© Glencoe/McGraw-Hill
0
4
36
22
4
5
A5
© Glencoe/McGraw-Hill
4.
1.
5
14
3
1
0
17
2
35
0
13
1
6
1 face
Total
Number
of Cubes
Figure
5.
2.
2
6
0
0
1
3
2 faces
673
19
16
16
18
18
17
16
14
20
10
16
12
6.
3.
4 faces
6
2
0
1
2
1
5 faces
1
0
0
0
0
0
6 faces
Mathematics: Applications and Concepts, Course 2
Answers
Mathematics: Applications and Concepts, Course 2
3 faces
Number of Faces with Glue on Them
Complete this chart for the figures below.
The figures on this page have been built by gluing cubes together. Use your
visual imagination to count the total number of cubes as well as the number
of cubes with glue on 1, 2, 3, 4, or 5, or 6 faces.
Counting Cubes
Enrichment
NAME ________________________________________ DATE ______________ PERIOD _____
➝
}
4m
There are
two layers.
3m
2m
Replace ᐉ with 5, w with 6, and h with 8.
Multiply.
V⫽5⭈6⭈8
V ⫽ 240
84 m3
3m
4m
7m
© Glencoe/McGraw-Hill
1.
2.
10 cm
674
630 cm3
9 cm
16.2 ft3
2.7 ft
6 in.
3 ft
2 ft
5 in.
8 in.
Mathematics: Applications and Concepts, Course 2
7 cm
3.
Find the volume of each rectangular prism. Round to the nearest
tenth if necessary.
The volume is 240 cubic inches.
Volume of a rectangular prism
V ⫽ ᐉwh
Find the volume of the rectangular prism.
V ⫽ Bh or V ⫽ ᐉwh
A rectangular prism is a solid figure that has two parallel and congruent sides, or bases, that are
rectangles. To find the volume of a rectangular prism, multiply the area of the base and the height, or
find the product of the length ᐉ, the width w, and the height h.
It takes 12 ⴢ 2 or 24 cubes to fill the box. So, the volume of the box is 24 cubic meters.
The bottom layer,
or base, has 4 ⴢ 3 ➝
or 12 cubes.
The volume of a solid is the measure of space occupied by it. It
is measured in cubic units such as cubic centimeters (cm3) or cubic
inches (in3). The volume of the figure at the right can be shown
using cubes.
Volume of Rectangular Prisms
Study Guide and Intervention
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lessons 12-1 and 12-2)
Lesson 12–1
© Glencoe/McGraw-Hill
A6
2
47ᎏ1ᎏ ft3
1
2 2 ft
180
3
4 4 ft
5 mm
4 ft
mm3
12 mm
5.
300
8.
2.8 in.
4.8 in.
9.
6.
3.
m3
7.2 cm
4m
1.2 cm
8.1 cm3
1.5 cm
4.5 cm
194.4 cm3
9 cm
96
6m
3 cm
4m
Mathematics: Applications and Concepts, Course 2
15 in.
9.5 in.
6 in.
675
691.2 in3
9.6 in.
`186.2 in3
7 in.
in3
cm3
3 mm
63
10 in.
5 in.
7 cm
2.
3 cm
3 cm
© Glencoe/McGraw-Hill
7.
4.
1.
Find the volume of each rectangular prism. Round to the nearest
tenth if necessary.
3
4
© Glencoe/McGraw-Hill
5. PACKAGING A box of tissues has a
length of 11.2 centimeters, a width of
11.2 centimeters, and a height of
13 centimeters. What is the volume of
the tissue box? 1,630.72 cm3
3. TRANSPORTATION The cargo-carrying
part of Billy’s truck has a length of
8.3 meters, a width of 3 meters, and a
height of 4.2 meters. What is the
maximum volume of sand that Billy’s
truck can carry? 104.58 m3
volume of the cereal box? 169ᎏᎏ in3
8
height of 12ᎏ1ᎏ inches. What is the
4
8 inches, a width of 1ᎏ3ᎏ inches, and a
118.35 cm3
Mathematics: Applications and Concepts, Course 2
4.5 cm
B = 26.3 cm2
6. GEOMETRY A pentagonal prism is a
prism that has bases that are
pentagons. Use V ⫽ Bh where B is the
area of the base, to find the volume of
the pentagonal prism below.
