of 1.75 meters.
a. Find the volume of water in the pool.
b. A cubic meter of water has a mass of 1,000 kilograms. Find the mass
of the water in the pool.
20. Analyze Relationships There are two glass pyramids at the Louvre
Museum in Paris, France. The outdoor pyramid has a square base with side
lengths of 35.4 meters and a height of 21.6 meters. The indoor pyramid has
a square base with side lengths of 15.5 meters and a height of 7 meters.
How many times as great is the volume of the outdoor pyramid than that
of the indoor pyramid?
21. Persevere in Problem Solving A small solid pyramid was installed
on top of the Washington Monument in 1884. The square base of the
pyramid is 13.9 centimeters on a side, and the height of the pyramid is
22.6 centimeters. The pyramid has a mass of 2.85 kilograms.
a. Find the volume of the pyramid. Round to the nearest hundredth.
b. Find the mass of the pyramid in grams.
c. Science The density of a substance is the ratio of its mass to its
volume. Find the density of the pyramid in grams per cubic
centimeter. Round to the nearest hundredth.
FOCUS ON HIGHER ORDER THINKING
Unit 5322
© H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany