NAME
DATE
NS1.3,
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MG2.1,
A
MG2.2,
N
D
MG2.4,
A
R
MG3.1,
D
S MR2.4
12-1 Area: Parallelograms,
Triangles, and Trapezoids (Pages 612–617)
When you find the area of a parallelogram, triangle, or trapezoid, you must
know the measure of the base and the height. The height is the length of
an altitude. Use the table below to help you define the bases and heights
(altitudes), and find the areas of parallelograms, triangles, and trapezoids.
Base: any side of the parallelogram
Height: the length of an altitude, which is a segment perpendicular to the base,
Parallelogram
with endpoints on the base and the side opposite the base
Area: If a parallelogram has a base of b units and a height of h units, then the
area A is b h square units or A b h.
Triangle
Base: any side of the triangle
Height: the length of an altitude, which is a line segment perpendicular to the
base from the opposite vertex
Area: If a triangle has a base of b units and a height of h units, then the area
1
1
A is b h square units or A b h.
2
2
Trapezoid
Bases: the two parallel sides
Height: the length of an altitude, which is a line segment perpendicular to
both bases, with endpoints on the base lines
Area: If a trapezoid has bases of a units and b units and a height of h units, then
1
1
the area A of the trapezoid is h (a b) square units or A h(a b).
2
2
PRACTICE
Find the area of each figure.
11 ft
1.
2.
6 ft
3.
17 cm
9 in.
4.
4.6 in.
5 in.
12 cm
3.2 in.
5 in.
Find the area of each figure described below.
5. trapezoid: height, 3 in.;
6. triangle: base, 9 cm;
bases, 4 in. and 5 in.
height, 8 cm
7. parallelogram: base, 7.25 ft;
height, 8 ft
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
9. Standardized Test Practice What is the area of a trapezoid whose
bases are 4 yards and 2 yards and whose height is 10 yards?
B 30 yd2
C 60 yd2
A 24 yd2
D 80 yd2
Answers: 1. 66 ft2 2. 102 cm2 3. 35 in2 4. 7.36 in2 5. 13.5 in2 6. 36 cm2 7. 58 ft2 8. 0.09 m2 9. B
3.
8. triangle: base, 0.3 m;
height, 0.6 m
© Glencoe/McGraw-Hill
111
CA Parent and Student Study Guide, Pre-Algebra
NAME
DATE
NS1.3, MG2.1,
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MG2.2,
A
MG2.4,
N
D
MR2.1,
A
R
MR2.4,
D
S MR2.7
12-2 Area: Circles (Pages 619–622)
When you find the area of a circle, you will have to use , which can be
22
approximated as 3.14 or . If you have a calculator that has a
7
key, you can use it when you calculate the area of a circle.
Area of
a Circle
If a circle has a radius of r units, then the area A of a circle is r 2 square units
or A r 2.
EXAMPLES
Find the area of each circle described below. Round decimal answers
to the nearest tenth.
A The radius is 5 cm.
B The diameter is 12 in.
A r 2
A
(3.14)(5)2
Formula for the area of a circle
A r 2
Formula for the area of a circle
The radius is 5 cm.
A
1
The radius is the diameter.
2
(3.14)(6)2
A (3.14)(25)
A (3.14)(36)
A 78.5
A 113.04
The area of the circle is about 78.5 cm2.
The area of the circle is about 113.0 in2. Note: If you use
the -button on your calculator, you will get an area of
about 113.097 in2, which rounds to 113.1 in2.
PRACTICE
Find the area of each circle. Round to the nearest tenth.
1.
2.
3.
5114– in.
30.9 cm
5.6 m
1
4. diameter, 19 mm
5. radius, 25 yd
6. diameter,9 ft
3
7. radius, 13.8 m
8. diameter, 46.2 cm
9. radius, 3 in.
4
1
10. Landscaping A sprinkler can spray water 10 feet out in all
directions. How much area can the sprinkler water?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
11. Standardized Test Practice What is the area of a half circle whose
diameter is 8 meters?