4. PLUMBING Alexia’s bathroom has a tub
in the shape of a rectangular prism
with a length of 1.5 meters, a width of
0.5 meter, and a height of 0.4 meter.
How many cubic feet of water can it
hold? 0.3 m3
her cooler hold? 2,772 in3
2
height of 10ᎏ1ᎏ inches, how much ice will
2. FOOD STORAGE Nara wants to
determine how much ice it will take to
fill her cooler. If the cooler has a length
of 22 inches, a width of 12 inches, and a
676
1. PACKAGING A cereal box has a length of
Volume of Rectangular Prisms
Practice: Word Problems
Practice: Skills
Volume of Rectangular Prisms
NAME ________________________________________ DATE ______________ PERIOD _____
Lesson 12–2
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 12-2)
Mathematics: Applications and Concepts, Course 2
© Glencoe/McGraw-Hill
r
h
B = ␲r 2
Replace r with 2 and h with 5.
Simplify.
V ⫽ ␲(2)2(5)
V ⬇ 62.8
5 in.
A8
12.9 ft
© Glencoe/McGraw-Hill
679
25.1 in3
2 in.
2 in.
2
6. diameter ⫽ 3 ᎏ m
5
1
height ⫽ 1 ᎏ
m
4
3
11.3 m
3.
Mathematics: Applications and Concepts, Course 2
311.0 cm3
648.4 ft3
4 ft
623.8 yd3
2.
5. diameter ⫽ 6 cm
height ⫽ 11 cm
5,654.9 mm3
18 mm
10 mm
4. radius ⫽ 9.5 yd
height ⫽ 2.2 yd
1.
Find the volume of each cylinder. Round to the nearest tenth.
The volume is approximately 62.8 cubic inches. Check by using estimation.
Volume of a cylinder
V ⫽ ␲r2h
Find the volume of the cylinder. Round
to the nearest tenth.
2 in.
V ⫽ Bh or V = ␲r 2h, where B ⫽ ␲r 2
230.9 yd3
3,078.8 cm3
20 cm
7 cm
8. radius ⫽ 4 ft
8.7 m
© Glencoe/McGraw-Hill
314.2 mm3
9. diameter ⫽ 10 mm
height ⫽ 4 mm
680
4 in.
Mathematics: Applications and Concepts, Course 2
39.6 in3
10. diameter ⫽ 7.1 in.
height ⫽ 1 in.
2
70.3 in3
6.2 in.
150.8 in3
6. 1.9 in.
125.7 ft3
191.9 m3
5.3 m
9 ft
3.
1,143.4 cm3
5.
1,809.6 ft3
8 ft
height ⫽ 2ᎏ1ᎏ ft
6 yd
1
3 2 yd
2.
height ⫽ 4.7 cm
7. radius ⫽ 8.8 cm
4.
1.
12 in.
Volume of Cylinders
Volume of Cylinders
A cylinder is a solid figure that has two congruent, parallel circles as its bases. The volume V of a
cylinder with radius r is the area of the base B times the height h.
Practice: Skills
Study Guide and Intervention
Find the volume of each cylinder. Round to the nearest tenth.
NAME ________________________________________ DATE ______________ PERIOD _____
Lesson 12–3
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 12-3)
Mathematics: Applications and Concepts, Course 2
© Glencoe/McGraw-Hill
A9
© Glencoe/McGraw-Hill
5. PAINT A can of paint is 15 centimeters
high and has a diameter of 13.6 cm.
What is the volume of the can? Round
to the nearest tenth. 2,179 cm3
27.2 in3
Mathematics: Applications and Concepts, Course 2
Answers
Mathematics: Applications and Concepts, Course 2
3 in.
1.7 in.
6. SPICES A spice manufacturer uses a
cylindrical dispenser like the one
shown. Find the volume of the
dispenser to the nearest tenth.
643.4 cm3
4. DESIGN Rodolfo is designing a new,
cylindrical drinking glass. If the glass
has a diameter of 8 centimeters and a
height of 12.8 centimeters, what is its
volume? Round to the nearest tenth.
