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8.6 Area of Trapezoids

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Page 1 of 5
8.6
Goal
Area of Trapezoids
Find the area of
trapezoids.
Recall that the parallel sides of a trapezoid are
called the bases of the trapezoid, with lengths
denoted by b1 and b2.
Key Words
The shortest distance between the bases is
the height of the trapezoid .
base, b1
height
base, b2
• trapezoid p. 332
• base of a trapezoid p. 332
• height of a trapezoid
Suppose that two congruent trapezoids with bases b1 and b2 and
height h are arranged to form a parallelogram as shown.
b1
b2
b2
b1
h
Student Help
LOOK BACK
To review more about
trapezoids, see p. 332.
The area of the parallelogram is h(b1 П© b2). Because the two trapezoids
are congruent, the area of one of the trapezoids is half the area of the
parallelogram.
AREA OF A TRAPEZOID
1
2
Words
Area П­ бЋЏбЋЏ(height)(sum of bases)
Symbols
b1
1
2
A П­ бЋЏбЋЏh(b1 П© b2)
b2
EXAMPLE
1
Find the Area of a Trapezoid
Find the area of the trapezoid.
Solution
1
A П­ бЋЏбЋЏh(b1 П© b2)
2
Polygons and Area
Formula for the area of a trapezoid
Substitute 5 for h, 6 for b1, and 8 for b2.
П­ бЋЏбЋЏ(5)(14)
1
2
Simplify within parentheses.
П­ 35
Simplify.
ANSWER
Chapter 8
8 in.
1
2
П­ бЋЏбЋЏ(5)(6 П© 8)
446
6 in.
б®Ј The area of the trapezoid is 35 square inches.
Page 2 of 5
Area of Trapezoids
Find the area of the trapezoid.
1.
2.
7m
3. 8 cm
16 ft
7m
5 cm
14 ft
12 cm
11 m
EXAMPLE
10 ft
2
Use the Area of a Trapezoid
Find the value of b2 given that the area
of the trapezoid is 96 square meters.
A П­ 96 m2
Student Help
STUDY TIP
The equation is easier
to solve without the
fraction. So, multiply
each side by 2 before
distributing.
9m
b2
Solution
1
2
Formula for the area of a trapezoid
1
2
Substitute 96 for A, 8 for h, and 9 for b1.
A П­ бЋЏбЋЏh(b1 П© b2)
96 П­ бЋЏбЋЏ(8)(9 П© b2)
192 П­ 8(9 П© b2)
Multiply each side by 2.
192 П­ 72 П© 8b2
Use the distributive property.
120 П­ 8b2
Subtract 72 from each side.
15 П­ b2
ANSWER
Divide each side by 8.
б®Ј The value of b2 is 15 meters.
Use the Area of a Trapezoid
A gives the area of the trapezoid. Find the missing measure.
4. A П­ 77 ft2
8 ft
5. A П­ 39 cm2
6. A П­ 84 in.2
b1
13 in.
6 cm
h
14 ft
8 in.
8 cm
b2
7. A trapezoid has an area of 294 square yards. Its height is 14 yards
and the length of one base is 30 yards. Find the length of the
other base.
8.6
Area of Trapezoids
447
Page 3 of 5
8.6 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Sketch a trapezoid. Label its height h and its bases b1 and b2.
Find the height and the lengths of the bases of the trapezoid.
2.
3.
13
6
3
7
4.
9
7
10
12
5
11
19
8
6
Match the trapezoid with the equation used to find the height.
1
2
1
2
C. A П­ бЋЏбЋЏh(5 П© 8)
5
7.
A. A П­ бЋЏбЋЏh(5 П© 13)
B. A П­ бЋЏбЋЏh(8 П© 13)
5.
6.
5
1
2
8
h
h
h
5
8
8
13
Practice and Applications
Extra Practice
Area of a Trapezoid Find the area of the trapezoid.
See p. 690.
8.
9.
11
7
10.
6
7
8
15
16
10
11.
12.
24
6.2
13.
14
13
19
15
4.6
8
14
4.4
Homework Help
Example 1: Exs. 8–13
Example 2: Exs. 18–22
448
Chapter 8
14.