B 50.2 m2
C 100.5 m2
D 201.0 m2
A 25.1 m2
Answers: Answers may vary slightly due to rounding. 1. 98.5 m2 2. 2062.9 in2 3. 2999.6 cm2 4. 283.5 mm2 5. 1963.5 yd2
6. 68.4 ft2 7. 598.3 m2 8. 1676.4 cm2 9. 33.2 in2 10. 314 ft2 11. A
3.
© Glencoe/McGraw-Hill
112
CA Parent and Student Study Guide, Pre-Algebra
NAME
DATE
12-3 Geometric Probability (Pages 623–627)
MG2.1
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Have you ever wondered about the probability of hitting the bull s eye
on a dartboard? This can be found using area.
P(bull s eye) area of the bull s eye
area of the whole target
Geometric probability uses ideas about area to find the probability
of an event.
EXAMPLES
Find the probability of hitting the shaded area on each dartboard below.
A
B
Because 2 of the 4 equal-sized areas on
shaded area
P(shaded) area of target
2
1
the circle are shaded, P(shaded) or 4
2
4
1
P(shaded) or 16
4
Try These Together
Each figure represents a dartboard. Find the probability of landing
in the shaded region.
1.
2.
3.
HINT: In some cases, you may have to calculate the shaded area using formulas you have learned
in previous lessons.
PRACTICE
Each figure represents a dartboard. Find the probability of landing in
the shaded region.
4.
5.
6.
B
C
C
B
C
7. Standardized Test Practice If you throw a dart at the square
target A B C D, what is the probability that the dart will land
in the shaded region?
1
4
D
7. C
© Glencoe/McGraw-Hill
113
2
C
6. 5
1
2
3
B
5. 8
A 2
A
1
B
A
1 cm
1 cm
B
1 cm
1 cm
1
8
4. 2
8.
D
5
A
7.
3. 12
B
6.
3
A
5.
2. 8
4.
C
5
Answers: 1. 8
3.
CA Parent and Student Study Guide, Pre-Algebra
NAME
DATE
12-4 Problem-Solving Strategy: Make
a Model or a Drawing (Pages 629–631)
MG2.2,
S
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MR2.1
A
N
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A
R
D
S
You can often use a drawing or a model to help you solve a problem.
EXAMPLE
When she is not home, Monique leaves her dog s 10 foot leash
connected to a 20 foot wire that is staked to the ground at the two
corners of the back of the house. The leash can slide along the full
length of the wire. How much playing area does her dog have?
Without a drawing, it is difficult to visualize the shape of the dog’s play area, much less
find the area of it. With the drawing, you can see that the leash and wire form a play
area that is a rectangle with two semi-circles on each end.
area of rectangle:
area of each semi-circle:
A w
1
A
r 2
2
A (20)(10) or 200 ft2
1
A
(10)2 or about 157 ft 2
2
wire
10 ft
20 ft
House
leash
Total area 200 2(157) or about 514 ft2
PRACTICE
Solve. Use any strategy.
1. Pets Laverne attaches her dog s leash to a corner of the garage
while she works in the garden. If the garage is 15 feet by 20 feet and
the rope is 25 feet long, how much playing area does her dog have?
2. Interior Design To make cleanup easier, Erin installs
6-inch tiles in front of her fireplace. How many tiles does
she use?
Fireplace
5 ft
2 ft
9 ft
3. Crafts Peter designs a 4-inch by 6-inch card for Mother s Day. He
3
1
wants a -inch yellow outer border and a -inch white inner border.
4
8
How much area is left for his image on the card?
4. Find the sum of the first 50 odd positive integers.
5. Pets Sidsel sets up a 10 foot-circular track with walls to race her
mice, Pinky and Twinkle. Pinky starts out at a rate of 8 feet per
minute. One minute late, Twinkle starts at the same place and
direction at 9 feet per minute. In which lap will Twinkle pass Pinky?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
6. Standardized Test Practice The perimeter of a rectangle is 20 inches.
Its area is 24 square inches. What are the dimensions of the rectangle?
