You can also find the perimeter of figures that are graphed ... So, the area of the
figure is 11. 1_. 2 square units. ... an irregular figure that is drawn on grid paper.
CHAPTE R
11
Perimeter and Area The
BIG Idea
connectED.mcgraw-hill.com
How do I find the perimeter of polygons? How can I use the area formula for a rectangle to find the area of other figures?
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Perimeter s Rectangle
Make this Foldable to help you organize information about geometric figures.
asoning Logical Re
grams Parallelo Triangles PSS ds Trapezoi
Audio Foldables
Practice Self-Check Practice eGames
Review Vocabulary
rilateral with four right Rectangle rectángulo a quad ual and parallel angles; opposite sides are eq
Worksheets Assessment
Key Vocabulary English parallelogram trapezoid triangle 538
Español
paralelogramo trapecio triángulo
When Will I Use This?
Your Turn! You will solve thhiis teerrr. problem in the chap
Perimeter and Area 539
Are You Ready for the Chapter?
Text Option
You have two options for checking Prerequisite Skills for this chapter.
Take the Quick Check below.
Add. 1. 15 + 20 + 25 + 7
2. 9_ + 11 + 14_
3. 8_ + 12 + 12
4. 5 + 13 + 19
5. 16.3 + 16.3 + 16.3
6. 4 + 9.1 + 3.2 + 8
1 2
1 4
7. The amount of money that Tyrone spent shopping is shown in the table. Find the total amount that he spent.
1 2
Item
Amount ($)
CD
14.99
T-shirt
26.30
Snack
5.20
Multiply. 8. 10 × 26
9. 12 × 14
10. 75 × 2
11. 25 × 48
12. 25 × 6
13. 5 × 32
14. 132 × 13
15. 45 × 45
16. Mrs. Ohlin sold 3 handmade bookshelves for $160 each. How much money did she earn in all? Multiply. 17. 12 × 3 × 5
18. 8 × 6 × 4
19. 14 × 10 × 3
20. 15 × 9 × 6
21. 13 × 9 × 11
22. 12 × 7 × 14
Online Option 540
Perimeter and Area
Take the Online Readiness Quiz.
Multi-Part Lesson
1
PART
Perimeter A
B
Perimeter of Rectangles Main Idea I will use models to find the perimeters of rectangles.
Materials color tiles
Get ConnectED
You can use color tiles to create rectangles with different lengths and widths. Each color tile is 1 unit long and 1 unit wide. The rectangle shown below is 1 unit wide and 2 units long. 2 units long
⎫ ⎬ ⎭
GLE 0506.1.1 Use mathematical language, symbols, and definitions while developing mathematical reasoning.
If you walk along the perimeter of Central Park in New York City, you’ll walk around the outside, or border, of the park. The perimeter is found by adding the lengths of the sides.
⎫
1 unit wide ⎬
⎭
Use color tiles to create each rectangle. Then copy and complete the table. Rectangle
Length ()
Width (w)
2
2w
Perimeter (P)
2
1
4
2
2+1+2+1=6
Lesson 1A Perimeter 541
About It E
1.
WRITE MATH Refer to the table on the previous page.
How are and w related to the perimeter of the rectangles? Use P, , and w to write an equation for the perimeter of a rectangle.
2. Use the formula you wrote in Exercise 1 to find the distance around the rectangle. Select and use appropriate units.
8 in. 5 in.
3. In Exercise 2, only two sides of the rectangle are labeled. Explain why this is enough information to find the perimeter.
4. Find 2 + 2w for the rectangle in Exercise 2. Then write an equation to describe the relationship between P, , and w.
and Apply It Find the distance around each rectangle or square. 5.
6.
7.
8.
9.
10.
11. Mr. Heist wants to surround his kitchen island with a tile border. Each tile on the top is 1 foot by 1 foot. What is the distance around the kitchen island?
542
Perimeter and Area
Multi-Part Lesson
1
Perimeter
PART
A
Main Idea I will find perimeters of polygons.
Vocabulary V perimeter
Get ConnectED GLE 0506.4.4 Solve problems that require attention to both approximation and precision of measurement. SPI 0506.4.6 Record measurements in context to reasonable degree of accuracy using decimals and/or fractions.
B
Perimeter of Polygons Recall that a polygon is made up of line segments that do not cross each other. The rectangle at the right has opposite sides that are congruent. You can find the perimeter of a figure by finding the distance around the figure. Perimeter is a measure of length. The perimeter of the rectangle is 6 + 4 + 6 + 4 or 20 centimeters.
6 cm 4 cm
Find the Perimeter by Adding Side Lengths A fence is being built around part of the dog park shown. How much fencing is needed?
To find the distance around the park, find the perimeter. 6m
6m
9m
9m
10.5 m
Estimate 10 + 10 + 10 + 10 + 10 = 50 m P = 6 + 6 + 9 + 10.5 + 9 Add the lengths of the sides. = 40.5 So, 40.5 meters of fencing is needed. This is close to the estimate, so the answer is reasonable.
Lesson 1B Perimeter 543
Mini Activity Copy and complete the table. A square has all sides congruent. All angles are right angles. A rectangle has both pairs of opposite sides parallel and congruent. All angles are right angles.
1
Square
Side Length (s)
1
Perimeter (P)
4
2
3
4
Describe the relationship between the perimeter of a square and side length. Then write an equation using P and s.
Perimeter of a Square
Words
The perimeter P of a square is 4 times the side length s.
Symbols
P = s + s + s + s or 4s
Model s
Perimeter of a Square ART Hai and his uncle tiled the kitchen floor using square tiles like the one shown at the right. What is the perimeter of the tile? P = 4s
Perimeter of a square
P = 4(2) Replace s with 2. P=8
2 ft
Multiply.
The perimeter of the tile is 8 feet. You can always find the perimeter of a rectangle or square by adding the lengths of the four sides.
Perimeter of a Rectangle
Words
Symbols
544
Perimeter and Area
The perimeter P of a rectangle is two times the length plus two times the width w. w P = + + w + w or 2 + 2w
Model w
Perimeter of a Rectangle CRAFTS Christa is sewing a lace border around the edges of her scrapbook. How many inches of lace will Christa need?
7 in.
Find the perimeter of the scrapbook. P = 2 + 2w
9 in.
Perimeter of a rectangle
P = 2(7) + 2(9) = 7 and w = 9 P = 14 + 18
Multiply.
P = 32
Add.
So, Christa will need 32 inches of lace.
Find the perimeter of each figure. figure See Example 1 1.
11 in.
2.
30 mm
1
18 mm
1
9 2 in.
9 2 in.
16 mm
7 in.
Find the perimeter of each square or rectangle. See Examples 2, 3 3.
4. 5 in.
5.
5 in. 1
14 cm
7 2 ft 5 in.
5 in.
20 cm 12 ft
6. A rectangular playground is 32 feet long and 14 feet wide. How many feet of edging are needed to enclose the playground? 7.
E
TALK MATH Describe two ways to find the
perimeter of a rectangle.
Lesson 1B Perimeter 545
EXTRA
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Begins on page EP2.
