8.5
Goal
Find the area of
parallelograms.
Key Words
Area of Parallelograms
Geo-Activity
Exploring the Area of a Parallelogram
1 Use a straightedge to draw a
●
line through one of the vertices
of an index card.
• base of a parallelogram
2 Cut out the triangle. Tape the
●
triangle to the opposite side to
form a parallelogram.
5
5
• height of a
parallelogram
3
• parallelogram p. 310
3
• rhombus p. 325
3 How does the area of the parallelogram compare to the area of the
●
rectangular index card? How do their bases compare? How do their
heights compare?
4 Write a conjecture about the formula for the area of a parallelogram.
●
Either pair of parallel sides of a parallelogram are called the
bases of a parallelogram .
The shortest distance between the bases of a parallelogram is called
the height of a parallelogram . The segment that represents the
height is perpendicular to the bases.
height
base
base
A height can be inside
the parallelogram.
A height can be outside
the parallelogram.
As the Geo-Activity suggests, the area of a parallelogram is found using
a base and its corresponding height.
AREA OF A PARALLELOGRAM
Words
Area (base)(height)
Symbols
A bh
b
8.5
Area of Parallelograms
439
EXAMPLE
Find the Area of a Parallelogram
1
Find the area of the parallelogram.
6m
9m
Solution
Use the formula for the area of a parallelogram.
Substitute 9 for b and 6 for h.
A bh
Formula for the area of a parallelogram
(9)(6)
Substitute 9 for b and 6 for h.
54
Multiply.
ANSWER
The parallelogram has an area of 54 square meters.
EXAMPLE
Find the Height of a Parallelogram
2
Find the height of the parallelogram
given that its area is 78 square feet.
h
A 78 ft 2
12 ft
Solution
A bh
78 12h
6.5 h
ANSWER
Formula for the area of a parallelogram
Substitute 78 for A and 12 for b.
Divide each side by 12.
The parallelogram has a height of 6.5 feet.
Area of Parallelograms
Find the area of the parallelogram.
1.
2.
3.
14 ft
7 mm
12 yd
14 ft
11 mm
8 yd
In Exercises 4–6, A gives the area of the parallelogram. Find the
missing measure.
4. A 72 in.2
h
12 in.
5. A 30 m2
6. A 28 cm2
5m
h
b
7 cm
440
Chapter 8
Polygons and Area
AREA OF A RHOMBUS
Words
1
2
Area (product of diagonals)
d1
1
2
A d 1d 2
Symbols
d2
Area of a Rhombus The formula for the area of a rhombus can be
justified using the area of a triangle. A specific case is given below.
The diagonals divide a rhombus into four congruent
right triangles. So, the area of the rhombus is 4 times
the area of one of the right triangles.
1
2
6
1
2
8
Area of 1 triangle bh (3)(4) 6 square units
3
Area of 4 triangles 4(6) 24 square units
1
2
4
1
2
Notice that d1d2, or (6)(8), also equals 24 square units.
IStudent Help
EXAMPLE
3
Find the Area of a Rhombus
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
Find the area of the rhombus.
a.
b.
6m
9m
14 in.
9m
6m
10 in.
Solution
1
2
a. A d1d2
Solution
1
2
b. A d1d2
1
2
1
2
(14)(10)
(6 6)(9 9)
70
(12)(18)
1
2
108
The area of the rhombus
is 70 square inches.
The area of the rhombus
is 108 square meters.
8.5
Area of Parallelograms
441
8.5 Exercises
Guided Practice
Vocabulary Check
1. The parallelogram at the right has a
base of 7 units and a height of 4 units.
Sketch the parallelogram and label its
base and height.
Skill Check
Match the quadrilateral with the corresponding area equation.
1
2
A. A (6)(10)
B. A (13)(6)
C. A (10)(6)
2.
3.
4.
5
6
3
5
6
3
3
13
10
Find the measures of the diagonals of the rhombus.
5.
6.
7.
4 cm
2m
8 ft
3m
5 cm
3m
11 ft
5 cm
2m
4 cm
Practice and Applications
Extra Practice
Area of Parallelograms Find the area of the parallelogram.
See p. 690.
8.
9.
10.
6
5
12
11
7
10
11.
12.
13.
2.8
3
14.