A 1 in. 24 in.
B 2 in. 12 in.
C 3 in. 8 in.
D 4 in. 6 in.
6. D
4.
© Glencoe/McGraw-Hill
Answers: 1. about 1,571 ft2 2. 56 tiles 3. 9 9\16 in2 4. 2500 5. A little after completing the 7th lap
3.
114
CA Parent and Student Study Guide, Pre-Algebra
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DATE
NS1.3, MG2.1,
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MG2.2, MG2.3,
A
MG3.5,
N
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MR2.1,
A
R
MR2.4,
D
S MR2.7
12-5 Surface Area: Prisms and
Cylinders (Pages 632–637)
In geometry, a solid like a cardboard box is
Back
called a prism.A prism is a solid figure that
Bottom
has two parallel, congruent sides, called bases.
Base
Base
A prism is named by the shape of its bases. For
Front
example, a prism with rectangular-shaped bases
A triangular prism
is a rectangular prism.A prism with triangularhas five faces.
shaped bases is a triangular prism.A cylinder is
a geometric solid whose bases are parallel, congruent circles. The surface area of a
geometric solid is the sum of the areas of each side or face of the solid. If you open up
or unfold a prism, the result is a net. Nets help you identify all the faces of a prism.
EXAMPLES
Find the surface area of the given geometric solids.
A a box measuring 6 in. 8 in. 12 in.
B a cylinder with a radius of 10 cm and
Find the surface area of the faces. Use the formula
a height of 24 cm
A w. Multiply each area by 2 because there are
two faces with each area.
Front and Back:
Top and Bottom:
Two Sides:
Total:
The surface area of a cylinder equals the area of the
two circular bases, 2r2, plus the area of the curved
surface. If you make a net of a cylinder, you see
that the curved surface is really a rectangle with a
height that is equal to the height h of the cylinder
and a length that is equal to the circumference of
the circular bases, 2r.
6 8 48 (each)
12 8 96 (each)
6 12 72 (each)
2(48) 2(96) 2(72) 432 in2
Surface area 2r 2 h 2r
Surface area 2(100) 48(10)
Surface area 628.3 1508.0
Surface area 2136.3 cm2
PRACTICE
Find the surface area of each solid. Round to the nearest tenth.
1.
2. 4.7 m
3. 29 mm
5 ft
8 ft
4.
3 ft
35 mm
12 m
3 cm
5.
11–2 in.
3 cm
5 in.
20 mm
21 mm
6.
13 m
12 m
10 cm
10 m
13 m
33 m
7. Pets A pet store sells nylon tunnels for dog agility courses. If a
1
tunnel is 6 feet long and 1 feet in diameter, how many square feet
2
of nylon is used?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
8. Standardized Test Practice The height of a cylinder is 10 meters and
its diameter is 4 meters. What is its surface area?
B 138.2 m2
C 150.8 m2
D 351.9 m2
A 75.4 m2
Answers: 1. 158 ft2 2. 211.9 m2 3. 2,870 mm2 4. 27.1 in2 5. 84.2 cm2 6. 1,308 m2 7. about 28.3 ft2 8. C
3.
© Glencoe/McGraw-Hill
115
CA Parent and Student Study Guide, Pre-Algebra
NAME
DATE
12-6 Surface Area: Pyramids and
Cones (Pages 638–642)
NS1.3,
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MG2.2,
A
MG3.5,
N
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MR2.1,
A
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MR2.4,
D
S MR2.7
A pyramid is a solid figure that has a polygon for a base and triangles
for sides, or lateral faces. Pyramids have just one base. The lateral faces
intersect at a point called the vertex. Pyramids are named for the
shapes of their bases. For example, a triangular pyramid has a
triangle for a base. A square pyramid has a square for a base. The
slant height of a pyramid is the altitude of any of the lateral faces of
the pyramid. To find the surface area of a pyramid, you must find the
area of the base and the area of each lateral face. The area of the lateral
surface of a pyramid is the area of the lateral faces (not including the
base). A circular cone is another solid figure and is shaped like some
ice cream cones. Circular cones have a circle for their base.
