Area: Parallelograms
PRE-ALGEBRA LESSON 10-1
It takes 126 ft of fence to enclose a field that is twice as long as it is
wide. What is the area of the field?
882 ft2
10-1
Area: Parallelograms
PRE-ALGEBRA LESSON 10-1
(For help, go to Lesson 3-4.)
Use A = w and find the third value.
1. A = 54 in.2, w = 6 in.
2.
= 35 m, w = 7 m
3. A = 25 cm2,
4.
= 7.2 ft, w = 7.2 ft
= 2.5 cm
Check Skills You’ll Need
10-1
Area: Parallelograms
PRE-ALGEBRA LESSON 10-1
Solutions
1. A = w
54 = 6
= 9 in.
2. A = w
A = 35 • 7
A = 245 m2
3. A = w
25 = 2.5w
w = 10 cm
4. A = w
A = 7.2 • 7.2
A = 51.84 ft2
10-1
Area: Parallelograms
PRE-ALGEBRA LESSON 10-1
Find the area of the rectangle.
Step 1 Change the units so that they are the same.
150 cm = 1.5 m
Change 150 centimeters to meters.
Step 2 Find the area.
A = bh
Use the formula for area of a rectangle.
= (4)(1.5)
Replace b and h with the dimensions 4
and 1.5.
= 6
Simplify.
The area of the rectangle is 6 m2.
10-1
Quick Check
Area: Parallelograms
PRE-ALGEBRA LESSON 10-1
Find the area of each parallelogram.
a.
b.
A = bh
area formula
A = bh
= (8)(2)
Substitute.
= (2.5)(6)
= 16
Simplify.
= 15
The area is 16 m2.
The area is 15 in.2.
10-1
Quick Check
Area: Parallelograms
PRE-ALGEBRA LESSON 10-1
Find each area.
1. rectangle ABFE
40 cm2
2. parallelogram ACFD
48 cm2
3. a rectangle with a base of 50 cm and a height of 5 cm
250 cm2
10-1
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
Find the area of a rectangle that is 3
1
ft wide and twice as high.
2
24 1 ft2
2
10-2
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
(For help, go to Lesson 5-4.)
Find each product.
1
• 16
2
1
3. 2 • 5 • 15
1.
1
• 14 • 6
2
1
1
4. 2 • 2 2 • 8
2.
Check Skills You’ll Need
10-2
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
Solutions
1. 1 • 16
2
1
8
21 • 16 = 8
3. 1 • 5 • 15
2
2. 1 • 14 • 6
2
1
7
21 • 14 • 6
7 • 6 = 42
4. 1 • 2 1 • 8
2
2
1 5
• •8
2 2
1
• 75 = 37 1
2
2
1
5
•
• 82
1
1
2
2
5 • 2 = 10
10-2
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
Find the area of the triangle.
A=
=
1
bh
2
Use the formula for area of a triangle.
1
• 13 • 6
2
Replace b with 13 and h with 6.
= 39
Simplify.
Quick Check
The area is 39 in.2.
10-2
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
Find the area of the figure.
Area of triangle
Area of rectangle
1
bh
2
1
=
• 45 • 20
2
A =
A = bh
= 45 • 30
= 450
= 1,350
Add to find the total: 450 + 1,350 = 1,800.
The area of the figure is 1,800 cm2.
10-2
Quick Check
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
Suppose that, through the years, a layer of silt and mud
settled in the bottom of the Erie Canal. Below is the resulting cross
section of the canal. Find the area of the trapezoidal cross section.
1
h(b1 + b2)
2
1
A =
• 3(31 + 40)
2
1
=
• 3(71)
2
1
=
• 213
2
A =
Use the formula for the area of a trapezoid.
Replace h with 3, b1 with 31, and b2 with 40.
Simplify.
= 106.5
The area of the cross section is 106.5
10-2
ft2.