You be the Judge A classmate states
that if you double the dimensions of the
trapezoid at the right, then its area doubles.
Do you agree? Explain your answer.
Polygons and Area
6
6
9
8
Page 4 of 5
15.
Student Help
LOOK BACK
To review the
midsegment of a
trapezoid, see p. 333.
Visualize It! Draw three different trapezoids with a height of
5 units and bases of 3 units and 7 units. Then find the areas of
the trapezoids. What do you notice?
Technology In Exercises 16 and 17, use geometry software.
1
в—Џ
2
в—Џ
Draw a trapezoid.
Draw the midsegment.
B
A
16. Find the length of the midsegment and
F
E
the height of the trapezoid. Multiply the
two measures.
C
D
17. Find the area of the trapezoid. How does
the area compare to your answer for
Exercise 16?
Using Algebra In Exercises 18–20, A gives the area of the
trapezoid. Find the missing measure.
18. A П­ 135 cm2
b1
10 cm
18 cm
IStudent Help
ICLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Exs. 21–22 is
at classzone.com
19. A П­ 132 in.2
20. A П­ 198 m2
b2
7 in.
h
12 m
17 in.
15 m
21. A trapezoid has an area of 50 square units. The lengths of the
bases are 10 units and 15 units. Find the height.
22. A trapezoid has an area of 24 square units. The height is 3 units
and the length of one of the bases is 5 units. Find the length of the
other base.
Bridges In Exercises 23–25, use the following information.
The roof on the bridge below, consists of four sides: two congruent
trapezoids and two congruent triangles.
137 ft
13 ft 2 in.
159 ft
Doe River Covered Bridge in Elizabethton, Tennessee
23. Find the combined area of the two trapezoids.
13 f t 7 in.
24. Use the diagram at the right to find the
combined area of the two triangles.
21 f t
25. What is the area of the entire roof ?
Detail of roof
8.6
Area of Trapezoids
449
Page 5 of 5
Student Help
Windows Find the area of the window.
VISUAL STRATEGY
26.
27.
16 in.
To find the area of a
complex polygon, you
can add the areas of
the simpler shapes that
make up the polygon,
as shown on p. 410.
28.
12 in.
30 in.
9 in.
12 in.
32 in.
12 in.
16 in.
21 in.
16 in.
20 in.
48 in.
Using the Pythagorean Theorem Find the height using the
Pythagorean Theorem and a calculator. Then find the area of
the trapezoid.
29.
30.
28
40
h
31.
100
91
11
24
Standardized Test
Practice
35
25
8
120
32. Multiple Choice What is the area of the trapezoid?
A
Й•
C
Й•
25 in.2
B
Й•
D
Й•
68 in.2
42 in.2
8 in.
4 in.
84 in.2
13 in.
33. Multiple Choice What is the area of the trapezoid?
F
Й•
H
Й•
88 ft2
G
Й•
J
Й•
152 ft2
128 ft2
16 ft
176 ft2
8 ft
6 ft
Mixed Review
Finding Area Match the region with a formula for its area. Use each
formula exactly once. (Lessons 8.3 – 8.6)
34. Region 1
A. A П­ s2
35. Region 2
B. A П­ бЋЏбЋЏd1d2
36. Region 3
C. A П­ бЋЏбЋЏbh
37. Region 4
D. A П­ бЋЏбЋЏh(b1 П© b2)
1
2
Algebra Skills
3
8
Chapter 8
2
5
3
1
2
4
E. A П­ bh
Fraction Operations Add or subtract. Write the answer as a fraction
in simplest form. (Skills Review, p. 658)
5
8
39. бЋЏбЋЏ П© бЋЏбЋЏ
450
1
1
2
38. Region 5
Polygons and Area
5
9
2
9
40. бЋЏбЋЏ ПЄ бЋЏбЋЏ
17
h
h
3
4
1
12
41. бЋЏбЋЏ П© бЋЏбЋЏ
4
7
1
5
42. бЋЏбЋЏ ПЄ бЋЏбЋЏ
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