Perimeter, Area, and Volume

Transcription

1 CHAPTER 9 connected.mcgraw-hill.com Investigate Animations Vocabulary Multilingual eglossary Perimeter, Area, and Volume The BIG Idea What is the relationship between the circumference and diameter of a circle? What is the relationship between area and surface area? Learn Personal Tutor Virtual Manipulatives Make this Foldable to help you organize your notes. Graphing Calculator Audio Perimeter Area Volume Foldables Practice Self-Check Practice Worksheets Assessment Review Vocabulary area área the number of square units needed to cover the surface enclosed by a geometric figure 5 units 3 units A = 5 units 3 units or 15 square units Key Vocabulary English Español base base circle círculo composite figure figura compleja volume volumen 518 Perimeter, Area, and Volume

2 When Will I Use This? Your Turn! You will solve this problem in the chapter. Perimeter, Area, and Volume 519

3 Two-Column Notes To take two-column notes, first fold your paper lengthwise into two columns. Make the right-hand column about 3 inches wide. I need advice for taking good notes. Can you help? When your teacher solves a problem in class, write all of the steps in the left-hand column. In the right-hand column, add notes in your own words that will help you remember how to solve the problem. Add a * by any step that you especially want to remember. Here is a sample from Lesson 1A, Example 1. How to Find the Area of a Parallelogram A = bh A = A =. Replace the variables with This is a sample from Lesson D, Example. Finding the Area of a Circle A = πr A A The area is about mm. My Notes Write the formula. Replace π with 3.14 and r with 1. Multiply. Since 3.14 was used for π, the answer is an approximation. Practice Refer to the following pages. Use the method above to write notes about each example. 1. page 530, Example. page 54, Example 3 3. page 554, Example 1 4. page 570, Example 1 GLE Use mathematical language, symbols, and definitions while developing mathematical reasoning. GLE Read and interpret the language of mathematics and use written/oral communication to express mathematical ideas precisely. Studying Math 51

4 Are You Ready for the Chapter? You have two options for checking prerequisite skills for this chapter. Text Option Take the Quick Check below. Refer to the Quick Review for help. Evaluate each expression _ 7 _ _ _ 7 7. _ 7 _ 7 8. _ _ 7 9. SUPPLIES Alano bought 4 packs of markers. Each pack cost $3.14. How much did Alano spend on markers? EXAMPLE 1 Evaluate Multiply as with whole numbers Place the decimal point two places to the left, since 3.14 has two decimal places. EXAMPLE _ 7_ 7 Evaluate _ 7 _ 7 1 = _ 7 _ = _ 11 6 or 1 _ 5 6 Divide 7 by 1. Divide and 1 by their GCF,. Multiply and simplify. Find the area of each rectangle cm in. 4.1 cm 15 in. EXAMPLE 3 Find the area of the rectangle. 6 ft 9 ft 1. BOARD GAMES The playing area of a board game is a rectangle with a length of 14 inches and a width of 0 inches. What is the area of the board game? A = lw Area of a rectangle A = 9 6 Replace l with 9 and w with 6. A = 54 Multiply. The area of the rectangle is 54 square feet. Online Option Take the Online Readiness Quiz. 50 Perimeter, Area, and Volume

5 Multi-Part Lesson 1 Area PART A B C D Main Idea Find the areas and missing dimensions of parallelograms. Area of Parallelograms Vocabulary base height Get ConnectED Step 1 Draw and then cut out a rectangle as shown. Step Cut a triangle from one side of the rectangle and move it to the other side to form a parallelogram. base (b) height (h) GLE Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas. GLE Use expressions, equations and formulas to solve problems. Also addresses GLE base (b) Step 3 Repeat Steps 1 and with two different rectangles. Step 4 Copy and complete the table below using the three rectangles and three corresponding parallelograms. height (h) Rectangle 1 Rectangle Rectangle 3 Base (b) Height (h) Parallelogram 1 Parallelogram Parallelogram 3 Base (b) Height (h) 1. How does the area of each parallelogram relate to the area of its original rectangle?. What part of the parallelogram corresponds to the base of the rectangle? 3. What part corresponds to the rectangle s height? 4. MAKE A CONJECTURE Write a formula that relates the area A of a parallelogram to its base b and height h. The area of a parallelogram is related to the area of a rectangle. Perpendicular The right angle symbol is used to show two lines are perpendicular. They form a right angle. The base of a parallelogram can be any one of its sides. height base The height is perpendicular from the base to the opposite side. 5 Perimeter, Area, and Volume

6 Everyday Use Parallel Having the same direction, course, or tendency. Math Use Parallel Lines that are the same distance apart. Words Symbols Area of a Parallelogram The area A of a parallelogram is the product of its base b and its height h. A = bh Model h b Find Areas of Parallelograms Find the area of each parallelogram. The base is 6 units, and the height is 8 units. A = bh Area of parallelogram Area Measurement An area measurement can be written using abbreviations and an exponent of. For example: square units = units square inches = in square feet = ft square meters = m A = 6 8 Replace b with 6 and h with 8. A = 48 Multiply. The area is 48 square units or 48 units. Estimate A 0 10 or 00 cm 11 cm 13 cm 0 cm A = bh Area of parallelogram A = 0 11 Replace b with 0 and h with 11. A = 0 Check for Reasonableness 0 00 The area is 0 square centimeters or 0 cm. a. b. 17 m 16 m 4 m Lesson 1A Area 53

7 Find Missing Dimensions Find the height of the parallelogram. A = bh Area of a parallelogram 9 in. Checking Your Work Checking Your Work To check your work, replace b and h in the formula with 9 and 5. A = bh A = 9 5 A = = 9 h Replace A with 45 and b with 9. 45_ 9 = _ 9 h 9 Divide each side by 9. 5 = h Simplify. So, the height is 5 inches. Find the missing dimension of each parallelogram. c. 6 m A = 48 m d. 8 yd A = 45 in A = 96 yd Height of Parallelograms For the parallelogram formed by the area shaded black in Example 4, its height, 1 inches, is labeled outside the parallelogram. FLAGS Romilla is painting a replica of the national flag of Trinidad and Tobago for a 1 in. research project. Find the area of the black stripe. The black stripe is shaped like a parallelogram. So, use the formula A = bh. A = bh Area of parallelogram A = 6 3_ 1 3_ Replace b with 6 and h with A = _ 1 = _ 7 1, or The area of the flag that is black is 81 square inches in. e. ART Guadalupe and her dad made a parallelogram-shaped picture frame to display her artwork. Find the area of the artwork that will be visible in the frame cm 18.4 cm 54 Perimeter, Area, and Volume

