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

42 Multi-Part Lesson 3 Composite Figures PART A B C D Area of Irregular Figures Main Idea Find and estimate the area of an irregular figure by counting squares. You can estimate the area of an irregular figure by using grid paper. How can you find the area of your footprint? 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. What do you need to find? the area of your footprint Trace your foot on grid paper. Count the number of whole squares of your footprint. Count the partial squares. Estimate the number of whole squares they cover. Add the areas of the whole squares and partial squares. the Results 1. Measure the length and width of your foot at its widest point. Use the formula for the area of a rectangle to estimate the area of your footprint. Compare this estimate to your answer using whole and partial squares.. Would your estimation be more or less accurate if you used smaller _ 1 -inch grid paper? 8 3. In the Activity, you counted partial squares to estimate the area. Describe another method of estimating the area of irregular figures on grid paper. 4. E WRITE MATH Describe some real-world examples of when it would be useful to estimate the area of figures. Lesson 3B Composite Figures 559

43 Another way to estimate the area of an irregular figure is to separate the figure into simpler shapes. Then find the sum of these areas. Estimate the area of Idaho. First, separate the figure into a triangle and a rectangle. 481 mi triangle 100 mi Find the area of each figure. Area of a triangle A = 1 _ bh = _ b = or 00 h = or 311 = 31,100 Simplify. Area of a rectangle A = lw IDAHO 300 mi 170 mi rectangle = or 51,000 l = 300 and w = 170 Add to find the total area. 31, ,000 = 8,100 The area of Idaho is about 8,100 square miles. Check for Reasonableness Solve the problem another way. How does it compare to the answer in the activity? the Results 5. How would your estimate change if you increased the number of shapes you used? 6. In the figure at the right, the area of Oklahoma is 170 mi separated into polygons. a. Explain how polygons can be used to estimate the total 35 mi land area. b. Estimate the area of each region. c. Estimate the total area of Oklahoma. 7. RESEARCH Use the Internet or another source to find the actual total area of Oklahoma. How does it compare to your answer in Exercise 6c? 8. RESEARCH Estimate the area of another state. Use the Internet or another source to compare your estimate with the actual area. 130 mi 90 mi Oklahoma Oklahoma City 305 mi 5 mi 560 Perimeter, Area, and Volume

44 Multi-Part t Lesson 3 3 Composite Figures PART A B C D Main Idea Find the areas of composite figures. 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 Composite Figures POOLS The dimensions of a pool at the recreation center are shown. 1. Describe the shape of the pool.. How could you determine the area of the pool s floor? 4 ft 6 ft ft To find the area of a composite figure, separate it into figures with areas you know how to find. Then add those areas. 8 ft Find the Area of a Composite Figure Find the area of the figure at the right. The figure can be separated into a rectangle and a triangle. Find the area of each. Area of Rectangle Area of Triangle 6 in. 6 in. 10 in. 14 ft 4 in. 6 in. A = lw 10 in. A = 10 6 or 60 4 in. A = 1 _ bh 4 in. The area is or 68 square inches. A = _ 1 (4)(4) or 8 The base of the triangle is 10-6 or 4 inches. Find the area of each figure. a. b. 10 ft 8 ft 4 ft 1.5 cm.5 cm 1 ft 9. cm Lesson 3C Composite Figures 561

45 POOLS The diagram of the pool from the beginning of the lesson is shown below. Find the area of the pool s floor. 8 ft 4 ft 6 ft 14 ft ft The figure can be separated into a rectangle and a trapezoid. The area of the rectangle is 8 14 or 39 square feet. The area of the trapezoid is _ 1 ()(4 + 6) or 10 square feet. So, the area of the pool s floor is or 40 square feet. Real-World Link The largest swimming pool in the world is located in Chile. It is one kilometer in length and is equivalent to approximately 6,000 standard pools. Find the area of the figure at the right. The figure can be separated into a square and a rectangle. The shared rectangle, however, will be counted twice if the areas are added, so the area of the small rectangle, 6 7 or 4, must be subtracted from the total. 1 cm 1 cm 15 cm 6 cm 8 cm The area of the square is 1 1 or 144 square centimeters. The area of the rectangle is 15 1 or 180 square centimeters. The sum of the areas is or 34 square centimeters. Since the small rectangle was counted twice, subtract its area from the total = 8 So, the area of the figure is 8 square centimeters. 1 cm c. DECKS Find the area of the d. Find the area of the figure deck shown. below. 0 ft ft 6 ft 0 ft 14 ft 14 ft 6 ft 36 ft 10 ft 8 ft 7 ft 1 ft 56 Perimeter, Area, and Volume