Find i d the h perimeter i off each h fi figure. See Example 1 8.
9.
16 mm
9 mm 10 mm
11.9 cm
10 cm
10 ft
8 ft
11 mm
10.
10 cm
15 mm 12 mm 20 mm
16.5 cm 6 ft
Find the perimeter of each square or rectangle. See Examples 2, 3 11.
12.
13. 16 yd
17 in.
8 cm 1
313 yd 4 in.
14.
15.4 cm
8 cm
15.
16.
15.4 cm
1
18 2 ft 12 ft
8.7 cm
12 ft
8.7 cm 1
18 2 ft
17. An octagon-shaped table has two sides measuring 4 feet and the other sides each measuring 1 foot each. What is the perimeter of the table? 18. A billiards table is twice as long as it is wide. If the perimeter of a billiards table is 24 feet, what is the length and width of the table? 19. Use a centimeter ruler to measure the side lengths of the rectangle at the right. Select and use appropriate units to find the perimeter of the rectangle.
20. OPEN ENDED Use a ruler to draw two different rectangles that have the same perimeter. 21.
E
WRITE MATH Write a real-world problem that can be solved
by finding the perimeter. Then solve the problem.
546
Perimeter and Area
Perimeter on the Coordinate Plane 10 y 9 8 7 6 A 5 4 3 2 1 B
You can also find the perimeter of figures that are graphed on a coordinate plane. For example, rectangle ABCD has vertices at A(1, 5), B(1, 2), C(7, 2), and D(7, 5). The length of the rectangle is 6 units, and the width is 3 units. The perimeter of the rectangle is 3 + 6 + 3 + 6 or 18 units.
O
6 units
D 3 units C
x
1 2 3 4 5 6 7 8 9 10
Find the perimeter of each polygon. 22. 10 y
23. 10 y
9 8 7 6 5 D 4 3 2 1 F
O
G H
x
1 2 3 4 5 6 7 8 9 10
O
O
X
Y
x W Z 1 2 3 4 5 6 7 8 9 10
25. 10 y
24. 10 y 9 8 7 6 5 4 3 2 1
9 8 7 6 5 4 3 2 1
S
R
Q
P
x N O 1 2 3 4 5 6 7 8 9 10
9 8 7 6 5 G F 4 I H D C 3 2 1 x A B O 1 2 3 4 5 6 7 8 9 10
For each set of coordinates, graph the figure with the following vertices. Then find the perimeter. 26. D(1, 1), E(1, 5), F(4, 5), G(4, 1) 27. A(1, 2), B(3, 2), C(3, 4), D(4, 4), E(4, 6), F(1, 6) 28. R(1, 7), S(1, 5), T(4, 5), U(4, 2), V(5, 2), W(5, 5), X(6, 5), Y(6, 7) To assess partial mastery of SPI 0506.4.6, see your Tennessee Assessment Book.
547
Multi-Part Lesson
2
PART
Area of Rectangles and Squares A
Main Idea I will find and estimate the areas of figures by counting squares.
B
C
D
Area Area is the number of square units that cover the surface of a closed figure.
Vocabulary V area
Get ConnectED GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures.
1 square unit
2 square units
4 square units
Find Area GAMES Find the area of a checkerboard. Count the number of square units. There are 64 squares. So, the area of a checkerboard is 64 square units.
If the figure is not a square or a rectangle, count the number of whole squares and the number of half squares.
Find Area Find the area of the figure. Step 1
Count the number of whole squares in the figure. 9 whole squares = 9 square units
Step 2
Count the number of half squares in the figure. 1 5 half squares = 2_ square units 2
Step 3
Add the number of whole and half squares. 1 1 9 square units + 2_ square units = 11_ square units 2
1 So, the area of the figure is 11_ square units. 2
548
Perimeter and Area
2
When you cannot count square units or half square units exactly, you can estimate the area.
of Some common units area are square inch, square foot, square yard, square mile, square centimeter, and square meter.
Estimate Area TREE HOUSES The diagram shows the floor plan for a tree house. One square on the grid represents 1 square foot. About how many square feet is the area of the floor?
1 1 2 3 4 5 6 10 9 2 7 8 9 1011 12 13 8 4 3 14 15161718192021 5 2223242526272829 7 30 31 32 33 34 35363738 6
Step 1 Count the number of whole squares in the diagram. 38 whole squares = 38 square feet Step 2
Count the partial squares circled in the diagram. 10 partial squares is about 5 square feet
Step 3
Add the number of whole and partial squares. 38 + 5 = 43 square feet
The tree house floor has an area of about 43 square feet.
LANDSCAPING A landscape 1 2 3 4 architect designed the pond at 26 1 2 3 4 5 25 5 6 7 8 6 the right. Each square represents 222324 9 1011 12 13 7 1 square meter. Estimate the 2114 15161718192021 8 area of the pond. 202223242526272829 9
s The areas of the figure on the previous page are exact. The areas of the figures in Examples 3 and 4 are estimates.
Step 1
Count the number of whole squares. There are 45 whole squares, which is 45 square meters.
Step 2
Count the partial squares.
193031 32 33 34 3510 18363738394011 17 41 42 434412 1645 15 14 13
26 partial squares is about 13 square meters. Step 3
Add the whole squares and partial squares. 45 + 13 = 58 square meters
The area of the pond is about 58 square meters.
Lesson 2A Area of Rectangles and Squares
549
Estimate the area of each figure figure. Each square represents 1 square centimeter. See Examples 1–4 1.
2.
3.
4. A cake decorator is drawing a heart on a cake. Each square represents 1 square inch. Estimate the area of the heart. 5.
E
TALK MATH Describe one way to estimate the area of
an irregular figure that is drawn on grid paper.
EXTRA
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Begins on page EP2.
Estimate i the h area off each h figure. fi Each h square represents 1 square centimeter. See Examples 1–4 6.
7.
8.
9.
10.
11.
12. Isaiah made a sign for his yard sale shown at the right. If each square represents 1 square inch, estimate the area of the sign.
550
Perimeter and Area
YARD SALE
13. The flower patch at the right is on Cindy’s backpack. One square represents 1 square centimeter. Estimate the total area of the patch.
14. OPEN ENDED Draw a figure on grid paper with an area of about 38 square units. 15. REASONING When estimating area, explain why a smaller grid size gives a more accurate estimate than a larger grid size. 16.
E
WRITE MATH Describe some real-life examples of when
it would be useful to know how to estimate the area of figures.
Test Practice 17.
SHORT RESPONSE Student Council is making a wooden sign 8 feet long and 3 feet wide. They want to paint the front of the sign red. To determine how much paint they will need, should they find the perimeter or area?
18. Which is the best estimate for the area of the figure? A. 12 square units B. 15 square units C. 18 square units D. 24 square units
Find the perimeter of each figure. (Lesson 1B) 19.
20.
11 ft 3 ft
6.5 m
7 ft 11 ft
13 m
21.
8 in.
8 in.
8 in.
8 in 8 in.
Lesson 2A Area of Rectangles and Squares
551
Multi-Part Lesson
2
PART
Area of Rectangles and Squares A
Main Idea I will explore finding the areas of rectangles and squares using models.