Surface
Area of a
Circular
Cone
The surface area of a cone is equal to the area of the base, plus the lateral area of
the cone. The surface area of the base is equal to r 2. The lateral area is equal to
r, where is the slant height of the cone. So, the surface area of the cone, SA,
is equal to r 2 r.
EXAMPLES
Find the surface area of the given geometric solids.
B a cone with a radius of 4 cm and
A a square pyramid with a base that is 20 m
a slant height of 12 cm
on each side and a slant height of 40 m
Use the formula.
Find the surface area of the base and the lateral faces.
Base:
Each triangular side:
A
s2
or
A
(20)2
A 400
SA 400 4(400)
SA 2000 m2
1
bh
2
or
SA r2 r
SA (4)2 (4)(12)
SA 50.3 150.8
SA 201.1 cm2
1
(20)(40)
2
A 400
Area of the base plus area
of the four lateral sides.
PRACTICE
Find the surface area of each solid. Round to the nearest tenth.
1.
2.
3.
18.2 m
28.7 ft
10 m
4.
12.3 mm
10 ft
5.
11.4 in.
8 mm
10 ft
8 mm
6.
21 cm
3.1 in.
15.3 in.
B
4.
C
B
C
B
A
7.
8.
17 cm
C
A
5.
6.
2.2 in.
B
A
7. Standardized Test Practice What is the surface area of a square
pyramid where the length of each side of the base is 10 meters and
the slant height is also 10 meters?
B 400 m2
C 500 m2
D 1000 m2
A 300 m2
Answers: 1. 885.9 m2 2. 674 ft2 3. 260.8 mm2 4. 548.0 in2 5. 18.5 in2 6. 2029.5 cm2 7. A
3.
2.2 in.
© Glencoe/McGraw-Hill
116
CA Parent and Student Study Guide, Pre-Algebra
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DATE
NS1.3,
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AF3.2,
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MG2.1,
N
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MR2.4,
A
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MR2.7
D
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12-7 Volume: Prisms and
Cylinders (Pages 644–648)
The amount a container will hold is called its capacity,or volume.
Volume is often measured in cubic units such as the cubic centimeter
.
(cm3) and the cubic inch (in3)
Volume
of a Prism
If a prism has a base area of B square units and a height of h units, then the
volume V is B h cubic units, or V Bh.
Volume of
a Cylinder
If a circular cylinder has a base with a radius of r units and a height of h units,
then the volume V is r 2h cubic units, or V r 2h.
EXAMPLES
Find the volume of the given figures.
A a rectangular prism with a length of 3 cm,
a width of 4 cm, and a height of 12 cm
V Bh
V (w)h
V r 2h
Formula for the volume of a cylinder
V (5) 2(18) The diameter is 10 in., so the
radius is 5 in.
V (25)(18)
V 1413.7 in3
Formula for the volume of a prism
Since the base of the prism is a
rectangle, B w.
V (3)(4)(12)
V 144 cm3
B a circular cylinder with a diameter
of 10 in. and a height of 18 in.
Try These Together
Find the volume of each solid. Round to the nearest tenth.
1.8 m
1. 2 m
2. 7 ft
3.
8 cm
1.2 m
11 ft
1.6 m
24 cm
8 cm
HINT: Find the area of the base first, then multiply by the height to get the volume.
PRACTICE
Find the volume of each solid. Round to the nearest tenth.
4.
5.
6.
29 in.
29 in.
2 mm
21 in.
2 mm
40 m
9 cm
5 mm
60 in.
1.8 cm
7. Landscaping Nat buys mulch for his flower gardens each fall. How
many cubic feet of mulch can he bring home if his truck bed is 5 feet
by 8 feet by 2 feet?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
8. Standardized Test Practice What is the height of a cylindrical prism
whose volume is 141.3 cubic meters and whose diameter is 10 meters?