Quick Check
Area: Triangles and Trapezoids
PRE-ALGEBRA LESSON 10-2
Find each area.
1. trapezoid PQRU
2. triangle PTU
192 ft2
20 ft2
3. triangle QRS
28 ft2
4. trapezoid PQSU
164 ft2
10-2
Area: Circles
PRE-ALGEBRA LESSON 10-3
Explain how this pattern works.
(15,873  7)  2 = (222,222)
(15,873  7)  3 = (333,333)
(15,873  7)  4 = (444,444)
because (15,873  7)  1 = (111,111)
10-3
Area: Circles
PRE-ALGEBRA LESSON 10-3
(For help, go to Lesson 4-7.)
Simplify each expression.
1. 3.14 • 42
2. 3.14 • 52
3. 3.14 • 92
4. 3.14 • 0.52
Check Skills You’ll Need
10-3
Area: Circles
PRE-ALGEBRA LESSON 10-3
Solutions
1. 3.14 • 42
= 3.14 • 4 • 4
= 50.24
2. 3.14 • 52
= 3.14 • 5 • 5
= 78.5
3. 3.14 • 92
= 3.14 • 9 • 9
= 254.34
4. 3.14 • 0.52
= 3.14 • 0.5 • 0.5
= 0.785
10-3
Area: Circles
PRE-ALGEBRA LESSON 10-3
Find the exact area of a circle with diameter 20 in.
A =
=
r2
(10)2
= 100
The area is 100
r=
1
d; r = 10
2
Simplify.
in.2.
Quick Check
10-3
Area: Circles
PRE-ALGEBRA LESSON 10-3
A TV station’s weather radar can detect precipitation in
a circular region having a diameter of 100 mi. Find the area of
the region.
A =
=
r2
(50)2
= 2,500
(2,500)(3.14)
= 7,850
r=
1
d; r = 50
2
exact area
Use 3.14 for
.
approximate area
The area of the region is about 7,850 mi2.
Quick Check
10-3
Area: Circles
PRE-ALGEBRA LESSON 10-3
A pound of grass seed covers approximately 675 ft2. Find the
area of the lawn below. Then find the number of bags of grass seed
you need to buy to cover the lawn. Grass seed comes in 3-lb bags.
Area of region that is one fourth of a circle:
area of circle =
r2
area of quarter circle = 1
A
4
1 (3.14)(15)2
4
r2
Replace
= 176.625 ft2
10-3
with 3.14 and r with 15.
Area: Circles
PRE-ALGEBRA LESSON 10-3
(continued)
Area of region that is a rectangle:
area of rectangle = bh
A = 45 • 25
Replace b with 45 and h with 25.
= 1,125 ft2
The area of the lawn is about 177 ft2 + 1,125 ft2 = 1,302 ft2.
1,302 ÷ 675
1.93
Divide to find the number of pounds of seed.
You need to buy one 3-lb bag of grass seed.
10-3
Quick Check
Area: Circles
PRE-ALGEBRA LESSON 10-3
Find the area.
1. Find the exact area of a circle with diameter 32 in.
256 in.2
2. A 5-ft-diameter round table is in a 12 ft-by-15 ft room.
a. What is the area covered by the table? Round to the nearest unit.
20 ft2
b. What is the area of the rest of the room? Round to the nearest unit.
160 ft2
10-3
Space Figures
PRE-ALGEBRA LESSON 10-4
Find the area of a square 15.7 mm on each side.
246.49 mm2
10-4
Space Figures
PRE-ALGEBRA LESSON 10-4
(For help, go to Lesson 9-3.)
Judging by appearance, classify each polygon.
1.
2.
3.
4.
Check Skills You’ll Need
10-4
Space Figures
PRE-ALGEBRA LESSON 10-4
Solutions
1. triangle
2. square
3. rectangle
4. hexagon
10-4
Space Figures
PRE-ALGEBRA LESSON 10-4
For each figure describe the bases and name
the figure.
a.
b.