8 Examples 1 and Find the area of each parallelogram ft 3. 5 ft 8 m 7 m 11 m Example 3 4. Find the height of a parallelogram if its base is 35 centimeters and its area is 700 square centimeters. 5. Find the area of a parallelogram with base 15 yards and height 1 _ 3 yards. Example 4 6. TANGRAMS The size of the parallelogram piece in a set of tangrams is shown at the right. Find the area of the piece. 6 cm 5.1 cm.6 cm Examples 1 and Find the area of each parallelogram m = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. 9 8 cm 4 m 9 cm 1 cm Example base, 6 millimeters; 11. base, 1 inches; 1. base, 37 feet; height, 4 millimeters height, 15 inches height, feet 13. Find the base of a parallelogram with an area of 4 square feet and height 3 feet. 14. Find the height of a parallelogram with base 6.75 meters and an area of 18.7 square meters. Example PARKING Find the area of the 16. MAPS What is the area of the parking space below. region shown on the map? mi 18 ft Livingston Ontario Yates Seneca 61.5 mi Steuben ft Lesson 1A Area 55

9 B 17. Find the area of the shaded region in each figure. 5 ft 4 ft 1 ft 11 ft cm 8 cm 6 cm 15 cm 19 BUILDINGS The base of a building is shaped like a parallelogram. The first floor has an area of 0,000 square feet. If the base of this parallelogram is 50 feet, can its height be 70 feet? Explain. Dividing Mixed Numbers When dividing mixed numbers, rewrite each as an improper fraction before dividing. 0. PATIOS An architect Patio Base (ft) Height (ft) Area (ft ) _ designed three different parallelogram-shaped brick patios. Find the missing dimensions in the table. 11 1_ _ 4 Draw and label each figure. Then find the area _ _ a parallelogram with a base twice as long as the height and an area less than 60 square inches. a parallelogram with an equal base and height and an area greater than 64 square meters 3. a parallelogram with a base four times the height and an area less than 00 square feet 4. PATTERN BLOCKS What is the height of the A = 55 mm parallelogram-shaped pattern block shown at the right? 1 mm 5. WALLPAPER The design of the wallpaper border below contains parallelograms. If each parallelogram covers 5 square inches, what is the base of each parallelogram? 5 in. 6. MULTIPLE REPRESENTATIONS Draw five parallelograms that each have a height of 4 centimeters and different base measurements on centimeter grid paper. a. TABLE Make a table with a column for base, height, and area. b. GRAPH Graph the ordered pairs (base, area). c. WORDS Describe the graph. 4 cm base 56 Perimeter, Area, and Volume

10 C 7. REASONING Refer to parallelogram KLMN at the right. If the area of parallelogram KLMN is 35 square inches, what is the area of triangle KLN? 8. OPEN ENDED On grid paper, draw three different parallelograms that each have an area of 4 square units and a height of 4 units. Compare and contrast the parallelograms. 9. CHALLENGE If x = 5 and y < x, which figure has the greater area? Explain your reasoning. x 30. E WRITE MATH Explain how the formula for the area of a parallelogram is related to the formula for the area of a rectangle. y x y Test Practice 31. Robert used a piece of poster board shaped like a parallelogram to make a sign. The base of the poster board is 5 inches, and the area is 1,87 square inches. Find the height of the poster board. A. 884 in. C. 4 in. B. 176 in. D. 36 in. 3. A family has a flower garden in the shape of a parallelogram in their backyard. They planted grass in the rest of the yard. What is the area of the backyard that is planted with grass? 75 ft 1 ft Backyard 5 ft 10 ft F. 390 sq ft H. 9,060 sq ft G. 8,940 sq ft I. 9,144 sq ft 33. What is the area of the kite shown below? A. 8.3 c m B c m C. 83 c m D. 1,664 c m 6 cm 64 cm 34. SHORT RESPONSE A wallpaper design uses 15 parallelogram-shaped pieces of paper, each with a base of 3 inches and a height of inches. How much paper is used to make the 15 pieces? Lesson 1A Area 57

11 Multi-Part Lesson 1 Area PART A B C D Area of Triangles Main Idea Discover the formula for the area of a triangle using the properties of parallelograms and a table of values. Get ConnectED GLE Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas. GLE Move flexibly between concrete and abstract representations of mathematical ideas in order to solve problems, model mathematical ideas, and communicate solution strategies. In this activity, you will discover the formula for the area of a triangle using the properties of parallelograms and a table of values. Copy the table shown. Parallelogram A B C D E Base, b Height, h Draw Parallelogram A on grid paper using the dimensions given in the table. Draw a diagonal as shown. Cut out the parallelogram. Then calculate its area. Record this measure. Area of Parallelogram Area of Each Triangle Cut along the diagonal to form two triangles. the Results 1. Compare the base and height of each triangle to the base and height of the original parallelogram. What do you notice?. Compare the two triangles formed. How are they related? 3. What is the area of each triangle? Record your answer in the table. 4. Repeat Steps through 4 for Parallelograms B through E. Calculate the area of each triangle formed and record your results in the table. 5. MAKE A CONJECTURE Write a formula that relates the area A of a triangle to the length of its base b and height h. 58 Perimeter, Area, and Volume

12 Multi-Part Lesson 1 Area PART A B C D E Main Idea Find the areas and missing dimensions of triangles. Get ConnectED GLE Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas. GLE Use expressions, equations and formulas to solve problems. Also addresses GLE Area of Triangles BIOSPHERE The Biosphere complex, located in Tucson, Arizona, is a center for research about Earth and its living systems. The structure of the different sections is made of interlocking triangles that are all the same size. 1. Compare the two outlined triangles.. What figure is formed by the two triangles? 3. How many small triangles make up the outlined parallelogram? 4. How many small triangles make up each outlined triangle? 5. MAKE A CONJECTURE Describe the relationship that exists between the area of one triangle and the area of the parallelogram. Recall that congruent figures are figures that are the same shape and size. A parallelogram can be formed by two congruent triangles. Since congruent triangles have the same area, the area of a triangle is one half the area of the parallelogram. The base of a triangle can be any one of its sides. The height is perpendicular from that base to the opposite vertex. height (h) base (b) Area of a Triangle Words Symbols A = 1_ The area A of a triangle is one half the product of the base b and its height h. _ bh or A = bh Model h b Lesson 1C Area 59