46 Example 1 Find the area of each figure. Round to the nearest tenth if necessary m 14 m 7 m 4 m 6 ft 15 ft 6 ft 4 m 15 m 4 m 10 m 10 m Examples and 3 4. APARTMENTS The manager of an apartment complex will install new carpeting in a studio apartment. The floor plan is shown at the right. What is the total area that needs to be carpeted? 1 ft 8 ft 5 ft 5 ft 10 ft 5. TILING The floor plan of a kitchen is shown at the right. If the entire kitchen floor is to be tiled, how many square feet of tile are needed? ft 6 ft 6 ft 11 ft 1 ft 16 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 cm 7 7 cm 4 in. 8 in. 5 yd 10 cm 8 in. 7 yd mm ft m 4 m 5. m 0 mm 8 ft 4.3 ft m 5. m Examples and 3 1. BLUEPRINTS On a blueprint, a rectangular room 14 ft 14 feet by 1 feet has a semicircular sitting area attached with a diameter of 1 feet. What is the 1 ft total area of the room and the sitting area? 13. POOLS The diagram at the right gives 18 ft the dimensions of a swimming pool. If a cover is needed for the pool, what will be 0 ft the approximate area of the cover? 36 ft Lesson 3C Composite Figures 563

47 14. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a b. Let s just find the area of the larger circle and take out the area of the smaller circle! a. Find the area of the dirt bike track. Use 3.14 for π. b. Suppose it costs $4.99 to cover one square foot of the dirt bike track with dirt. How much will it cost to cover the track with dirt? Round to the nearest cent. B 15 PAINTING The diagram shows one side of a storage barn. a. This side needs to be painted. Find the total area to be painted. b. Each gallon of paint costs $0 and covers 350 square feet. Find the total cost to paint this side once. Justify your answer ft.8 ft 6.5 ft CHALLENGE Describe how to separate each state into simpler figures. Then use these figures to estimate the area of each state. One square unit equals,400 square miles. Justify your answer NEVADA MISSOURI 18. E WRITE MATH Describe how you would find the area of the figure shown at the right. 5 cm 9 cm 7 cm 564 Perimeter, Area, and Volume

48 Test Practice 19. What is the area of the window shown? Use 3.14 for π. 0. The shaded part of the grid represents the plans for a fish pond. 48 in. 36 in. A.,36.68 in C in B. 1,78 in D. 168 in If each square on the grid represents 5 square feet, what is the approximate area of the fish pond? F. 175 square feet G. 165 square feet H. 150 square feet I. 33 square feet 1. EXTENDED RESPONSE To promote recycling, the ground of the neighborhood playground shown is being covered by shredded tires. The sandbox will NOT be covered. Part A What is the area, in square feet, of the shredded tire portion of the playground? Part B If shredded tires cost $.99 per square foot, at the required depth, how much will it cost to cover the playground? sandbox 30 ft 80 ft 15 ft 45 ft Find the perimeter of each figure. (Lesson 3A). 4.3 m 4.3 m 4.3 m 4.3 m 4.3 m 4.3 m 3. 3 ft 3 ft 3 ft 3 ft 3 ft cm 8 cm 4 cm 4 cm 4 cm.6 cm Find the area of each circle. Use 3.14 for π. Round to the nearest tenth. (Lesson D) 5. radius = 4.7 cm 6. radius = 1 in. 7. diameter = 15 in. Lesson 3C Composite Figures 565

49 Multi-Part t Lesson 3 Composite Figures PART A B C D Problem-Solving Investigation Main Idea Solve problems by making a model. Make a Model D.J.: I m helping set up 7 rows of chairs for a school assembly. There are eight chairs in the first row. Each row after that has two more chairs than the previous row. If I have 100 chairs, can I set up enough rows? YOUR MISSION: Make a model to find whether D.J. has enough chairs to set up all 7 rows. Understand You know that each row has two more chairs than the previous row. The first row has 8 chairs. There are 7 rows. You need to determine if 100 chairs are enough. Plan Solve Make a model to see if there are enough chairs. Use counters to show the layout of the chairs. Row 1 Row Row 3 Row 4 Row 5 Row 6 Row 7 8 chairs 10 chairs 1 chairs 14 chairs 16 chairs 18 chairs 0 chairs Add: = 98 Since 98 < 100, there are enough chairs. Check The average number of chairs in the first and last row is 8 _ + 0 = _ 8 or 14. Since there are 7 rows and 7 14 = 98, the answer is reasonable. 1. Tell how making a model helped D.J. solve the problem. GLE Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution. 566 Perimeter, Area, and Volume