B
C
D
E
Area of Rectangles and Squares You can use color tiles to find area. Each color tile represents 1 square unit.
Materials color tiles
T Miller family wants to build a walkway that is 6 feet The llong and 4 feet wide. Determine the walkway’s area.
Step 1
Represent the walkway using tiles. Each tile represents one square foot.
Get ConnectED GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures.
Step 2
Count the number of tiles. The area of the walkway is 24 square feet.
Find the area of a square with a side length of 5 units. F
5 units
Step 1
Step 2
Represent the square using color tiles.
Count the number of tiles. The area is 25 square units.
552
Perimeter and Area
You can use the length and width of a rectangle to find area. The length of a rectangle is the number of units it is long. The width is the number of units it is wide.
Determine a Formula Copy and complete the table below. Use color tiles to create and measure the rectangles shown.
Rectangle length ()
3
width (w)
1
Area (A)
3
About It 1. Study the pattern in the table in Activity 3. How are the length and width of each rectangle related to its area? 2. Use A, and w to write a formula for the area of a rectangle. 3. Give the length and width of three different rectangles that each have an area of 24 square units.
and Apply It Find the area of each shape without using models. 5.
4.
6. 30 ft
9 in.
16 cm
9 in.
50 ft
12 cm
7. A parking space in front of the library is 14 feet long and 9 feet wide. Determine the area of the parking space. 8.
E
WRITE MATH Suppose s represents the length of a side
of a square. Write a formula you can use to find the area of the square. Justify your formula.
Lesson 2B Area of Rectangles and Squares
553
Multi-Part Lesson
2
PART
Area of Rectangles and Squares A
Main Idea I will find the area of rectangles and squares.
Get ConnectED
B
C
D
E
Area of Rectangles and Squares You can use multiplication to find the area of rectangles and squares.
GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures.
Area of a Rectangle
Words
To find the area of a rectangle, multiply the length by the width.
Symbols
FLAGS The American Flag shown has a length of 114 feet and a width of 65 feet. What is the area of the flag?
65 ft
114 ft Formula for area of a rectangle
A = 114 × 65 Replace with 114 and w with 65.
A = 7,410
Multiply.
The area of the flag is 7,410 square feet.
554
Perimeter and Area
w
A = w
A = w
Model
Recall that a square is a rectangle with four congruent sides. Each side length is represented by s. So, you can replace and w in the formula A = w with s. Area of a Square
Words
To find the area of a square, multiply the length of one side by itself.
s2
is The expression d read s square because its model forms a square with
Model s
A = s × s or A = s2
Symbols
side s.
Area of a Square SPORTS A baseball diamond is actually a square. Find the area of the baseball diamond. A = s2
Formula for area of a square
A = 90 × 90
Replace s with 90.
A = 8,100
Multiply.
äCQ
The area of the baseball diamond is 8,100 square feet.
Find the area of each rectangle or square. square See Examples 1 and 2 1.
2.
3.
13 yd
15 in
17 m
3m
22 yd
15 in.
4. = 9 km, w = 1 km
5. = 8 cm, w = 6 cm
6. The Parthenon of ancient Greece had the rectangular floor plan shown. How much area does the building cover? 7.
E
TALK MATH Write the formulas for the area
69 m
31 m
of a rectangle and a square. Explain what each variable represents.
Lesson 2C Area of Rectangles and Squares
555
EXTRA
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Begins on page EP2.
Find Fi d th the area off each h rectangle t l or square. See Examples l 1 and d2 8.
9.
10. 88 ft
5 mi
11 mm 50 ft
7 mi
11 mm
11. = 18 m
12. = 24 m
w=5m
13. = 12 in.
w = 37 m
w = 10 in.
14. Use a centimeter ruler to draw two different rectangles and one square that each have an area of 16 square centimeters. 3 cm
15. Find the area of the figure at the right by dividing it into a square and a rectangle.
2 cm 3 cm
2 cm
16. A square has an area of 196 square inches. What is the side length?
1 cm
17. A soccer field has to be 100 to 130 yards long and 50 to 100 yards wide. Find the least and greatest areas for the soccer field.
5 cm
18. The door of a new building measures 7 feet by 3 feet. It is to be covered with 12-inch square metal tiles that cost $15 each. How much will it cost to cover the door? Explain.
All of the license plates in the United States may have a different design, but they all come in one standard size. License Plate Size Customary Units
Metric Units
12 in. × 6 in.
30 cm × 15 cm
Find the area of the license plate using each type of unit. 19. square inches
556
Perimeter and Area
20. square centimeters
21. OPEN ENDED Give the dimensions of a rectangle whose area is between 100 and 200 square centimeters. Find the area. 22. CHALLENGE Suppose you double the length and the width of a rectangle. Would the area also double? Explain. 23.
E
WRITE MATH Write about a real-life situation that can
be solved by finding the area of a rectangle. Then solve.
Test Practice 24. Jamie wants to make a banner with a length of 7 feet and a width of 2 feet. How many square feet of paper will he need?
25. What is the area of the figure? 27 ft 11 ft 21 ft
A. 24 ft2 B. 22
10 ft
ft2
38 ft
C. 19 ft2 D. 14 ft2
F. 297 ft2
H. 798 ft2
G. 677 ft2
I. 1,026 ft2
Estimate the area of each figure. Each square represents 1 square centimeter. (Lesson 2A) 26.
27.
28. Kaden wants to place a string of lights around the roof of his gazebo shaped like the octagon shown. Each side is 8 feet long. The lights come in lengths of 20 feet. How many strings of lights will he need? (Lesson 1B)
Lesson 2C Area of Rectangles and Squares
557
Multi-Part Lesson
2
Area of Rectangles and Squares
PART
A
B
C
D
Problem-Solving Strategy:
Logical Reasoning
Main Idea I will solve problems by using logical reasoning. Jorge, Adam, Iesha, and Nicole each bought different sized birthday cards. The areas of the cards are 21 square inches, 24 square inches, 40 square inches, and 45 square inches. Use the clues to determine which person bought each card. 1. Jorge’s card is larger than Adam’s. 2. Iesha’s card has a length of 9 inches and a width of 5 inches. 3. The card that has the least area was purchased by a girl.
Understand What facts do you know? • The clues that are listed above. What do you need to find? • Which person has each card.
Plan
You can use logical reasoning to find which person has each card. Make a table to help organize the information.
Solve
Place an “×” in each box that cannot be true. • Clue 2 shows that Iesha’s card has an area of 45 in2.
A = 21 in2 A = 24 in2 A = 40 in2 A = 45 in2
Jorge Adam Iesha Nicole
× × × yes
× yes × ×
yes × × ×
× × yes ×
• Clue 3 shows that Adam’s card could not have an area of 21 in2. • Clue 1 shows that Jorge’s card could not be the smallest. So, Nicole’s card has an area of 21 square inches, Adam’s card has an area of 24 square inches, Jorge’s card has an area of 40 square inches, and Iesha’s card has an area of 45 square inches.
Check
Since all of the answers match the clues, the solution is reasonable.
GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.
558
Perimeter and Area
Refer to the problem on the previous page. 1. If you did not know that a girl had the smallest card, would it be possible to determine who had each card? Explain your reasoning.
3. The area of a garden is 16 square feet. If the length and width are whole numbers, is the garden definitely a square? Explain.
2. Suppose Adam’s card is larger than Jorge’s. Who has which card?
4. Explain when to use the logical reasoning strategy to solve a problem. EXTRA
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Begins on page EP2.
Solve the problem. Use logical reasoning. 5. There is a red, a green, and a yellow bulletin board hanging in the hallway. All of the bulletin boards are rectangular with a height of 4 feet. Their lengths are 6 feet, 5 feet, and 3 feet. The red bulletin board is the largest and the yellow one is the smallest. What is the area of the green bulletin board? 6. Algebra If the pattern below continues, how many pennies will be in the fifth figure?
Figure 1 Figure 2
9. Ethan has $1.25 in change. He has twice as many dimes as pennies, and the number of nickels is one less than the number of pennies. How many dimes, nickels, and pennies does he have? 10. There are 4 more girls in Mrs. Pitt’s class than Mr. Brown’s class. Five girls moved from Mrs. Pitt’s class to Mr. Brown’s class. Now there are twice as many girls in Mr. Brown’s class as there are in Mrs. Pitt’s. How many girls were in Mr. Brown’s class before the move? 11. Geometry Set up 12 toothpicks as shown below. Move three toothpicks so that you form four squares.
Figure 3
7. A cafeteria table has an area of 21 square feet. If three tables are pushed together, what is the combined area of the tables? 8. Three dogs are sitting in a line. Rocky is not last. Coco is in front of the tallest dog. Marley is sitting directly behind Rocky. List the dogs in order from first to last.
12.
E
WRITE MATH How did you use
logical reasoning to determine which dog was first in line in Exercise 8?
Lesson 2D Area of Rectangles and Squares
559
Mid-Chapter Check Find the perimeter of each figure. (Lesson 1B) 17 in.
1.
Estimate the area of each figure. Each square represents 1 square foot. (Lesson 2A) 6.
7.
22 in.
2. 5 cm
13 cm 12 cm
8. Mr. Kelly has a garden shaped like the figure below. Each square represents 1 square foot. Estimate the total area of the garden.
3. A regular pentagon has side lengths of 11 yards. What is the perimeter of the figure? (Lesson 1B) 4. MULTIPLE CHOICE The perimeter of an equilateral triangle is 36 millimeters. What is the length of each side? (Lesson 1B) A. 9 millimeters B. 12 millimeters
Find the area of each rectangle or square. (Lesson 2C)
9.
10. 3 cm
C. 18 millimeters D. 24 millimeters 5. An outdoor ice skating rink is surrounded by a fence as shown. The fence needs to be replaced. How much new fencing is needed? (Lesson 1B)
10 ft
3 cm 8 ft
11. MULTIPLE CHOICE A square has a length of 20 inches. What is the area? (Lesson 2C) F. 40 square inches G. 80 square inches H. 200 square inches I. 12.
E
400 square inches WRITE MATH Describe how to
estimate the area of the figure in Exercise 6. (Lesson 2A)
560
Perimeter and Area
Multi-Part Lesson
3
PART
Area of Parallelograms A
B
C
D
E
Area of Parallelograms Main Idea I will explore using models to find the area of parallelograms.
In Lesson 2C, you found the areas of rectangles and squares. Another kind of quadrilateral (4-sided figure) is a parallelogram. Parallelograms
Not Parallelograms
Materials grid paper
Finding the area of a parallelogram is related to finding the area of a rectangle.
Get ConnectED GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. SPI 0506.4.1 Solve contextual problems that require calculating the area of triangles and parallelograms. Also addresses GLE 0506.1.3, GLE 0506.1.4.
Step 1
Cut out a rectangle using grid paper like the one shown below.
Step 2
Cut a triangle from one side of the rectangle and move it to the other side to form a parallelogram.
Step 3
Repeat Steps 1 and 2 with two other rectangles of different dimensions.
Step 4
Copy and complete the table below using the three rectangles and three related parallelograms you created. Length (l ) Rectangle 1 Rectangle 2 Rectangle 3
Base (b)
Width (w)
Height (h)
Parallelogram 1 Parallelogram 2 Parallelogram 3
Lesson 3A Area of Parallelograms 561
Find the area of the parallelogram below.
Step 1
Draw the parallelogram on grid paper and cut it out.
Step 2
Fold and cut along the dotted line.
Step 3
Move the triangle to the right to make a square.
Step 4
Count the number of units in the square.
The area is 81 square units.
About It 1. How is the area of each parallelogram related to the area of its corresponding rectangle? 2. What part of the parallelogram corresponds to the length of the rectangle? 3. What part corresponds to the rectangle’s width? 4. What do you think is the formula for the area of a parallelogram?
and Apply It Find the area of each parallelogram. 6.
5.
7. On grid paper, draw three different parallelograms with a base of 6 units and a height of 4 units. Compare the areas. 8.
E
WRITE MATH How is the formula for the area of a parallelogram
found from the area of a rectangle?
562
Perimeter and Area
Multi-Part Lesson
3
PART
Area of Parallelograms A
Main Idea I will find the area of parallelograms.
Vocabulary V base height
B
D
E
Area of Parallelograms Remember that a parallelogram is a four-sided figure in which each pair of opposite sides are parallel and congruent. The base of a parallelogram can be any one of its sides.
height
Get ConnectED GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. SPI 0506.4.1 Solve contextual problems that require calculating the area of triangles and parallelograms.
base
The shortest distance from the base to the opposite side is the height of the parallelogram.
To find the area of a parallelogram, multiply the measures of the base and the height.
Area of a Parallelogram
Model Words
Symbols
To find the area of a parallelogram, multiply the base and the height.
b h
A = bh
LANDSCAPING A landscape architect is designing the flower garden shown. What is the area of the flower garden? 9 ft
Since the design of the flower garden is a parallelogram, use the formula A = bh. A = bh
Area of a parallelogram
A = (6)(9)
Replace b with 6 and h with 9.
A = 54
Multiply.
6 ft
The area of the flower garden is 54 square feet.
Lesson 3B Area of Parallelograms 563
Area of a Parallelogram Find the area of the parallelogram. t An area measuremen can be written using abbreviations and an exponent of 2. For example: 2 square units = units 2 square inches = in 2 square feet = ft 2 square meters = m
15 in. 10 in.
The base is 15 inches, and the height is 10 inches.
A = bh
Area of a parallelogram
A = 15 × 10
Replace b with 15 and h with 10.
A = 150
Multiply.
The area is 150 square inches or 150 in 2.
Find the area of each parallelogram parallelogram. See Examples 1 and 2 1.
2.
3. 6m
10 ft 8m
5 ft
4.
5.
12 m
6. 15 in.
4m 12 in.
7. Find the area of a parallelogram that is 52 feet wide and 63 feet high. 8. What is a reasonable estimate for the area of a parallelogram with 3 1 a base of 18_ inches and a height of 16_ inches? 4
9.