13 Find the area of each triangle. Find the Area of a Triangle height base Mental Math You can use mental math to multiply 1_ (6)(4). Think: Half of 6 is 3, and 3 4 is 1. By counting, you find that the measure of the base is 6 units and the height is 4 units. A = _ 1 bh Area of a triangle A = _ 1 (6)(4) Replace b with 6 and h with 4. A = _ 1 (4) Multiply. A = 1 Multiply. The area of the triangle is 1 square units. 1.1 m 6.4 m A = _ 1 bh Area of a triangle A = _ 1 (1.1)(6.4) Replace b with 1.1 and h with 6.4. A = _ 1 (77.44) Multiply. A = 38.7 Divide. 1_ (77.44) = 77.44, or 38.7 The area of the triangle is 38.7 square meters. To estimate the area of the triangle, round the base to 1 meters and the height to 6 meters. The area is then _ 1 6 or 36 square meters. Since 38.7 is close to 36, the answer is reasonable. Find the area of each triangle. a. b. 9 ft 7 ft 530 Perimeter, Area, and Volume

14 Find Missing Dimensions Area of a Triangle Recall that _ 1_ either A = bh or A = bh can be used to find the area of a triangle. Find the base of the triangle. A = _ bh Area of a triangle 4 = _ b 6 4() = _ b 6 48 = b 6 Simplify. _ 48 6 = _ b 6 6 Replace A with 4 and h with 6. () Multiply each side by. Divide each side by 6. 6 cm b A = 4 cm 8 = b Simplify. So, the base is 8 centimeters. Check A = _ bh A = _ 8 6 A = _ 48 Multiply. A = 4 Divide. Area of a triangle Replace b with 8 and h with 6. Find the missing dimension of each triangle. c. d. A = 7 yd 8 m b A = 40 m 1 yd h TENTS The front of a camping tent has the dimensions shown. How much material was used to make the front of the tent? A = _ 1 bh Area of a triangle A = _ 1 (5)(3) Replace b with 5 and h with 3. A = _ 1 (15) or 7.5 Multiply. The front of the tent has an area of 7.5 square feet. 5 ft 3 ft e. SNACKS A triangular cracker has a height of 4 centimeters and a base of 5 centimeters. Find the area of the cracker. Lesson 1C Area 531

15 Examples 1 and Find the area of each triangle ft m 1 ft 15.6 m Examples 3 and 4 4. ART Tayshan designs uniquely 5. CRAFTS Consuela made a shaped ceramic floor tiles. triangular paper box as shown. What is the base of the What is the area of the top tile shown? of the box? b 6 in. = 1 in 9 cm 10 cm Examples 1 and Find the area of each triangle = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. 10 in. 9 in cm 4.8 cm m 5 m 36 ft 1 41 ft Example 3 Find the missing dimension of each triangle described. 1. height: 14 in., area: 45 in 13. base: 7 cm, area: 56.5 cm Example ROOFING Ansley is going to help his father shingle the roof of their house. What is the area of the triangular portion of one end of the roof? 4 yd 7 yd B 15. ARCHITECTURE An architect is designing a building on a triangular plot of land. If the base of the triangle is feet and the height is 96.3 feet, find the available floor area of the building. 53 Perimeter, Area, and Volume

16 16. FLOWER BEDS A flower bed in a parking lot is shaped like a triangle as shown. a. Find the area of the flower bed in square feet. b. If one bag of topsoil covers 10 square feet, how many bags are needed to cover this flower bed? 3 yd yd 17 MULTIPLE REPRESENTATIONS The table Area of Triangles shows the areas of a triangle where the base of the triangle stays the same but the height changes. a. EXPRESSION Write an algebraic expression that can be used to find the area of a triangle that has a base of 5 units and a height of n units. b. GRAPH Graph the ordered pairs (height, area). c. WORDS Describe the graph. Base (units) Height (units) Area (units ) n 18. FLAGS What is the area of the triangle on the flag of the Philippines in inches? 30 in. 3 ft 5 ft C 19. FIND THE ERROR Dwayne is finding the base of the triangle shown. Its area is 100 square meters. Find his mistake and correct it. 0 m 100 = (b)0 100 = 0b 5 = b 0. CHALLENGE How can you use triangles to find the area of the hexagon shown? Draw a diagram to support your answer. 1. E WRITE MATH Draw a triangle and label its base and height. Draw another triangle that has the same base, but a height twice that of the first triangle. Find the area of each triangle. Then write a ratio that expresses the area of the first triangle to the area of the second triangle. Lesson 1C Area 533

17 Test Practice. The table shows the areas of a triangle where the height of the triangle stays the same but the base changes. Height (units) Areas of Triangles Base (units) Area (square units) _ _ 7 x? Which expression can be used to find the area of a triangle that has a height of 7 units and a base of x units? A. 7x C. _ 7 B. 7x_ D. _ x 3. GRIDDED RESPONSE The triangle has an area of 640 square millimeters. What is the length of the triangle in millimeters? 4. Norma cut a triangle out of construction paper for an art h cm project. The area of the triangle is 84.5 square centimeters. What is the height of the triangle? F. 6.5 cm H. 6 cm G. 13 cm I. 169 cm 13 cm 5. A piece of metal is cut in the shape of the right triangle below. 3 ft 3 ft What is the area of the piece of metal? A. 3 _ 1 4 ft C. 6 _ 3 8 ft B. 4 1 _ ft D. 9 f t ft 3 mm Find the area of each parallelogram. (Lesson 1A) m 4 in. 6.8 cm 5 yd 9 yd 5 m 1 in. 5.7 cm 30. Find the area of a parallelogram with base 15 inches and height 10 inches. (Lesson 1A) 534 Perimeter, Area, and Volume