50 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Make a model. Look for a pattern. Draw a diagram. Choose an operation. Use the make a model strategy to solve Exercises 4.. GEOMETRY For a school assignment, Santiago has to give three different possibilities for the dimensions of a rectangle that has a perimeter of 8 feet and an area greater than 30 square feet. One of the models he made is shown below. What are two other possibilities for the dimensions of the rectangle? 4 ft 10 ft 3. DESIGN A designer wants to arrange 1 square glass bricks into a rectangular shape with the least perimeter possible. How many blocks will be in each row? 4. PAPER Timothy took a piece of notebook paper and cut it in half. Then he placed the pieces on top of each other and cut them in half again to have 4 pieces of paper. If he could keep cutting the paper, how many pieces of paper would he have after 6 cuts? Use any strategy to solve Exercises SKATES Of 50 students surveyed, have a skateboard, and 18 have roller shoes. Of those, 6 students have both. How many students have neither a skateboard nor roller shoes? 6. BOOSTERS In 010, 5 parents participated in the band booster organization at King Middle School. Participation increased to 40 parents in 011 and 55 parents in 01. If the trend continues, about how many parents can be expected to participate in 013? 7. Meghan sends four friends an . Each friend then forwards the to another four friends, and so on. If four friends forward the to another four friends each hour, how long will it take for 64 friends to receive the ? 8. GEOMETRY The base and height of each triangle are half their length than in the previous triangle. What will be the area of the fourth triangle? 16 cm 4 cm 9 WATER PARKS What is the total price for two adult and three children one-day passes to a local water park? One-Day Pass Two-Day Pass Adults $40 $45 Child $30 $ DOLLS Rosario is arranging her doll collection on 5 shelves. The top shelf has dolls on it, and each shelf has 3 more dolls than the one above it. How many dolls does Rosario have in all? 11. SOCCER Refer to the graph. How many more boys signed up for soccer in 010 than 008? Number of Kids Newtown Soccer League Sign-Ups Boys 08 Girls 09 Year E WRITE MATH Write a problem that can be solved by making a model. Lesson 3D Composite Figures 567

51 Multi-Part Lesson 4 Volume and Surface Area of Rectangular Prisms PART A B C D E Volume of Rectangular Prisms Main Idea Find the volume of rectangular prisms. The figures shown are rectangular prisms. Get ConnectED GLE Develop and use formulas for surface area and volume of 3-dimensional figures. SPI Determine the surface area and volume of prisms, pyramids and cylinders. Copy the table below. Prism A B C Number of Cubes Height of Prism Length of Base Width of Base Area of Base D E Using centimeter cubes, build five different rectangular prisms. For each prism, record the dimensions and the number of cubes used. the Results 1. Examine the rows of the table. What do you notice?. MAKE A CONJECTURE Describe the relationship between the number of cubes needed and the dimensions of the prism. 3. Make a rectangular prism with a height of, a length of 3, and a width of 3. a. What is the area of the base? b. How many cubes did you use to make your prism? 4. List the dimensions of as many possible prisms that each are made of 36 cubes. 5. Suppose the dimensions of the rectangular prism in Exercise 3 were doubled. How would that affect the volume? 6. Suppose the dimensions of the rectangular prism in Exercise 3 were tripled. How would that affect the volume? 568 Perimeter, Area, and Volume

52 Multi-Part Lesson 4 Volume and Surface Area of Rectangular Prisms PART A B C D E Main Idea Find the volume of rectangular prisms. Vocabulary three-dimensional figure rectangular prism volume cubic units Get ConnectED GLE Develop and use formulas for surface area and volume of 3-dimensional figures. SPI Determine the surface area and volume of prisms, pyramids and cylinders. Volume of Rectangular Prisms On a piece of grid paper, cut out a square that is 10 centimeters on each side. Cut a -centimeter square from each corner. Fold the paper and tape the corners together to make a box. 1. What is the area of the base, or bottom, of the box? What is the height of the box?. How many centimeter cubes fit in the box? 3. What do you notice about the product of the base area and the height of the box? A three-dimensional figure has length, width, and height. A rectangular prism is a three-dimensional figure with two parallel bases that are congruent rectangles. rectangular bases Volume is the amount of space inside a three-dimensional figure. Volume is measured in cubic units. Decomposing the prism tells you the number of cubes of a given size it will take to fill the prism. The volume of a rectangular prism is related to its dimensions. Volume of a Rectangular Prism Words The volume V of a rectangular Model prism is the product of its length l width w, and height h. Symbols V = lwh l w h Lesson 4B Volume and Surface Area of Rectangular Prisms 569