E
8
TALK MATH Suppose both the base and height of the
parallelogram in Exercise 2 are doubled. Explain how the area will change.
564
Perimeter and Area
EXTRA
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Begins on page EP2.
Find Fi d the th area off each h parallelogram. ll l See Examples 1 and 2 10.
11.
12. 8 ft 7 ft 15 ft 5 ft
14.
13.
15. 12 mm
7 in.
36 mm
8 ft 9 in.
4 ft
16. Find the area of a parallelogram with a base of 75 meters and a height of 4 meters. 17. Evan made a parallelogram-shaped picture frame to display a piece of his artwork. The base of the frame is 14 inches and the height is 9 inches. Find the area enclosed by the frame.
18. OPEN ENDED Draw and label two different parallelograms each with area of 16 square units. 19. FIND THE ERROR Sonia is finding the area of the parallelogram below. Find her mistake and correct it.
5 ft
4 ft
The base is 8 and the height is 5. A = 40 ft 2
8 ft
20.
E
WRITE MATH Explain how to find the height of a parallelogram
if its area is 216 square meters and its base is 12 meters.
Lesson 3B Area of Parallelograms 565
Multi-Part Lesson
4
PART
Area of Triangles and Trapezoids A
B
C
D
E
Area of Triangles Main Idea I will find the area of triangles using models.
How can you use the area of a related rectangle to find the area of a triangle?
Materials 4 × 6 index cards
Step 1 S
Find the area of the index card.
4 in. grid paper
6 in.
Step 2
Draw a diagonal line across the index card from one corner to another. Then cut across the line.
scissors
4 in. 6 in.
Get ConnectED
Step 3
Find the area of one of the remaining triangles.
GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. SPI 0506.4.1 Solve contextual problems that require calculating the area of triangles and parallelograms. Also addresses GLE 0506.1.3, GLE 0506.1.4.
The triangle is exactly half the size of the related rectangle. So, the area of the rectangle can be divided by 2 to find the area of one triangle. The area is 24 ÷ 2 or 12 square inches.
566
Perimeter and Area
You can also find the area of a triangle from the area of a related parallelogram.
Step 1
Copy the table shown. Parallelogram A B C D E
Base, b 4 2 3 6 8
Area of Height, Parallelogram h 6 5 4 3 5
Area of Each Triangle
Step 2
Draw Parallelogram A on grid paper using the dimensions given in the table.
Step 3
Draw a diagonal as shown by the red dashed line.
Step 4
Cut out the parallelogram. Then find its area. Record this measure in the table.
Step 5
Cut along the diagonal to form two triangles. Then find the area of one triangle. Record it in the table.
About It Refer to Activity 2 for Exercises 1–6. 1. Repeat Steps 2 through 5 for Parallelograms B through E. Find the area of each triangle formed and record your results in the table. 2. Compare the base and height of each triangle to the base and height of the original parallelogram. What do you notice? 3. Compare the two triangles formed. How are they related? 4. Compare the area of each triangle to the area of its related parallelogram. 5. What patterns do you notice in the columns of the table? 6.
E
WRITE MATH Write a formula that relates the area A of a
triangle to the length of its base b and height h.
Lesson 4A Area of Triangles and Trapezoids 567
Multi-Part Lesson
4
Area of Triangles and Trapezoids
PART
A
Main Idea I will find the areas of triangles.
Get ConnectED GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. SPI 0506.4.1 Solve contextual problems that require calculating the area of triangles and parallelograms. SPI 0506.4.2 Decompose irregular shapes to find perimeter and area. Also addresses GLE 0506.1.7.
B
C
D
E
Area of Triangles In Lesson 4A, you learned that the area of a triangle is related to the area of a rectangle and parallelogram. To find the area of a triangle, multiply the measures of the base and the height and divide by two. You can also find half of the base, then multiply by the height. The base of a triangle can be any one of its sides. The height is the shortest distance from a base to the opposite vertex.
height (h) base (b)
Area of a Triangle
Words
Symbols
The area of a triangle A is one-half the product of the base b and its height h.
Model
bh 1 A = _ or A = _ bh 2
b
2
FLAGS What is the area of the triangle on the flag?
4 in.
1 A = _ bh
Area of a triangle
1 A = _ (6)(4)
Replace b with 6 and h with 4.
A = (3)(4)
1 _ of 6 is 3.
A = 12
Multiply.
2
2
h
2
The area of the triangle on the flag is 12 square inches.
568
Perimeter and Area
6 in.
Find the Area of a Triangle Find the area of the triangle. Multiply 2-digit numbers. 34 12 × −−−− 68 0 34 + −−−− 408
34 ft
12 ft
Place a zero.
bh A=_
2 34 × 12 A=_ 2 408 A=_ 2
A = 204
Area of a triangle. Replace b with 34 and h with 12. Multiply 34 and 12. Divide by 2.
The area of the triangle is 204 square feet.
Find the area of each triangle. triangle See Examples 1 and 2 1.
2.
3. 10 in.
9 in.
4.
5. 8 ft 7 ft
6. 3m 6m
16 cm 24 cm
7. The side of a triangular boat sail measures 9 feet and the height measures 6 feet. How much fabric was used to make the sail? 8.
E
TALK MATH Explain how two different triangles can have an
area of 24 square feet.
Lesson 4B Area of Triangles and Trapezoids 569
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Find i d the h area off each h triangle. i l See Examples 1 and 2 9.
10.
11.
12 in.
8 in.
12.
2 ft
13.
14. 8 mm
13 ft
3 mm
13 cm
16 cm
15. base = 13 ft height = 6 ft area =
16. base = 22 cm height = 7 cm area =
Use the figure at the right. 17. Find the area of the largest triangle.
24 m
18. Find the area of the green triangle. 19. What is the area of one small triangle? 30 m
Use the information to solve the problem.
20. What are the areas of the triangles and the rectangle on Desiree’s sketch?
570
Perimeter and Area
21. CHALLENGE The area of a triangle is 16 square feet, and its height is 4 feet. What is the length of the base? 22. WHICH ONE DOESN’T BELONG? Which figure does not belong with the other two? Explain your reasoning. A.
B.
C. 3 ft
3 ft
10 ft
6 ft 4 ft 2 ft
23.
E
WRITE MATH Write about a real-world problem that could be
solved by finding the area of a triangle.
Test Practice 25. Which formula will give you the area of a triangle?
24. Find the area of the triangle. 3 units
F. A = b × h 1 G. A = _ bh
18 units
A. 27 units
2
B. 36 units 2
C. 40 units
2
2
H. A = (b × h) × 2
D. 54 units 2
I. A = (b × b) ÷ 2
Area of Regular Polygons The regular pentagon at the right was created using five congruent triangles. To find the area of the pentagon, find the area of one triangle and multiply by 5. 6×4 The area of one triangle is _ or 12 square inches. So, the area 2
8 in. 4
6 in.
of the regular pentagon is 12 × 5 or 60 square inches. Find the area of each regular polygon. 26.