18 Multi-Part Lesson 1 Area PART A B C D Main Idea Find the area of trapezoids. Get ConnectED GLE Use expressions, equations and formulas to solve problems. GLE Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes. Area of Trapezoids Step 1 On grid paper, cut out two identical trapezoids. Label the bases b 1 and b, respectively, and label the heights h. Step Then turn one trapezoid upside down and tape it to the other trapezoid as shown. 1. Write an expression to represent the base of the parallelogram.. Write a formula for the area A of the parallelogram using b 1, b, and h. 3. How does the area of each trapezoid compare to the area of the parallelogram? b 1 h b b b1 h 4. MAKE A CONJECTURE Write a formula for the area A of a trapezoid with height h, and bases b 1 and b. A trapezoid has two bases, b 1 and b. The height of a trapezoid is the distance between the bases. The height is perpendicular to the bases. h b 1 b The two bases are parallel. They will always be the same distance apart. Area of a Trapezoid Words The area A of a trapezoid is one half the product of the height h and the sum of the bases b 1 and b. Symbols A = 1_ h(b 1 + b ) Model h b 1 b When finding the area of a trapezoid, it is important to follow the order of operations. In the formula, the bases are to be added before multiplying by _ 1 of the height h. Lesson 1D Area 535

19 Find the Area of a Trapezoid Find the area of the trapezoid. The bases are 5 inches and 1 inches. The height is 7 inches. A = _ 1 h(b 1 + b ) Area of a trapezoid A = _ 1 ( 7)(5 + 1) Replace h with 7, b 1 with 5, and b with 1. A = _ 1 (7)(17) Add 5 and 1. A = 59.5 Multiply. The area of the trapezoid is 59.5 square inches. 1 in. 5 in. 7 in. a. 11 cm b..5 m c. 1 ft 8 cm 14 cm 4 m 4.8 m 0.3 ft 0.5 ft 1_ Mental Math To multiply (51)(64), it is easier to use the Commutative Property to 1_ reorder the factors as (64)(51) and take half of 64 instead of half of 51. GEOGRAPHY The shape of Osceola County resembles a trapezoid. Find the approximate area of this county. A = _ 1 h(b 1 + b ) Area of a trapezoid A = _ 1 ( 51)( ) Replace h with 51, b 1 with 48, and b with 16. A = _ 1 (51)(64) Add 48 and 16. A = 1,63 Multiply. St. Cloud Kissimmee 48 mi OSCEOLA Lake COUNTY Kissimmee 16 mi So, the approximate area of the county is 1,63 square miles. 51 mi d. GEOGRAPHY The shape of Arkansas resembles a trapezoid. Find the approximate area of Arkansas. 35 mi 60 mi ARKANSAS Little Rock 10 mi 536 Perimeter, Area, and Volume

20 Example 1 Find the area of each figure. Round to the nearest tenth if necessary ft m 8 ft 15.6 ft 1.1 cm 3.4 cm cm 8 m 10.5 m Example 4. HOCKEY In the National Hockey League, goaltenders can play the puck behind the goal line only in a trapezoid-shaped area, as shown at the right. Find the area of the trapezoid. 11 ft 18 ft 8 ft Example 1 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Find the area of each figure. Round to the nearest tenth if necessary in. 7 5 yd 1 8 ft 15 ft 3 ft 10 ft 10 in. 7 in. 1 yd 3 yd 8. 6 m cm ft 8 m 11 cm 13.4 ft 11 m 9 cm 10.7 ft Example 11. PATIOS Find the area of the patio shown. ft 1. GEOGRAPHY A county is shaped like a trapezoid. Its northern border is about 9.6 miles across, and the southern border is approximately 5 miles across. The distance from the southern border to the northern border is about 90 miles. Find the approximate area of the county ft 5 ft 1 B 13. FIREPLACE Tiles are being placed in front of a fireplace to create a trapezoidal hearth. The hearth will have a height of 4 inches and bases that are 48 inches and 60 inches. If the tiles cover 16 square inches, how many tiles will be needed? 14. REASONING A trapezoid has an area of 150 square meters. If the bases are 14 meters and 16 meters, what is the height of the trapezoid? Lesson 1D Area 537

21 15. TENTS A play tent is shown. How much fabric was used to make the front and back of the play tent? Draw and label each figure. Then find the area. 16. a trapezoid with no right angles and an area less than 1 square centimeters 17. a trapezoid with a right angle and an area greater than 40 square inches 18. a trapezoid with no right angles and an area less than 5 square meters 3 in in. 3 in. 19 LANDSCAPING Use the diagram that 100 ft shows the lawn that surrounds an office building. a. What is the area of the lawn? 80 ft b. If one bag of grass seed covers,000 square feet, how many bags are needed to seed the lawn? 50 ft 140 ft 5 ft Each figure below is made up of congruent trapezoids. Find the area of each figure cm 1. 1 cm 6 cm 1 cm 18 cm 18 cm 4 cm 4 cm 7 cm C. CHALLENGE Apply what you know about rounding to explain how to estimate the height h of the trapezoid shown if the area is 35.5 m m h 6.75 m 3. OPEN ENDED Find the possible lengths of the bases of a trapezoid with a height of 1 foot and an area of 9 square feet. Explain how you found your answer. 4. E WRITE MATH Compare and contrast the formula for the area of a parallelogram and a trapezoid. 538 Perimeter, Area, and Volume