Homework Help
Example 1: Exs. 8–13
Example 2: Exs. 18–22
Example 3: Exs. 23–28
442
Chapter 8
22
19
You be the Judge One of your
classmates states that it is possible to
draw a parallelogram that is different
from the one at the right, but has the
same area. Do you agree or disagree?
Explain your reasoning.
Polygons and Area
12
13
6 ft
7 ft
15. Stained Glass The piece of stained
glass at the right is made up of eight
congruent parallelograms. Each
parallelogram has a base of
8 centimeters and a height of
3 centimeters. Find the area of the
entire piece.
Crafts
Error Analysis In Exercises 16 and 17, students were asked to find the
area of the parallelogram. Describe and correct any errors.
STAINED GLASS The
16.
17.
1
2
1
(14)(11)
2
A bh
A bh
amount of glass needed for a
stained glass design depends
on the overall area of the
piece, while the amount of
metal needed to connect the
pieces depends on their
individual perimeters.
(7)(5)
5
35
4
11
14
77
3
Using Algebra In Exercises 18–20, A gives the area of the
parallelogram. Find the missing measure.
18. A 88 cm2
19. A 30 in.2
8 cm
20. A 39 ft2
6 ft
h
b
b
5 in.
21. A parallelogram has a base of 13 meters and an area of 104 square
meters. Find the height.
22. A parallelogram has a height of 12 feet and an area of 132 square
feet. Find the base.
Area of Rhombuses Find the area of the rhombus.
23.
24.
25.
4
12
13
5
5
4
7
16
26.
27.
7
28.
9
5
13
7
5
8.1
9
13
12.3
8.5
Area of Parallelograms
443
Penrose Rhombuses The rhombuses shown below are named after
the British physicist Roger Penrose. He found that these rhombuses
can be used to cover a surface without any gaps and without having
to repeat the same pattern. An example is shown below.
Puzzles
15.7
18.5
11.4
6
29. Find the area of the yellow rhombus.
PENROSE TILINGS
The jigsaw puzzle above
illustrates a Penrose tiling
that involves six figures. The
design will cover a surface
without gaps and without
repeating the same pattern.
30. Find the area of the red rhombus.
31. The pattern above contains 32 yellow and 21 red rhombuses.
Find its area.
Area of Complex Polygons Find the area of the entire polygon by
adding the areas of the smaller polygons.
32.
33.
34.
8
4 ft
6
13 cm
6
5 ft
3 cm
9 ft
EXAMPLE
12 cm
8
5m
10 m
Area on the Coordinate Plane
Find the area of the parallelogram.
y
Solution
5
Count the squares to find the base and the
height. The base is 4 units and the height is
5 units.
A bh
1
Formula for area of a parallelogram
(4)(5)
Substitute 4 for b and 5 for h.
20
Multiply.
4
1
x
Area on the Coordinate Plane Find the area of the parallelogram.
35.
36.
y
1
Chapter 8
Polygons and Area
x
y
1
1
1
444
37.
y
1
x
1
x
Challenge Find the area of the given quadrilateral.
38. Parallelogram ABCD
B
39. Rhombus FGHJ
C
G
26
10
F
15
A
Standardized Test
Practice
18
D
H
10
9
J
40. Multiple Choice What is the area of the parallelogram?
A
C
12 cm2
18 cm2
B
D
5 cm
15 cm2
30 cm2
3 cm
41. Multiple Choice What is the area of the rhombus?
F
H
42 ft2
72 ft2
G
J
3 ft
63 ft2
7 ft
84 ft2
7 ft
3 ft
42. Multiple Choice What is the area of the shaded polygon?
A
C
30 in.2
50 in.2
B
D
40 in.2
5 in.
60 in.2
4 in.
Mixed Review
8 in.
Similar Triangles Find the value of x. (Lesson 7.5)
43.
44.
x
45.
x4
2x
32
18
2
Convex and Concave Polygons Decide whether the polygon
is convex or concave. (Lesson 8.1)
46.
Algebra Skills
47.
48.
Simplifying Radicals Simplify the expression. (Skills Review, p. 668)
49. 1
2
50. 4
5
51. 5
2
52. 8
0
53. 3
0
0
54. 1
2
5
55. 1
8
0
56. 4
3
2
8.5
Area of Parallelograms
445