681
3. CONTAINERS Tionna wants to determine
the maximum capacity of a cylindrical
bucket that has a radius of 6 inches
and a height of 12 inches. What is the
capacity of Tionna’s bucket? Round to
the nearest tenth. 1,357.2 in3
198.6 m3
is three-dimensional
CP
© Glencoe/McGraw-Hill
682
Mathematics: Applications and Concepts, Course 2
7. Work with a partner. Bring an object that is a cylinder to school. Take the
measurements and determine the volume of your cylindrical object.
Exchange objects with your partner, but do not share the calculations.
Determine the volume of your partner’s object. Then compare your results
with those of your partner. See students’ work.
Helping You Remember
6. What is the formula for the area of the base of a cylinder? B ⴝ ␲r2
5. What shape is the base of a cylinder? circle
has volume
CP
has bases that are circular
C
is a solid
has sides and bases that are polygons
P
CP
has bases that are parallel and congruent
CP
4. Write C if the phrase is true of a cylinder, P if the phrase is true of a
prism, or CP if the phrase is true of both.
Reading the Lesson
height.
3. Make a conjecture about how you could find the volume of the soup
can. Sample answer: Multiply the area of the base and the
work.
2. How many layers would it take to fill the cylinder? See students’
1. Estimate the number of centimeter cubes that would fit at the bottom of
the can. Include parts of cubes. See students’ work.
Write your answers below.
Pre-Activity Complete the Mini Lab at the top of page 524 in your textbook.
Volume of Cylinders
Volume of Cylinders
2. PACKAGING A can of corn has a
diameter of 6.6 centimeters and a
height of 9.9 centimeters. How much
corn can the can hold? Round to the
nearest tenth. 338.7 cm3
Reading to Learn Mathematics
Practice: Word Problems
1. WATER STORAGE A cylindrical water
tank has a diameter of 5.3 meters and
a height of 9 meters. What is the
maximum volume that the water tank
can hold? Round to the nearest tenth.
NAME ________________________________________ DATE ______________ PERIOD _____
Lesson 12–3
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 12-3)
© Glencoe/McGraw-Hill
length
width
height
length
Non-right Prism
width
height
radius
Right Cylinder
A10
7 yd
6 yd
180 yd3
3 yd
7.9
m3
2.5 m 3 m
5 yd
2m
© Glencoe/McGraw-Hill
4.
1.
10 yd
5.
2.
3 in.
2 cm
683
125.7 cm3
10 cm
60 in3
2 in.
11 in.
6.
10 in.
4 cm
240 cm3
4 cm
785.4 in3
5 in.
12 in.
15 cm
Mathematics: Applications and Concepts, Course 2
10 in.
3.
radius
Non-right Cylinder
height
Find the volume of each solid figure. Round to the nearest tenth.
height
Right Prism
The diagrams below show prisms and cylinders that
have the same volume but do not have the same shape.
Imagine a stack of ten pennies. By pushing against the
stack, you can change its shape as shown at the right.
But, the volume of the stack does not change.
24 ⫹ 16 ⫹ 12 ⫽ 52
Sum of the areas
3m
2m
side
2m
4m
top
front
bottom
back
4m
side
2m
3m
4m
3 cm
102 cm2
3 cm
© Glencoe/McGraw-Hill
1.
7 cm
2.
232 in2
2 in.
684
8 in.
10 in.
3.
286 ft2
5 ft
7 ft
9 ft
Mathematics: Applications and Concepts, Course 2
Find the surface area of each rectangular prism.
So, the surface area of the rectangular prism is 52 square meters.
2m
3m
Alternatively, replace ᐉ with 4, w with 3, and h with 2 in the formula for surface area.
S ⫽ 2ᐉw ⫹ 2ᐉh ⫹ 2wh
⫽2⭈4⭈3⫹2⭈4⭈2⫹2⭈3⭈2
Follow order of operations.
⫽ 24 ⫹ 16 ⫹ 12
⫽ 52
(2 ⭈ 3) ⫹ (2 ⭈ 3) ⫽ 12
two sides
(4 ⭈ 2) ⫹ (4 ⭈ 2) ⫽ 16
(4 ⭈ 3) ⫹ (4 ⭈ 3) ⫽ 24
top and bottom
front and back
Area
top and bottom
front and back
two sides
Faces
䊉
䊉
䊉
You can use the net of the rectangular
prism to find its surface area. There are
three pairs of congruent faces in a
rectangular prism:
Find the surface area of the rectangular prism.