22 Test Practice 5. SHORT RESPONSE A piece of sod is shaped like a trapezoid as shown. What is the area of the piece of sod? 50 cm 10 cm 140 cm 80 cm 5 cm 6. Barrington cuts a piece of wood in the shape of a trapezoid. The height is 4 feet. The top is 3 feet across and the bottom is 10 feet across. Which equation could be used to find the area of the piece of wood? A. 10 = _ 1 h(4 + 3) B. A = _ 1 10(4 + 3) C. A = _ 1 3(4 + 10) D. A = _ 1 4(3 + 10) 7. Find the area of a trapezoid with a height of 4 yards and bases of 5 _ 1 yards and 6 _ 1 yards. F. 16 square yards G. 4 square yards H. 8 _ 1 square yards I. 143 square yards 8. GRIDDED RESPONSE Serina designed the bag shown. How many square inches of fabric will be needed to make the front of the bag? 13 in. 10 in. 9 in. 9. What is the area of a triangle with a base of 5 feet and a height of 38 feet? (Lesson 1C) Find the missing dimension of each triangle described. (Lesson 1C) 30. height: 7 in., area: 1 i n 31. height: 11 m, area: m 3. base:.7 cm, area: 5.65 c m 33. base: 6 ft, area: 5 _ 3 4 ft 34. MEASUREMENT Find the area of the parallelogram at the right. Round to the nearest tenth. (Lesson 1A) 35. Find the height of a parallelogram with an area of 104 square yards and a base of 8 yards. (Lesson 1A) 36. Find the base of a parallelogram with a height of 3. meters and an area of square meters. (Lesson 1A).3 cm 1.6 cm Lesson 1D Area 539

23 Multi-Part t Lesson Circles PART A B C D Circumference Main Idea Describe the relationship between the diameter and circumference of a circle. In this activity, you will investigate how the distance around a circle (circumference) is related to the distance across a circle through its center (diameter). Get ConnectED GLE Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes. SPI Calculate with circumferences and areas of circles. Also addresses GLE Make a table like the one shown. Object C d Cut a piece of string the length of the distance C around a circular object such as a jar lid. Use a centimeter ruler to measure the length of the string to the nearest tenth of a centimeter. C d Measure the distance d across the lid. Record this measurement in the table. Use a calculator to find the ratio of the distance around each circle to the distance across the circle. Repeat Steps through 4 for other circular objects. the Results 1. Describe the ratio _ C for the values in the table above. d. MAKE A PREDICTION Measure the diameter of a different circular object. Predict its circumference. Then check your prediction by measuring. 3. MAKE A CONJECTURE Write a formula that relates the circumference C of a circle to its diameter d. 540 Perimeter, Area, and Volume

24 Multi-Part t Lesson Circles PART A B C D Main Idea Estimate and find the circumference of circles. Vocabulary circle center diameter circumference radius pi Get ConnectED GLE Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes. SPI Calculate with circumferences and areas of circles. Also addresses GLE Circumference DREAMCATCHERS The table shows the approximate circumference, diameter, and radius of several dreamcatchers. Circumference (in.) Diameter (in.) Radius (in.) Describe the relationship between the diameter and radius of each hoop.. Describe the relationship between the circumference and diameter of each hoop. A circle is the set of all points in a plane that are the same distance from a point, called the center. center diameter the distance across a circle through its center circumference the distance around a circle radius the distance from the center to any point on a circle Words Radius and Diameter The diameter d of a circle is twice its radius r. The radius r of a circle is half of its diameter d. Symbols d = r r = d _ Find the Radius and Diameter The diameter of a circle is 14 inches. Find the radius. r = _ d Radius of circle 14 in. r = _ 14 Replace d with 14. r = 7 Divide. The radius is 7 inches. Lesson B Circles 541

25 The radius of a circle is 8 feet. Find the diameter. d = r Diameter of circle 8 ft The diameter is 16 feet. d = 8 Replace r with 8. d = 16 Multiply. Find the radius or diameter of each circle with the given dimension. a. d = 3 cm b. r = 3 in. c. d = 16 yd Symbols The symbol means approximately equal to. In Lesson A, you learned that _ C 3. The exact ratio is d represented by the Greek letter π (pi). The exact value of π is The decimal never ends, but it is often approximated as Circumference Words The circumference of a circle is equal to π times its diameter or π times twice its radius. Model d r Symbols C = πd or C = πr Estimation To estimate the circumference of a circle, you can use 3 for π since π = 69 ft LANDMARKS Big Ben is a famous clock tower in London, England. Find the circumference of the clock face. C = πd Circumference of a circle C 3.14(3) Replace π with 3.14 and d with 3. C 7. Multiply. So, the distance around the clock is about 7. feet. d. FENCES A small circular fence is being placed to surround a young tree. The diameter of the circular fence is 4 feet. How much fencing is used? Use 3.14 for π. Round to the nearest tenth if necessary. 54 Perimeter, Area, and Volume

26 Another approximation for π is _. Use this value when the 7 radius or diameter is a multiple of 7 or has a multiple of 7 in its numerator if the radius is a fraction. Find Circumference Technology You can use a calculator to find the circumference. To find π(1), press [π] 1. The circumference is about Find the circumference of a circle with a radius of 1 inches. Since 1 is a multiple of 7, use _ for π. 7 C = πr Circumference of a circle C _ 1 Replace π with _ and r with C _ 7 _ 1 1 C 13 1 Divide by the GCF, 7. Simplify. The circumference of the circle is about 13 inches. _ Find the circumference of each circle. Use for π. 7 e. f. 70 in. 7 8 ft Examples 1 and Examples 3 and 4 Find the radius or diameter of each circle with the given dimension. 1. d = 3 m. r = 14 ft 3. d = 0 in. _ Find the circumference of each circle. Use 3.14 or for π. Round to the 7 nearest tenth if necessary in ft 11 m 7 13 cm 8. 7 yd in Find the circumference of a circular fountain with a diameter of 15 meters. Round to the nearest tenth. Lesson B Circles 543

27 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Examples 1 and Examples 3 and 4 Find the radius or diameter of each circle with the given dimensions. 11. d = 5 mm 1. d = 4 ft 13. r = 17 cm 14. r = 36 in. _ Find the circumference of each circle. Use 3.14 or for π. Round to the 7 nearest tenth if necessary ft m 3.5 mi 18. r = 15 yd 19. d = 35 ft 0. d = 56 cm 1. a button with a radius of 1 millimeters. a dunk tank with a radius of 36 inches B 3. MUSIC The diameter of a music CD is 1 centimeters. Find the circumference of a CD to the nearest tenth. 4. VOLCANOES The Belknap shield volcano is located in Oregon. The volcano is circular and has a diameter of 5 miles. What is the circumference of this volcano to the nearest tenth? 5 TREES The largest tree in the world by volume is The General Sherman Tree in Sequoia National Park. The diameter at the base is 36 feet. If a person with outstretched arms can reach 6 feet, how many people would it take to reach around the base of the tree? Real-World Link In California and Oregon, many shield volcanoes have diameters of three or four miles. 6. WALKING At a local park, Sarala can choose between two circular paths to walk. One path has a diameter of 10 yards, and the other has a radius of 45 yards. How much farther can Sarala walk on the longer path than the shorter path if she walks around the path once? 7. ESTIMATION Refer to the circle at the right. a. Find the circumference of the circle. Use 3 as the estimate of π. b. Find the circumference of the circle using 3.14 for π. c. Another estimate of π is Find the circumference using this estimate. d. What do you notice about the estimate used for π and the circumference of the circle? 10 mm 8. REASONING The diagram at the right is made up of circles with the same center. The innermost circle has a diameter of 1 unit. Each circle moving outward has a diameter one more unit than the previous. Without calculating, how much longer is the circumference of each circle? 544 Perimeter, Area, and Volume