27. 9 cm 10 cm
5 ft 4 ft
Lesson 4B Area of Triangles and Trapezoids 571
Multi-Part Lesson
4
PART
Area of Triangles and Trapezoids A
B
C
D
E
Problem-Solving Investigation Main Idea I will choose the best strategy to solve a problem.
AMANDA: To make a quilt pattern, I pieced together triangles to make squares of different sizes. The first square has 2 triangles, the second square has 8 triangles, and the third square has 18 triangles. The quilt will have squares of five different sizes.
YOUR MISSION: Find how many triangles are in the fifth square.
Understand You know how many triangles are in the first, second, and third squares. You need to find how many triangles are in the fifth square.
Plan Solve
Look for a pattern to find the number of triangles. Each square has twice as many triangles as small squares. First square Second square Third square
2 × 1 or 2 triangles 2 × 4 or 8 triangles 2 × 9 or 18 triangles
Continuing the pattern, the fourth square has 2 × 16 or 32 triangles. The fifth square has 2 × 25 or 50 triangles.
Check
Draw the fifth square and count the number of triangles. Since there are 50 triangles in the fifth square, the answer is correct.
GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.
572
Perimeter and Area
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Begins on page EP2.
• • • •
Draw a diagram. Look for a pattern. Use logical reasoning. Choose an operation.
7. A family has four cats. Fluffy is 8 years old and is 4 years younger than Tiger. Tiger is 2 years older than Max, and Max is 3 years older than Patches. List the cats from oldest to youngest.
Use any strategy shown to solve each problem. 1. Mr. Toshi’s fifth grade class sold containers of popcorn and peanuts. If each day they sold 25 fewer containers of peanuts than popcorn, how many containers of popcorn and peanuts did they sell in all? Day 1 Day 2 Day 3 Day 4 Popcorn Peanuts
225
200
150
300
2. Algebra Find the fifteenth term in the pattern shown below. 5, 4, 7, 6, 9, 8, 11, . . . 3. Nigel is drawing a pattern with parallelograms. He drew the first parallelogram with an area of 12 square inches, the second with an area of 20 square inches, and the third with an area of 28 square inches. If the pattern continues, what is the area of the fifth parallelogram? 4. BAR DIAGRAM T here are 8 girls for every 7 boys on a field trip. If there are 56 girls on the trip, how many students are on the trip? 5. Measurement When Cheryl goes mountain climbing, she rests 5 minutes for every 15 minutes that she climbs. If Cheryl climbs for 2 hours, how many minutes does she rest? 5 a 6. The fraction _ is equivalent to _, and b
20
b - a = 3. Find the values of a and b.
8. BAR DIAGRAM T he number of fifth grade students who helped clean the park this year was 5 less than twice as many as last year. If 39 fifth graders helped clean up the park this year, how many cleaned the park last year? 9. Five friends go to a batting cage. Andrea bats after Daniel and before Jessica. Juwan bats after Andrea and before Jessica and Filipe. Jessica always bats immediately after Juwan. Who bats last? 10. Madeline has 2 times the number of games as Paulo. Paulo has 4 more games than Tyler. If Tyler has 9 games, how many games are there among the 3 friends? 11. Algebra The first three triangular numbers are shown below. How many dots will be in the sixth triangular number?
1
12.
E
3
6
WRITE MATH In addition to logical
reasoning, what is another strategy that you could use to solve Exercise 11?
Lesson 4C Area of Triangles and Trapezoids 573
Multi-Part Lesson
4
Area of Triangles and Trapezoids
PART
A
B
D
C
E
Area of Trapezoids Main Idea
Another kind of quadrilateral is a trapezoid.
I will find the area of trapezoids using models.
Trapezoids
Not Trapezoids
Materials grid paper
How can you use the area of a parallelogram to find the area of a trapezoid? scissors
Step 1 S Get ConnectED
Draw the trapezoid shown below on grid paper. Label the height as h, and label the bases as b 1 and b 2.
GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures. Also addresses GLE 0506.1.3, GLE 0506.1.4.
A trapezoid has two bases, b 1 and b 2. The height (h) of a trapezoid is the distance between the bases.
b1 h b2
Step 2
Draw and label another trapezoid that is identical to the one in Step 1.
Step 3
Turn one trapezoid upside down. Tape it to the other trapezoid as shown. b2
b1 h b2
574
Perimeter and Area
b1
h
About It 1. What figure is formed by the two trapezoids? 2. Write an expression to represent the base of the figure. 3. Write a formula for the area A of the parallelogram using b 1 and b 2 and h. Then, find the area of the parallelogram. 4. How does the area of each trapezoid compare to the area of the parallelogram? 5. Find the area of the trapezoid in the Activity.
and Apply It Find the area of each trapezoid. 7.
6.
6 units
8.
9.
8 units
7 units
14 units
12 units
10.
3 units
8 units
E
11.
10 units
9 units
7 units
12.
11 units
6 units
WRITE MATH Write a formula for the area A of a trapezoid
with bases b 1 and b 2, and height h.
Lesson 4D Area of Triangles and Trapezoids 575
Multi-Part Lesson
4
Area of Triangles and Trapezoids
PART
A
Main Idea I will find the areas of trapezoids.
Get ConnectED GLE 0506.4.1 Use basic formulas and visualization to find the area of geometric figures.
B
C
E
D
Area of Trapezoids You can find the area of a trapezoid using a formula. Remember, a trapezoid is a quadrilateral with exactly one pair of parallel sides. Area of a Trapezoid
Words
Symbols
The area A of a trapezoid equals the product of the height h and the sum of the bases b1 and b2 divided by two. (b1 + b2)h A=_
Model b1 h b2
2
KITES Paida is flying a kite like the one shown. The kite is in the shape of a trapezoid. What is the area of the kite? 6 ft
3 ft
4 ft
(b1 + b2)h A=_ 2 (4 + 6)3 _ A= 2 10 × 3 A=_ 2 30 A=_ 2
A = 15
Area of a trapezoid Replace h with 3, b1 with 4, and b2 with 6. Add 4 and 6. Multiply 10 and 3. Divide by 2.
So, the area of the kite is 15 square feet.
576
Perimeter and Area
Use a Formula Find the area of the trapezoid. (b1 + b2)h A=_
Area of a trapezoid
(5 + 7)3 A=_ 2
Replace h with 3, b1 with 5, and b2 with 7.
12 × 3 A=_
Add 5 and 7.
36 A=_
Multiply 12 and 3.
A = 18
Divide by 2.
2
You may need to use the lengths of other sides to determine the length of a missing measurement.
2
2
5 3 7
So, the area of the trapezoid is 18 square units.
Find the area of each trapezoid. trapezoid See Examples 1 and 2 1.
12 m
2.
11 cm
3.
8 cm
8m
17 cm
15 m
4.
12 ft
14 mm
5.
3 ft
6. 14 ft
16 mm
12 ft
4 ft
12 mm
5 ft
10 mm 6 ft
7. A trapezoid has bases of 15 meters and 18 meters, and a height of 10 meters. What is the area of the trapezoid?
8.
E
TALK MATH Could you find the area of the trapezoid in Exercise 2
by finding the area of a triangle and the area of a rectangle? Explain.