The bases are circles.
The bases are rectangles.
The figure is a cylinder.
The figure is a rectangular prism.
Quick Check
10-4
Space Figures
PRE-ALGEBRA LESSON 10-4
Name the space figure you can form from each net.
a.
b.
With two hexagonal bases
and rectangular sides, you
can form a hexagonal prism.
With a rectangular base and
triangular sides, you can form a
rectangular pyramid.
Quick Check
10-4
Space Figures
PRE-ALGEBRA LESSON 10-4
Describe the base(s) and name each solid.
1.
2.
circular base; cone
triangular bases;
triangular prism
3. Name the solid you can form from this net.
octagonal pyramid
10-4
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
A prism has 7 faces and 10 vertices. How many edges does the
prism have?
15
10-5
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
(For help, go to Lesson 9-6.)
Find the circumference of each circle with the given radius or diameter.
1. r = 5 in.
2. r = 4.2 cm
3. d = 8 ft
4. d = 6.8 in.
Check Skills You’ll Need
10-5
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
Solutions
1. C = 2 r
C = 2(3.14)(5 in.)
C = 31.4 in.
2. C = 2 r
C = 2(3.14)(4.2 cm)
C = 26.4 cm
3. C = d
C = (3.14)(8 ft)
C = 25.1 ft
4. C = d
C = (3.14)(6.8 in.)
C = 21.3 in.
10-5
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
Find the surface area of the rectangular prism
using a net.
Draw and label a net.
Find the area of each
rectangle in the net.
60 + 60 + 150 + 90 + 150 + 90 = 600
Add the areas.
Quick Check
The surface area is 600 cm2.
10-5
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
Find the surface area of the rectangular prism.
Step 1
Find the lateral area.
L.A. = ph
= (5 + 6 + 5 + 6)20
Use the formula
for lateral area.
p=5+6+5+6
and h = 20
= 440
Step 2 Find the surface area.
S.A. = L.A. + 2B
= 440 + 2(5 • 6)
L.A. = 440 and B = 5 • 6
= 440 + 60
= 500
The surface area of the rectangular prism is 500 in.2.
10-5
Quick Check
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
Find the surface area of the cylindrical water tank.
Step 1
Find the lateral area.
L.A. = 2
rh
Use the formula
for lateral area.
2(3.14)(8)(15)
754
Step 2
Find the surface area.
S.A. = L.A. + 2B
Use the formula for surface area.
= L.A. + 2( r 2)
754 + 2(3.14)(8)2
1,156
The surface area of the water tank is about 1,156 ft2.
10-5
Quick Check
Surface Area: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-5
Find the surface area of each figure rounded to the nearest whole unit.
1. triangular prism with base perimeter 24 cm, base area 24 cm2, and
height 15 cm
408 cm2
2. rectangular prism with base perimeter 30 cm, base area 50 cm2, and
height 150 cm
4,600 cm2
3. cylindrical candle with radius 2 cm and height 16 cm
about 226 cm2
10-5
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
Give the number of faces, edges, and vertices of a rectangular
prism.
6 faces, 12 edges, 8 vertices
10-6
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
(For help, go to Lesson 5-4.)
Use the Order of Operations to simplify each expression.
1.
2
(9
3
3. 1 (24
6
)+
1
(8
2
) + 1 (3
3
)
2.
)
3
(12
4
4. 5 (32
8
2
(15
5
)
) + 1 (14
)
)+
7
Check Skills You’ll Need
10-6
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
Solutions
1. 21 • 93
3
+ 11• 84
2. 31 • 123
2
4
=6 +4
= 10
3. 11 • 244
6
=4
=5
+ 21 • 153
5
=9 +6
= 15
+ 11 • 31
4. 51 • 324
3
8
+
= 20
= 22
10-6
+ 11 • 142
7
+2
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
Find the surface area of the square pyramid.