28 9. ESTIMATION Without calculating, determine if the circumference of a circle with a radius of 4 feet will be greater or less than 4 feet. Explain. 30. MATH IN THE MEDIA Find an example of a circular object in a newspaper or magazine, on television, or on the Internet. Write a real-world problem in which you would estimate the circumference. 31 ESTIMATION Catalina is giving pillar candles as favors at her birthday party. She wants to glue a piece of ribbon around each candle. The diameter of each candle is 4 inches. She has 8 candles and yards of ribbon. Does she have enough ribbon? Explain. Multiplication Equations When solving multiplication equations, such as = 3.14d, use the inverse operation, and divide by Find the diameter given the circumference of each object. Use 3.14 for π. 3. a satellite dish with a circumference of meters 33. a basketball hoop with a circumference of 56.5 inches 34. a nickel with a circumference of about millimeters Find the distance around each figure. Use 3.14 for π cm 5 ft 5 ft C 37. OPEN ENDED Draw and label a circle that has a diameter more than 5 inches, but less than 10 inches. Estimate its circumference and then find its circumference. Then compare your estimate to the value you found on your calculator. 38. CHALLENGE Analyze how the circumference of a circle would change if the diameter was doubled. Provide an example to support your explanation. 39. E WRITE MATH A circle has a circumference of about 15.7 meters and a diameter of about 5. meters. What is the relationship between the circumference and diameter of this circle? Lesson B Circles 545

29 Test Practice 40. A circle with center at point O is shown below. Which line segment is half the length of diameter QM? A. Segment ON B. Segment PM C. Segment QP D. Segment OL 41. An above-ground circular swimming pool is 18 feet in diameter. How does the pool s diameter d compare to its circumference C? F. d _ 1 C G. d C H. d 3C I. d _ 1 3 C 4. GRIDDED RESPONSE The circumference of the Ferris wheel at the county fair is 78.5 feet. What is the diameter of the Ferris wheel, in feet? Use 3.14 for π. 43. A bicycle wheel has spokes for support. Each spoke extends from the center of the wheel to the rim. Which method can be used to find the circumference of the bicycle wheel? 1 in. A. Multiply the diameter by π and by. B. Divide the diameter by π. C. Multiply the radius by π. D. Multiply the radius by π and by. Find the area of each trapezoid. (Lesson 1D) cm m 14 cm.8 m 34 cm 3.7 m mm 31 mm 47 mm 47. Find the area of a triangle with a base of 5 inches and a height of 30 inches. (Lesson 1C) 98 ft 48. BUILDING Find the area of glass used on the side of the parallelogram-shaped building shown. (Lesson 1A) 377 ft 546 Perimeter, Area, and Volume

30 Multi-Part t Lesson Circles PART A B C D Area of Circles Main Idea Develop a formula for the area of a circle. In this activity, you will explore how the formula for the area of a circle is related to the formula for the area of a parallelogram. Get ConnectED GLE Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes. SPI Calculate with circumferences and areas of circles. Also addresses GLE Fold a paper plate in half four times to divide it into 16 equal-sized sections. Label the radius r as shown. Let C represent the circumference of the circle. Cut out each section. Reassemble the sections to form a parallelogram-shaped figure. r (height) r 1 C (base) 1 C the Results 1. What are the measurements of the base and the height?. Substitute these values into the formula for the area of a parallelogram. Write the new formula. 3. Replace C with the expression for the circumference of a circle, πr. Simplify the equation and describe what it represents. 4. MULTIPLE REPRESENTATIONS Use the formula A = πr and 3.14 for π. a. TABLE Copy and complete Radius the table. (in.) b. GRAPH Graph the ordered 1 pairs (radius, circumference) and (radius, area) on the same 3 coordinate plane. 4 c. WORDS Describe the graph. 5 Circumference (in.) Area (in ) Lesson C Circles 547

31 Multi-Part t Lesson Circles PART A B C D Main Idea Find the areas of circles. Vocabulary semicircle Get ConnectED GLE Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes. SPI Calculate with circumferences and areas of circles. Also addresses GLE Area of Circles PETS Adrianne bought an 8-foot leash for her dog. 1. Adrianne wants to find the distance the dog runs when it runs one circle with the leash fully extended. Should she calculate the circumference or area? Explain.. Suppose she wanted to find out the amount of running room the dog has with the leash. Should she calculate the circumference or area? Explain. In Lesson C, the formula for the area of a parallelogram was used to develop a formula for the area of a circle. Area of a Circle Words The area A of a circle equals the product of π and the square of its radius r. Model r Symbols A = πr Find the Area of a Circle Find the area of the circle. Use 3.14 for π. Estimate 3 = 1 A = πr Area of a circle in. A 3.14 Replace r with. A = = 4 A 1.56 Multiply. Check for Reasonableness The area of the circle is approximately 1.56 square inches. a. Find the area of a circle with a radius of 3. centimeters. Round to the nearest tenth. 548 Perimeter, Area, and Volume