S ⫽ 2ᐉw ⫹ 2ᐉh ⫹ 2wh
The sum of the areas of all the surfaces, or faces, of a three-dimensional figure is the surface area.
The surface area S of a rectangular prism with length ᐉ, width w, and height h is found using the
following formula.
Surface Area of Rectangular Prisms
Study Guide and Intervention
Enrichment
Volumes of Non-Right Solids
NAME ________________________________________ DATE ______________ PERIOD _____
Lesson 12–3
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lessons 12-3 and 12-4)
Mathematics: Applications and Concepts, Course 2
© Glencoe/McGraw-Hill
A11
396 ft2
7 ft
522
12 ft
mm2
10 mm
6 ft
9 mm
12 cm
396 cm2
7 cm
9 mm
6 cm
8.
5.
2.
1 ft
179.7 in2
8.3 in.
8.5 cm
4.5 in.
4.1 in.
3 cm
4 ft
143 cm2
4 cm
78 ft2
7 ft
9.
6.
3.
9 in.
6 in.
685
Mathematics: Applications and Concepts, Course 2
Answers
Mathematics: Applications and Concepts, Course 2
11. Find the surface area of a rectangular prism that has a length of
8 inches, a width of 3 inches, and a height of 6 inches. 180 in2
© Glencoe/McGraw-Hill
7.3 mm
6.4 mm
4.3 in.
15 in.
7 in.
205.8 mm2
4.1 mm
127.8 in2
3.7 in.
606 in2
10. A cube has a surface area of 126 square feet. What is the area of one
face? 21 ft2
7.
4.
1.
Find the surface area of each rectangular prism. Round to the
nearest tenth if necessary.
© Glencoe/McGraw-Hill
130.4 m2
5. CONTAINERS What is the total surface
area of the inside and outside of a
container in the shape of a rectangular
prism with length of 5 meters, width of
3 meters, and height of 2.2 meters?
4 cm
96 cm2
3. ICE Suppose the length of each edge of
a cube of ice is 4 centimeters. Find the
surface area of the cube.
1,032 in2
1. PACKAGING A packaging company
needs to know how much cardboard
will be required to make boxes
18 inches long, 12 inches wide, and
10 inches high. How much cardboard
will be needed for each box if there is
no overlap in the construction?
686
2.3 in.
Mathematics: Applications and Concepts, Course 2
66.7 in2
6.1 in.
2.3 in.
6. TOYS Oscar is making a play block for
his baby sister by gluing fabric over the
entire surface of a foam block. How
much fabric will Oscar need?
4. ICE Suppose you cut the ice cube from
Exercise 3 in half horizontally into two
smaller rectangular prisms. Find the
surface area of one of the two smaller
prisms. 64 cm2
insulation should Jane buy if all inside
surfaces except the floor are to be
insulated? 465 ft2
2
wide, and 7ᎏ1ᎏ feet high. How much
2. INSULATION Jane needs to buy
insulation for the inside of a truck
container. The container is a
rectangular prism 15 feet long, 8 feet
Surface Area of Rectangular Prisms
Practice: Word Problems
Practice: Skills
Surface Area of Rectangular Prisms
NAME ________________________________________ DATE ______________ PERIOD _____
Lesson 12–4
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 12-4)
The surface area
of a cylinder
equals
© Glencoe/McGraw-Hill
冎
r
C = 2␲r
r
h
The rectangle
makes up the
curved surface.
C
C = 2␲r h
(2␲r)h
A13
Simplify.
Replace r with 6 and h with 20.
Surface area of a cylinder.
...make up the
two bases.
408.4 in2
8 in.
10 in.
© Glencoe/McGraw-Hill
1.
2.
689
508.9 ft2
24 ft
3 ft
440.4 cm2
12 cm
4.3 cm
20 m
6m
20 m
6m
Mathematics: Applications and Concepts, Course 2
Answers
Mathematics: Applications and Concepts, Course 2
3.
Find the surface area of each cylinder. Round to the nearest tenth
The surface area is about 980.2 meters.
⬇ 980.2
S ⫽ 2␲(6)2 ⫹ 2␲(6)(20)
S ⫽ 2␲r2 ⫹ 2␲rh
Find the surface area of the cylinder.
Round to the nearest tenth.