Step 1
L.A. =
=
Step 2
1
p
2
Use the formula for lateral area.
1
• 20 • 8 = 80
2
p = 4(5) and
S.A. = L.A. + B
= 80 + 52
= 80 + 25 = 105
= 8.
Lateral area = 80 and B = 52.
The surface area of the pyramid is 105 m2.
10-6
Quick Check
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
Find the surface area of the cone.
Step 1 L.A. =
=
r
Use the formula for lateral area.
3.14(3)(7)
65.94
r = 3 and
Step 2 S.A. = L.A. + B
65.94 + 3.14(3)2
= 65.94 + 28.26
= 94.2
Use the formula for surface area.
L.A.
65.94 and B =
(3)2.
The surface area of the cone is about 94 m2.
10-6
= 7.
Quick Check
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
Earth has an average radius of 3,963 mi. What is
Earth’s approximate surface area to the nearest 1,000 mi2?
Assume that Earth is a sphere.
S.A. = 4
r2
4(3.14)(3,963)2
= 197,259,434.64
197,259,000
Use the formula for surface area.
r
3,963
Multiply.
Round to nearest 1,000.
The surface area of Earth is about 197,259,000 mi2.
10-6
Quick Check
Surface Area: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-6
Find the surface area of each space figure. Round to the nearest whole unit.
1. a square pyramid with base edge 80 m and slant height 100 m
22,400 m2
2. a cone with slant height 22 cm and radius 7 cm
about 637 cm2
3. a sphere with radius 12 cm
about 1,809 cm2
10-6
Volume: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-7
Use graph paper. Design and draw a diagram to determine which
has the greater area—a square with sides 10 cm or a circle with a
diameter 10 cm?
Draw a circle within the square to prove that the square has the greater area.
10-7
Volume: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-7
(For help, go to Lesson 10-3.)
Find the area of each circle.
1.
2.
3.
Check Skills You’ll Need
10-7
Volume: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-7
Solutions
1. A = r2
= (82)( )
= 64(3.14)
201 cm2
2. A = r2
= (122)( )
= 144(3.14)
452.2 cm2
3. r = 1 d
2
1
= 2 (20)
= 10
A = r2
= (102)( )
= 100(3.14)
314 cm2
10-7
Volume: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-7
Find the volume of the triangular prism.
V = Bh
Use the formula for volume.
1
• 9 • 14 = 63 cm2
2
= 63 • 20
B=
= 1,260
Simplify.
The volume is 1,260 cm3.
Quick Check
10-7
Volume: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-7
Find the volume of the juice can, to the nearest cubic
centimeter.
V = Bh
V =
r 2h
3.14 • 3.42 • 16
= 580.7744
The volume is about 581
Use the formula for volume.
B =
r2
Replace r with 3.4, and h with 16.
Simplify.
cm3.
10-7
Quick Check
Volume: Prisms and Cylinders
PRE-ALGEBRA LESSON 10-7
Find the volume of each space figure.
1. rectangular prism with base 12 m by 14 m and height 50 m
8,400 m3
2. cylindrical pool with diameter 24 ft and height 4 ft
about 1,808.64 ft3
3. right triangular prism with base legs 8 cm and 10 cm and height 20 cm
800 cm3
10-7
Problem Solving Strategy: Make a Model
PRE-ALGEBRA LESSON 10-8
1
A roll of wallpaper is 24 in. wide. You used all but 4 2 ft of one roll.
1
Another roll has 7 3 ft of wallpaper on it. What is the total area you can
cover with the remaining wallpaper?
23 2 ft2
3
10-8
Problem Solving Strategy: Make a Model
PRE-ALGEBRA LESSON 10-8
(For help, go to Lesson 9-3.)
Draw each figure described below.