32 Calculating with π When evaluating expressions involving π, using the π key on a calculator will result in a different approximation. COINS Find the area of the face of the Virginia quarter with a diameter of 4 millimeters. Use 3.14 for π. Round to the nearest tenth if necessary. The radius is _ 1 (4) or 1 millimeters. Estimate = 43 A = πr Area of a circle A Replace r with 1. A Multiply. Check for Reasonableness The area is approximately 45. square millimeters. b. POOLS The bottom of a circular swimming pool with a diameter of 30 feet is painted blue. How many square feet are blue? A semicircle is half of a circle. The formula for the area of a semicircle is A = 1 _ πr. Area of Semicircles Find the area of the semicircle. Use 3.14 for π. Round to the nearest tenth if necessary. A = 1 _ πr Area of a semicircle A = 1 _ π8 Replace r with 8. A 0.5(3.14)(8 ) Multiply. Use 3.14 for π. A Simplify. 16 in. The area of the semicircle is approximately square inches. c. Find the approximate area of a semicircle with a radius of 6 centimeters. Lesson D Circles 549

33 Example 1 Find the area of each circle. Round to the nearest tenth. Use 3.14 or _ 7 for π cm. 7 in. 3. diameter = 16 m 4. diameter = 4 ft Example 5. SPRINKLERS A rotating sprinkler with a radius of 11 feet is used to water a lawn. Find the area of the lawn that is watered. Use 3.14 for π. Example 3 6. ART Rondell draws the semicircle shown at 14 yd the right. What is the area of the semicircle? Use 3.14 for π. Example 1 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Find the area of each circle. Round to the nearest tenth. Use 3.14 or _ 7 for π cm 8. 8 in ft 10. diameter = 8.4 m 11. diameter = 1.6 cm 1. radius = 4 1 _ in. Example 13. PATCHES Find the area of the Girl Scout patch shown if the diameter is 1.5 inches. Round to the nearest tenth. 14. PETS Refer to the pets problem at the beginning of this lesson. Find the area, to the nearest tenth, of grass that Adrianne s dog may run in if the leash is 9 feet long. Example TUNNELS The tunnel opening shown is a semicircle. Find the area, to the nearest tenth, of the opening of the tunnel enclosed by the semicircle. 3 ft 550 Perimeter, Area, and Volume

34 B 16. FINANCIAL LITERACY Harry s Pizzeria is having a sale on medium and large pizzas. Medium pizzas are 10 inches in diameter and cost $7.99. Large pizzas are 14 inches in diameter and cost $ Which size pizza is the better deal? Explain. (Hint: Find the cost per square inch of each pizza.) 17 Which has a greater area, a triangle with a base of 100 feet and a height of 100 feet or a circle with diameter of 100 feet? Justify your selection. 18. RADIO SIGNALS A radio station sends a signal in a circular area with an 80-mile radius. Find the approximate area in square kilometers that receives the signal. (Hint: 1 square mile.6 square kilometers) 19. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a b. Use 3.14 for π. Help me find the area inside the dirt bike track. a. What is the area of the circle with the 5-foot radius? b. What is the area of the outside circle with the 40-foot radius? C 0. REASONING If the length of the radius of a circle is doubled, how does that affect the circumference and area? Explain. CHALLENGE Find the area of the shaded region in each figure. Round to the nearest tenth m 3.5 cm 1 m 5.5 in. 1.5 cm 4. E WRITE MATH Write and solve a real-world problem in which you would solve the problem by finding the area of a circle. Lesson D Circles 551

35 Test Practice 5. A distance measuring wheel is used to measure long distances by rotating 360 degrees. 7. Which two figures have the same area shaded? 8 m 7.5 m 1 m 1 m Figure I Figure II 10 m Which of the following best describes the distance in one 360-degree rotation? A. the area of the wheel B. the radius of the wheel C. the diameter of the wheel D. the circumference of the wheel 6. Which equation could be used to find the area in square inches of a circle with a radius of 1 inches? F. A = 6 π H. A = 1 π G. A = π 6 I. A = π 1 1 m 1 m 6 m Figure III Figure IV A. Figure I and Figure IV B. Figure I and Figure II C. Figure II and Figure IV D. Figure II and Figure III 8. MEASUREMENT What is the circumference of a circle that has a radius of 8 yards? Use 3.14 for π and round to the nearest tenth if necessary. (Lesson B) 9. MEASUREMENT A frame for a collage of pictures is in the shape of a trapezoid. The two bases are 15 inches and 0 inches. The height of the trapezoid is 1 inches. What is the area enclosed by the frame? (Lesson 1D) Find the area of each parallelogram. Round to the nearest tenth if necessary. (Lesson 1A) in. 5 cm 8.7 m 1 in. 7.9 cm 11.5 m 55 Perimeter, Area, and Volume

36 Mid-Chapter Check Find the area of each parallelogram. (Lesson 1A) cm 5 cm. 6 ft 1 8 ft Find the missing dimension of each parallelogram. (Lesson 1A) 3. height, 5 _ 1 ft; area, 1 ft 4 4. base, 5.65 m; area, m Find the area of each triangle. (Lesson 1C) m 1 m 8 ft 7. MULTIPLE CHOICE What is the height of a triangle with a base of 14 centimeters and an area of 56 square centimeters? (Lesson 1C) A. 11 centimeters B. 56 centimeters C. 8 centimeters D. 7 centimeters 8. FURNITURE A corner 45 in. table is in the shape of a trapezoid. Find the area of the tabletop. (Lesson 1D) 30 in. 8 in. Find the radius or diameter of each circle with the given dimensions. (Lesson B) 9. d = 7 in. 10. r = 3 ft 11. r = 16 yd 1. d = 18 cm Find the _ circumference of each circle. Use 3.14 or for π. (Lesson B) cm MULTIPLE CHOICE Ernesto knows the circumference of a DVD but would like to find its diameter. Which method can Ernesto use to find the diameter of the DVD? (Lesson B) F. Multiply the circumference of the DVD by its radius. G. Divide the circumference of the DVD by π and then divide by. H. Divide the circumference of the DVD by π. I. Multiply the circumference of the DVD by. Find the area of each circle with the given dimension. Use 3.14 or for π. Round to 7 the nearest tenth if necessary. (Lesson D) 16. r = 14 cm 17. d = 3 ft 18. d = 3.1 m 19. r = _ 3 4 in. 0. FOOD Josie is baking a pie for a family reunion. What is the approximate area of the pie if the diameter is 9 inches? (Lesson D) 7 8 yd Mid-Chapter Check 553