A 0.45 m
B 0.9 m
C 1.8 m
D 2.25 m
Answers: 1. 1.7 m3 2. 1693.3 ft3 3. 1536 cm3 4. 25,200 in3 5. 27.9 mm3 6. 114.5 cm3 7. 80 ft3 8. C
3.
© Glencoe/McGraw-Hill
117
CA Parent and Student Study Guide, Pre-Algebra
NAME
DATE
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MR2.4,
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MR2.7
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D
A
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12-8 Volume: Pyramids and
Cones (Pages 649–653)
When you find the volume of a pyramid or cone, you must know the height
h. The height is not the same as the lateral height, which you learned in
an earlier lesson. The height h of a pyramid or cone is the length of a
segment from the vertex to the base, perpendicular to the base.
Volume of
a Pyramid
If a pyramid has a base of B square units, and a height of h units, then the
Volume
of a Cone
If a cone has a radius of r units and a height of h units, then the volume V
1
1
volume V is B h cubic units, or V Bh.
3
3
1
1
r 2h.
is r 2 h cubic units, or V 3
3
EXAMPLES
Find the volume of the given figures.
A a square pyramid with a base side
length of 6 cm and a height of 15 cm
1
V
Bh
3
B a cone with a radius of 3 in. and a
height of 8 in.
1 2
V
r h
3
Formula for the volume of a pyramid
Formula for the volume of a cone
1 2
V
s h Replace B with s2.
3
1
V
(3)2(8) r 3 and h 8
3
1
V
(6)2(15) or 180 cm3
3
1
V
(9)(8) or about 75.4 in3
3
PRACTICE
Find the volume of each solid. Round to the nearest tenth.
1.
2.
3.
8 cm
4.
5 cm
50 mm
21 in.
40 mm
18 in.
8 ft
4 cm
A = 36 ft2
5. Cooking A spice jar is 3 inches tall and 1.5 inches in diameter.A
funnel is 2 inches tall and 2.5 inches in diameter. If Hayden fills the
funnel with pepper to put into the spice jar, will it overflow?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
6. Standardized Test Practice A square pyramid is 6 feet tall and with
the sides of the base 8 feet long. What is the volume of the pyramid?
B 128 ft3
C 192 ft3
D 384 ft3
A 96 ft3
Answers: 1. 53.3 cm3 2. 20,944.0 mm3 3. 1781.3 in3 4. 96 ft3 5. No; jar volume 5.3 in3, funnel volume 3.3 in3 6. B
3.
© Glencoe/McGraw-Hill
118
CA Parent and Student Study Guide, Pre-Algebra
NAME
DATE
MG2.1, MR1.1,
S
T
MR2.1,
A
MR2.5,
N
D
MR2.6,
A
R
MR2.8,
D
S MR3.1
Chapter 12 Review
Product Design
As a product designer, you are challenged to analyze the four
cardboard popcorn containers below from the perspectives of a
manufacturer, a movie theater, and a consumer.
7 in.
4 in.
5 in.
6 in.
6 in.
10 in.
A
10 in.
B
C
5 in.
D
Height = 10 in.
Height = 10 in.
Slant height
= 5
5 in.
1. Which container will hold the most popcorn? Justify your answer.
2. Which popcorn container will require the most cardboard to make?
Justify your answer. (Note that the tops of the containers are open.
For example, the rectangular prism will require cardboard for the
bottom and four sides only.
)
3. Which of the four popcorn containers might a manufacturer prefer
to make? Explain your reasoning.
4. Which of the four popcorn containers might a movie theater want to
use? Explain your reasoning.
5. Which of the four might be easiest for the consumer to carry around
at an outdoor event? Explain your reasoning.
Answers are located on page 140.
© Glencoe/McGraw-Hill
119
CA Parent and Student Study Guide, Pre-Algebra