1. a rectangle with small squares drawn in each corner
2. a rectangle divided into eight congruent rectangles
3. two parallelograms that have different shapes but the same perimeter
Check Skills You’ll Need
10-8
Problem Solving Strategy: Make a Model
PRE-ALGEBRA LESSON 10-8
Solutions
Answers may vary. Samples:
1.
2.
3.
10-8
Problem Solving Strategy: Make a Model
PRE-ALGEBRA LESSON 10-8
A can company rolls rectangular pieces of
metal that measure 8 in. by 10 in. to make the sides of
cans. Which height, 8 in. or 10 in., will make the can
with the greater volume?
Build two cans using 8 in.-by-10 in. pieces of paper. You do not
need to make the bases, just the sides.
not to scale
10-8
Problem Solving Strategy: Make a Model
PRE-ALGEBRA LESSON 10-8
(continued)
not to scale
Measure your models to find approximate radii.
Radius of 10-in. high can 1.3 in.
Radius of 8-in. high can 1.6 in.
Find the volumes.
V =
r 2h
V =
r 2h
(3.14)(1.32)(10)
(3.14)(1.62)(8)
53.1 in.3
64.3 in.3
The volume of the can with height 8 in. is greater.
10-8
Quick Check
Problem Solving Strategy: Make a Model
PRE-ALGEBRA LESSON 10-8
Solve.
1. You cut square corners from a piece of cardboard that has
dimensions 32 cm by 40 cm. You then fold the cardboard to create a
box with no lid. To the nearest centimeter, what are the dimensions of
the box that will have the greatest volume?
20 cm by 28 cm by 6 cm
10-8
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
Multiply. Write each answer in simplest form.
a. 3
1
3
2
4
17
b. 1
1
1
2
8
6
27
c. 1  2
7
5
8
16
2
35
10-9
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
(For help, go to Lesson 5-4.)
Multiply.
1
(3.14)(2)2(5)
3
4
3. 3 (3.14)(2)3
1.
1
(4)2(6)
3
4
4. 3 (3.14)(0.5)3
2.
Check Skills You’ll Need
10-9
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
Solutions
1. 1 (3.14)(22)(5) = 1 (3.14)(4)(5)
3
3
= 1 (62.8)
3
1
2. 1 (42)(6) = 3 (16)(6)
3
3
= 20.93
= 32
4
4. 1 (3.14)(0.5)3 = (3.14)(0.125)
3. 4 (3.14)(23) = 4 (3.14)(8)
3
= 1 (96)
3
= 4 (25.12)
3
3
= 33.493
3
= 4 (0.3925)
3
= 0.523
10-9
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
Find the volume of the cone.
V =
1
Bh
3
Use the formula for volume.
V =
1
3
B =
r 2h
1
(3.14)(2)2(12)
3
= 50.24
r2
Replace r with 2 and h with 12.
Simplify.
The volume of the cone is about 50 in.3.
10-9
Quick Check
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
Find the volume of the square pyramid.
1
Bh
3
1
V = s 2h
3
1
= (8)2(12)
3
V =
= 256
Use the formula for volume.
B = s2
Replace s with 8 and h with 12.
Simplify.
The volume of the pyramid is 256 in.3.
10-9
Quick Check
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
Earth has an average radius of 3,963 mi. What is
Earth’s approximate volume to the nearest 1,000,000 mi3?
Assume that Earth is a sphere.
V =
4
3
r3
Use the volume formula.
4
(3.14)(3,963)3
3
Replace r with 3,963.
260,579,713,159
Simplify.
The volume of the Earth is about 260,580,000,000 mi3.
Quick Check
10-9
Volume: Pyramids, Cones, and Spheres
PRE-ALGEBRA LESSON 10-9
Find the volume of each space figure to the nearest unit.
1. a cone with diameter 9 cm and height 12 cm
about 254 cm2
2. a square pyramid with base edges 12 m and height 18 m
864 m3
3. a basketball with diameter 10 in.
about 523 in.3
10-9