Lesson 4E Area of Triangles and Trapezoids 577
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Begins on page EP2.
Find Fi d th the area off each h ttrapezoid. id See Examples 1 and 2 9.
10.
12 ft
11.
5 ft
5 cm
17 m
30 cm
13 m 8 ft 20 cm 11 m 11 ft
12.
13.
14. 6 ft
19 cm
6 ft
15 cm 16 ft
25 ft 13 cm
4 ft
11 cm 15 ft
11 ft
Find the area of each trapezoid given the dimensions. 15. b1 = 12 in. b2 = 24 in. h = 5 in.
16. b1 = 8 ft b2 = 20 ft h = 9 ft
17. b1 = 4 m b2 = 6 m h = 32 m
18. The surface of Marlo’s desk is a trapezoid. What is the area of his desk if the bases of the trapezoid are 26 inches and 74 inches, and the height is 30 inches? 19. The blueprints for a back patio are shown. If the cost of the patio is $4 per square foot, what will be the total cost of the patio?
20. OPEN ENDED Draw and label two trapezoids that each have an area of 100 square inches. 21. CHALLENGE Determine the area of the figure using a combination of area formulas. 22.
E
WRITE MATH Explain how the order of operations
4 ft
6 ft 5 ft
7 ft
helps you find the area of trapezoids.
578
To assess mastery of SPI 0506.4.1 and SPI 0506.4.2, see your Tennessee Assessment Book.
What’s the Area? Measuring Areas Get Ready! Players: 2 to 4
Get Set! Shuffle and turn the cards facedown in a pile. Give paper and pencil to each player.
You will need: 15 cards with sketches of different-sized shapes, centimeter ruler, paper, pencils Each player compares the actual area to their estimate. The player with the closest estimate earns 1 point. Continue playing until all the cards are used. The player with the most points wins.
Go! Turn the first card face up. Each player estimates the area of the figure in square centimeters and records that number. Players work together to measure the figure and calculate the area. Select and use appropriate units and formulas.
Game Time What’s the Area? 579
Pompeii, an Ancient Roman City, had houses that were designed in classic Roman style. This style was characterized by using a variety of geometrical shapes and patterns. Many of the houses included murals painted on the walls, decorative fountains, and patterned mosaic floors.
The city of Pompeii was accidentally rediscovered by an Italian architect named Fontana in 1599.
580
Perimeter and Area
In the year 79, nearby Mount Vesuvius erupted violently, spewing lava and ash throughout Pompeii. For 1,600 years, the city was lost under Mount Vesuvius’ ashes. Today, scientists continue to uncover buildings and artworks at the site.
Use the image above of the design on a Pompeii building to solve each problem. Use a metric ruler for Exercises 5–8. 1. Where do you see rectangles in the design? 2. Where do you see triangles in the design? 3. Where do you see parallelograms in the design? 4. Are there any trapezoids in the design? If so, where? 5. Measure the rectangle in the photograph to the nearest millimeter. What is the estimated area of the rectangle?
6. Measure the triangle in the photograph to the nearest millimeter. What is the estimated area of the triangle? 7. Measure the parallelogram in the photograph to the nearest millimeter. What is the estimated area of the parallelogram? 8. How can you use the formula for the area of a trapezoid to determine the area of the hexagon found in the center of the design?
Problem Solving in Art 581
Chapter Study Guide and Review Be sure the following Key Concepts are noted in your Foldable.
Vocabulary area base height
Perimeter Rectangles
perimeter
Logical Reasoning
Parallelograms Triangles PSS Trapezoids
Vocabulary Check Key Concepts
Choose the correct word or number that completes each sentence.
Area of Rectangles (Lesson 2) • The area A of a rectangle is the length () times the width (w).
1. The formula for the area of a (triangle, square) is A = s2.
Area of Parallelograms (Lesson 3)
2. (Area, Perimeter) is the number of square units needed to cover a surface.
• The area A of a parallelogram is the base b times the height h.
3. The formula for the area of a (trapezoid, parallelogram) is
Area of Triangles (Lesson 4) • To find the area A of a triangle, multiply the base b times the height h. Then, divide by 2. Or, find half the base and multiply by the height. Area of Trapezoids (Lesson 4) • The area A of a trapezoid equals the product of the height h and the sum of the bases b1 and b2, divided by two.
582
Perimeter and Area
(b1 + b2)h A = _. 2
4. To find the (perimeter, area) of a square, you can use the formula 4s.
b1
5. You can form a (triangle, parallelogram) by putting two trapezoids together.
b2
6. The formula for the area of a (rectangle, triangle) is A = bh.
h
Multi-Part Lesson Review Lesson 1
Perimeter
Perimeter of Polygons
(Lesson 1B)
Find the perimeter of each figure.
Find the perimeter of the rectangle.
8.
7.
EXAMPLE 1
31 cm 15 yd
18 yd
9 in. 31 cm
10 yd
9. A garden in the shape of a square is 15 feet on each side. What is the perimeter?
16 in.
P = 2 + 2w
Perimeter of rectangle
P = 2(16) + 2(9)
= 16, w = 9
P = 50 in.
Simplify.
The perimeter is 50 inches. Lesson 2
Area
Area of Rectangles and Squares (Lesson 2A)
Estimate the area of each figure. Each square represents 1 square centimeter. 10.
EXAMPLE 2
Estimate the area of the figure.
11.
12. Marco wants to estimate the area of his pool. He drew a sketch on grid paper. Each square represents 1 square foot. Estimate the area of his pool.
7 whole squares = 7 square centimeters 11 partial squares is about 6 square centimeters. The area is about 7 + 6 = 13 square centimeters.
Chapter Study Guide and Review 583
Chapter Study Guide and Review
Lesson 2
Area of Rectangles and Squares
Area of Rectangles and Squares
(Lesson 2C)
Find the area of each rectangle or square. 13.
5 ft
14.
(continued)
EXAMPLE 3
Find the area of the rectangle.
9 cm 4 cm
7 cm 10 ft
Find the area of each square.
6m
EXAMPLE 4
Find the area of the square.
16.
15.
8 cm
A = w A = 8 × 4 = 8, w = 4 A = 32 square centimeters
12 yd 3 in.
17. A new building measures 42 feet by 30 feet. How much land does the building cover?
A = s2 A = 32 s = 3 A = 9 square inches
Problem-Solving Strategy: Logical Reasoning Solve. Use the logical reasoning strategy. 18. Kristina, Jamel, and Phillip each created a rectangular poster for the book fair. Kristina’s poster had a length of 4 feet and a width of 3 feet. Jamel’s poster had a length of 5 feet and a width of 4 feet. Phillip’s poster had an area of 15 square feet. Which student’s poster had the smallest area?
(Lesson 2D)
EXAMPLE 5
Bella, Mato, and Camille each play a different sport: basketball, soccer, or football. Bella does not like soccer. Mato’s sport is not played with a round ball. Who plays each sport? Make a table to organize the information. Basketball Bella Mato Camille
yes
Soccer
Football
×
×
×
×
yes
×
yes
×
Bella likes basketball, Mato likes football, and Camille likes soccer.