In the diagram above, the length of the rectangle is the same as the
circumference of the circle. Also, the width of the rectangle is the same
as the height of the cylinder.
Two congruent
circles...
冎
⫹
冎
2(␲r 2)
冎
⫽
the area of the
curved surface.
冎
S
the area of
two bases plus
The diagram below shows how you can put two circles and a rectangle
together to make a cylinder.
Surface Area of Cylinders
Study Guide and Intervention
NAME ________________________________________ DATE ______________ PERIOD _____
355.9 yd2
7 yd
1,608.5 in2
24 in.
8 in.
4.8 yd
5.
2.
107.3 ft2
1
4 3 ft
1
2 2 ft
603.2 mm2
6 mm
10 mm
6.
3.
506.7 cm2
© Glencoe/McGraw-Hill
690
Mathematics: Applications and Concepts, Course 2
9. Find the area of the curved surface of a D battery with a diameter of
3.2 centimeters and a height of 5.6 centimeters. 56.3 cm2
8. Find the surface area of the outside of a cylindrical barrel with a
diameter of 10 inches and a height of 12 inches. 534.1 in2
7m
6.5 cm
12.6 cm
276.5 m2
4m
7. Find the surface area of a can with a radius of 4 centimeters and a height
of 11 centimeters. 377.0 cm2
4.
1.
Find the surface area of each cylinder. Round to the nearest tenth.
Surface Area of Cylinders
Practice: Skills
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 12-5)
Lesson 12–5
© Glencoe/McGraw-Hill
A14
© Glencoe/McGraw-Hill
29.8 ft2
tenth. (Hint: Do not include the top.)
4
height of 4ᎏ1ᎏ feet? Round to the nearest
5. MANUFACTURING How much sheet
metal is required to make a cylindrical
trash can with a diameter of 2 feet and
691
Mathematics: Applications and Concepts, Course 2
282.7
in2
6. PLUMBING How much steel is needed to
make a hollow pipe with a radius of
3 inches and a height of 15 inches?
Round to the nearest tenth.
480.3 cm2
4. CANS A cylindrical can has a height of
14 centimeters and a radius of 4.2
centimeters. Find the surface area of
the can. Round to the nearest tenth.
3. CANS A cylindrical can has a diameter
of 6 inches and a height of 7.3 inches.
What is the surface area of the can?
Round to the nearest tenth.
194.2 in2
2. TIRES Betty wants to know the total
surface area of the tread on one of her
tires. If the diameter of the tire is
18 inches and the width of the tire is
5 inches, what is the total surface area
of the tire’s tread? Round to the nearest
tenth. 282.7 in2
1. PACKAGING What is the area of the
label on a box of oatmeal with a radius
of 9.3 centimeters and a height of
16.5 centimeters? Round to the nearest
tenth. 964.2 cm2
Surface Area of Cylinders
Practice: Word Problems
NAME ________________________________________ DATE ______________ PERIOD _____
Sample answer:
692
ⴙ
plus
(2␲r)h
the area of the
curved surface.
Mathematics: Applications and Concepts, Course 2
2(␲r 2)
ⴝ
S
Symbols
© Glencoe/McGraw-Hill
the area of
two bases
equals
The surface area
of a cylinder
Words
9. Complete the table.
Helping You Remember
the base and add it to the area of the curved surface;
S ⴝ ␲r 2 + 2␲rh.
8. How would you find the surface area of a cylinder with no top? Give your
answer in words and symbols. Sample answer: Find the area of
7. the area of a rectangle _______________ A ⴝ bh
6. the circumference of a circle _______________ C ⴝ 2␲r
5. the area of a circle _______________ A ⴝ ␲r 2
Write the formula to use to find each of the following.
Reading the Lesson
of both circles and the area of the rectangle.
4. Explain how to find the surface area of the cylinder. Add the areas
of the rectangle is the circumference of the circle. So, the
length is 2␲r.
3. How is the length of the rectangle related to the circles? The length
2. Name the shapes in the net. circle, rectangle
1. Make a net of the cylinder.
Write your answers below.
Pre-Activity Complete the Mini Lab at the top of page 538 in your textbook.
Surface Area of Cylinders
Reading to Learn Mathematics
NAME ________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 12-5)
Mathematics: Applications and Concepts, Course 2
Lesson 12–5