37 Multi-Part Lesson 3 Composite Figures PART A B C D Main Idea Find the perimeter of a composite figure. Vocabulary perimeter composite figure Get ConnectED GLE Use expressions, equations and formulas to solve problems. Perimeter of Composite Figures MOVIE THEATERS One of the largest movie theater screens is in St. Louis, Missouri. The 500-seat theater houses a 60-foot by 80-foot movie screen that is nearly twice the size of a traditional movie theater screen. 1. Suppose the theater owners wanted to put a fabric border around the screen. What would be the length of fabric?. How did you find the distance around the screen? 3. How could you use multiplication and addition to find the distance around the screen? Perimeter is the distance around a figure. Recall that a rectangle is a four-sided figure with four right angles and the opposite sides have equal lengths. Perimeter of Rectangles Words The perimeter P of a rectangle is twice the sum of the length l and width w. Model w Symbols P = l + l + w + w P = l + w or (l + w) l Find Perimeter of a Rectangle Find the perimeter of the rectangle. P = l + w Perimeter of a rectangle P = (7) + (13) Replace l with 7 and w with 13. P = or 40 Multiply. Then add. The perimeter is 40 yards. 7 yd 13 yd a. Find the perimeter of a rectangle with a length of 7.6 centimeters and a width of 4.9 centimeters. 554 Perimeter, Area, and Volume

38 Composite Figures Composite figures are also sometimes called complex figures or irregular figures. A composite figure is made of triangles, quadrilaterals, semicircle trapezoid semicircles, and other two-dimensional figures. To find the perimeter of a composite figure, add the distances around the closed figure. Find Perimeter of a Composite Figure Find the perimeter of the figure. Add all of the distances around the composite figure. P = Sum of all sides P = 11.4 m Add. The perimeter is 11.4 meters. 1. m 1. m 4 m 4 m 1 m Find the perimeter of each figure. b ft c. 5.5 ft 5.5 ft 5 cm 5 cm 5.5 ft 10.5 ft 5.5 ft 6 cm 6 cm 6 cm GARDEN Find the perimeter of the garden to the nearest tenth. The distance around the semicircle is unknown. So, you need to find the circumference of a circle. 1 ft 8 ft Step 1 Find the circumference of the circle. C = πd Circumference of a circle 5 ft 0 ft 15 ft x semicircle C = 3.14(15) Replace d with 15. Use 3.14 for π. C 47.1 Multiply. Step Since you only need half of the circumference, divide by. The distance around the semicircle is approximately 47.1 or 3.6. Step 3 Find the perimeter of the garden. The perimeter of the garden is or 88.6 feet. Lesson 3A Composite Figures 555

39 18 ft d. POOLS Find the perimeter of the pool shown. 1 ft 1 ft 18 ft Find Missing Measures to Find Perimeter Find the perimeter of the figure. Find the unknown lengths by breaking the figure into two shapes. 40 ft 45 ft 45 ft 40 ft 15 ft? 5 ft? 35 ft 5 ft The sum must be 40 feet. 15 ft 5 ft 15 ft 30 ft The sum must be 45 feet. So, the perimeter is or 170 feet. e. 16 cm 1 cm 7 cm f. 3.1 mm 5. mm cm 6.1 mm Examples 1 and 4 Find the perimeter of each figure yd. 1.6 cm m 5 yd.9 cm 0 m 11 m 4 m Examples and 3 4. FLAGS The flag of Ohio is the only state 5 ft flag that is not in the shape of a rectangle. Find the perimeter of the flag. 3 ft ft 1 ft 5 ft 556 Perimeter, Area, and Volume

40 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Examples 1 and Find the perimeter of each figure. Use 3.14 for π yd 6. 1 yd 1 yd 1.5 m 1.5 m 1 yd 1 yd m m m m 40 yd 1.5 m 1.5 m cm in. 14 cm 11 cm 14 cm 1 in. 15 in. Example 3 9. CARPENTRY Mr. Thomas wants to put a 10 ft baseboard in the room shown. Find the perimeter of the room to determine how many feet of baseboard he will need. 1 ft 4 ft 5 ft 15 ft 18 ft 10. NASCAR One of the shortest professional racetracks is the Bristol Motor Speedway in Tennessee. Find the length, in feet, of one lap of the track. 650 ft 580 ft Example 4 Find the perimeter of each figure yd 1 yd 4 yd m 17 yd 19 yd 7.8 m 30 yd 4.3 m 3.1 m.6 m B 13 BASKETBALL A basketball court measures 6 meters by 14 meters. Ten meters of seating is added to each side of the court. Find the perimeter of the new figure created by the seating area. 14. ALGEBRA Find the value of y given the perimeter P. How many segments y units long are needed for the perimeter? y P = 54 cm y y Lesson 3A Composite Figures 557

41 C 15. OPEN ENDED Draw and label a composite figure made up of a rectangle and semicircle with a perimeter between 100 and 00 centimeters. Find the perimeter. 16. CHALLENGE Find the perimeter of the figure at the right. 17. E WRITE MATH What is the difference between area and perimeter? 16 in. 8 in. 16 in. 8 in. Test Practice 18. GRIDDED RESPONSE Mao is framing a picture. The picture frame is shown. What is the perimeter of the outside of the picture frame in inches? 10 in. 1 1 in. 19. Phillipe draws an equilateral triangle. Next he draws a regular pentagon with the same perimeter as the triangle. How long is each side of the regular pentagon? 6.4 m 6.4 m 8 in. 6.4 m A. 3 m C. 6.4 m B. 5.6 m D m Find the area of each circle. Use 3.14 or for π. Round to the nearest tenth if 7 necessary. (Lesson D) _ 47 ft 19 yd 7 11 in. 3. TRACK AND FIELD The diameter of the circle that a shot-putter stands in is 7 feet. What is the circumference of the circle? Use _ for π. (Lesson B) 7 4. PURSE Morgan bought the purse shown. How much material was used to make the front and back of her purse? (Lesson 1D) 5. SAIL Sail Away Boats use canvas to make their triangular sails for boats. Find the amount of canvas needed to make a sail with a height of 14 feet and a base of 8 feet. (Lesson 1C) 6 in. 5 in. 11 in. 558 Perimeter, Area, and Volume