584
Perimeter and Area
Lesson 3
Area of Parallelograms
Area of Parallelograms
(Lesson 3B)
Find the area of each parallelogram. 20.
19.
EXAMPLE 6
Find the area of the parallelogram.
14 ft
7 in.
8 ft
3m
4 in. 12 m
A = bh A = 7 × 4 Replace b with 7 and h with 4. 21. What is the measure of the area of a parallelogram whose base is 82 inches and whose height is 6 inches?
Lesson 4
A = 28 square inches or 28 in2
Area of Triangles and Trapezoids
Area of Triangles
(Lesson 4B)
Find the area of each triangle.
EXAMPLE 7
Find the area of the triangle.
23.
22.
8 ft
4m 13 ft
5 ft
8m 14 ft
1 A = _ bh
24. A house has a triangular window with a base of 4 feet and a height of 3 feet. What is the area of the window?
2
1 A = _ (14)(5)
b = 14, h = 5
A = (7)(5)
1 _ of 14 is 7.
A = 35
Multiply.
2
2
A = 35 square feet or 35 ft2
Chapter Study Guide and Review 585
Chapter Study Guide and Review
Lesson 4
Area of Triangles and Trapezoids
(continued)
Problem-Solving Investigation: Choose the Best Strategy EXAMPLE 8
Solve. 25. Holly is 6 years younger than her sister. Their mother is 44 years old, and her age is twice the sum of her two children’s ages. How old is Holly?
Find the eighth number in the pattern below. 18, 19, 15, 16, 12, 13, . . . Find the pattern in the list of numbers. 18, 19, 15, 16, 12, 13, . . .
26. How many 2-inch squares fit inside a rectangle 6 inches by 8 inches? 27. Paloma uses one pencil the first week of drawing class, and twice as many pencils each week as she did the week before. How many pencils does she use the fifth week?
Area of Trapezoids
5m
+1 -4 +1 -4 +1
The pattern is to add 1, and then subtract 4. So, the seventh number in the pattern is 13 - 4 or 9. The eighth number is 9 + 1 or 10.
(Lesson 4E)
Find the area of each trapezoid. 28.
(Lesson 4C)
29.
12 cm
EXAMPLE 9
Find the area of the trapezoid. 10 m
3m
8 cm 4m 7m
15 cm 5m
30.
31. 9 mi
14 ft 11 ft
7 mi 10 ft 4 mi
32. A trapezoid has bases of 15 meters and 9 meters, and a height of 10 meters. What is the area of the trapezoid?
586
Perimeter and Area
(b1 + b2)h A=_ 2 (10 + 5)4 A=_ 2 15 × 4 A=_ 2 60 A=_ 2
A = 30
Area of a trapezoid. b1 = 10, b2 = 5, h = 4 Add 10 and 5. Multiply 15 and 4. Divide by 2.
A = 30 square meters or 30 m2
Practice Chapter Test Find the perimeter of each figure. 3.4 m
2.
1. 20 yd
Find the area of each figure.
20 yd
10 ft
8.
9.
5m
8m 7 ft
7m
30 yd
20 m
5m 3.4 m
Find the area of each rectangle or square. 3.
4. 15 in.
11 cm
10.
13 in.
11. 11 m
15 in.
10 in.
4 cm
8m
5. MULTIPLE CHOICE What is the area of the picture frame in square inches? 11 cm
12.
17 in.
13.
10 ft 5 ft
8 cm 11 in.
8 ft
17 cm
A. 56
C. 200
B. 187
D. 212
14. MULTIPLE CHOICE Which formula could you use to find the area of a trapezoid?
6. Estimate the area of the figure. Each square represents 1 square centimeter. 7. Theo’s smoothie contains 32 ounces. Tina’s smoothie contains half as many ounces as Theo’s. Pam’s smoothie contains 8 ounces less than Theo’s. How many ounces does each person’s smoothie contain?
bh F. A = _ 2
G. A = w H. A = bh I.
15.
E
(b1 + b2)h A=_ 2
WRITE MATH How is the area of
a triangle related to the area of a parallelogram?
Practice Chapter Test
587
Test Practice
The school’s parking lot is a parallelogram. What is the area, in square yards, of the parking lot?
48 yards
Sometimes, a problem may contain extra information. In the example, the measurement of 48 yards will not be used.
45 yards 50 yards
Read the Test Item
2 250
You need to find the area of the parking lot.
Solve the Test Item
Multiply the base of the parallelogram by its height.
A = bh A = 50 × 45 A = 2,250
So, the area in square yards is 2,250. Fill in the grid.
Read each question. Then fill in the correct answer on the answer sheet provided by your teacher or on a sheet of paper. 1. Look for the pattern in the sequence of numbers below.
2. Find the area, in square centimeters, of the rectangle.
5, 10, 7, 14, 11, 22, 19, . . . 14 cm
Which rule best describes the pattern?
7 cm
A. Add 5, subtract 3. B. Multiply by 2, subtract 3.
588
C. Add 5, multiply by 2.
F. 42 cm2
H. 105 cm2
D. Multiply by 2, add 3.
G. 98 cm2
I. 196 cm2
Perimeter and Area
7. Justin is finding the area of the trapezoid using the formula
3. Which of the following is the solution of the equation x + 4 = 24? A. 28
C. 8
B. 20
D. 6
(b1 + b2)h
A = _ . Which step 2 should he perform first? 9 ft
4. The wall of a building is in the shape of the triangle as shown. What is the area of the wall?
4 ft 5 ft
18 ft
8. 25 ft
5.
F. 200 square feet
H. 250 square feet
G. 225 square feet
I. 275 square feet
F. 5 × 4
H. 4 ÷ 2
G. 9 + 4
I. 9 + 5
GRIDDED RESPONSE Three runners ran a race in the following times. What is the combined total of their times? Runner
SHORT RESPONSE Gloria’s 7 cheerleading practice lasted _ hour 10 on Friday. Write this fraction as a decimal. 9.
6. The graph shows some areas around Anica’s home town. y 5 Park 4 3 Shop Zoo 2 1 School
O
Time (sec)
A
12.5
B
13.12
C
13.4
SHORT RESPONSE Which quadrilateral has a larger area: a square with a side length of 4 meters, or a rectangle with a length of 2 meters and a width of 6 meters?
10. What is the area of a parallelogram with a base of 14 inches and a height of 12 inches?
1 2 3 4 5x
Which ordered pair best represents the point on the graph labeled “School”?
A. 52 square inches
A. (1, 2)
C. (5, 2)
C. 168 square inches
B. (4, 1)
D. (1, 4)
D. 200 square inches
B. 75 square inches
NEED EXTRA HELP? If You Missed Question. . . Go to Chapter-Lesson. . . For help with. . .
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5
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7-3C
11-2C
9-3B
11-4B
7-4B
9-2A
11-4E
5-2C
11-2C
11-3B
GLE 1.2
GLE 4.1
GLE 3.4
SPI 4.1
SPI 2.7
GLE 4.3
GLE 4.1
SPI 2.5
GLE 4.1
SPI 4.1
Test Practice 589