53 Volume Measurement A volume measurement can be written using abbreviations and an exponent of 3. For example: cubic units = units 3 cubic inches = in 3 cubic feet = ft 3 cubic meters = m 3 Another method to decompose a rectangular prism is to find the area of the base (B) and multiply it by the height (h). V = Bh area of the base, or the number of cubes needed to cover the base number of rows of cubes needed to fill the prism h Find the volume of the rectangular prism. Estimate V 10 cm 10 cm 6 cm or 600 cm 3 Find the Volume of a Rectangular Prism 1 cm In the figure, the length is 1 centimeters, the width is 10 centimeters, and the height is 6 centimeters. 6 cm 10 cm Method 1 V = lwh Use V = lwh. Volume of rectangular prism V = Replace l with 1, w with 10, and h with 6. V = 70 Multiply. Decomposing Figures You can think of the volume of the prism as consisting of six congruent slices. Each slice contains the area of the base, 10 cm, multiplied by a height of 1 cm. Method Use V = Bh. B, or the area of the base, is 10 1 or 10 square centimeters. V = Bh Volume of rectangular prism V = 10 6 Replace B with 10 and h with 6. V = 70 Multiply. The volume is 70 cubic centimeters. Check for Reasonableness Since you underestimated, the answer should be greater than the estimate. 70 > 600 Find the volume of each prism. a. 5 in. b. 5 in. 5 in. 4 ft 10 ft 6 ft 570 Perimeter, Area, and Volume

54 Real-World Link Almost half of the people in the U.S. start their day with a bowl of cereal. PACKAGING A cereal box has the dimensions shown. What is the volume of the cereal box? Estimate = 300 Find the volume. V = lwh V = 8 3 1_ 1_ V = _ 8 1 _ 13 4 _ Replace l with 8, w with 3 1_ 4, and h with 1 1_. 8 in. Write as improper fractions. Then divide by the GCFs. V = _ 35 or 35 Multiply. 1 The volume of the cereal box is 35 cubic inches. Check for Reasonableness in in. c. CONTAINERS A storage container measures 4 inches as its length, 5 inches high, and 8 _ 1 inches wide. Find the volume of the storage container. Find Missing Dimensions Find the height of the rectangular prism. V = lwh Volume of rectangular prism 84 = 6 4 h Replace V with 84, l with 6, and w with 4. 4 m 6 m h 84 = 4 h Multiply = 4 h Divide each side by 4. 4 V = 84 m = h Simplify. The height of the prism is 3.5 meters. Check = 84 Find the missing dimension of each prism. d. V = 44 ft 3, l = ft, w = 4 ft, h =? e. V = 94.5 km 3, l = 7 km, h = 3 km, w =? Lesson 4B Volume and Surface Area of Rectangular Prisms 571

55 Example 1 Find the volume of each prism ft. 5 ft 3 ft cm 8 cm 7 cm Example 3. SINKS A rectangular kitchen sink is 5.5 inches long, inches wide, and 10 inches deep. Find the amount of water that can be contained in the sink. Example 3 Find the missing dimension of each prism h 9 in. 3.7 m 4.5 m V = m 3 14 in. V =,50 in 3 w = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Example 1 Find the volume of each prism in m 10 m 3 m in. 6 in. 5 yd 1 yd 10 yd 9. 7 cm 10. ft m 9.8 m 4 cm 3 cm 5 ft 13 ft 6.3 m Example 1. FISHING A fishing tackle box is 13 inches long, 6 inches wide, and.5 inches high. What is the volume of the tackle box? 13. PETS Find the volume of the pet carrier shown at the right in. Example WATERMELON In Japan, farmers have created watermelons in the shape of rectangular prisms. Find the volume of a prism-shaped watermelon if its length is 10 inches, its width is 8 inches, and its height is 9 inches. 15. Find the length of a rectangular prism having a volume of,830.5 cubic meters, width of 18.5 meters, and height of 9 meters. 16. What is the width of a rectangular prism with a length of 13 feet, volume of 11,3 cubic feet, and height of 36 feet? 0 in in 57 Perimeter, Area, and Volume

56 B Find the missing dimension of each prism in..5 in. l w = 60 in 3 7 mm 5. mm V = 109. mm CANYONS The Palo Duro Canyon is 10 miles long, as much as 0 miles wide, and has a maximum depth of more than 0.15 mile. What is the approximate volume of this canyon? Replace each with <, >, or = to make a true statement ft 3 1 yd m 3 5 yd 3. 7 ft 3 1 yd 3 Real-World Link Palo Duro Canyon State Park in Canyon, Texas, opened on July 4, 1934, and contains 18,438 acres. 3 SAND ART The glass container shown is filled to a height of.5 inches. a. How much sand is currently in the container? b. How much more sand could the container hold before it overflows? c. What percent of the container is filled with sand? 5 in. 3 in. 1 4 in. 4. REASONING Which has the greater volume: a prism with a length of 5 inches, a width of 4 inches, and a height of 10 inches, or a prism with a length of 10 inches, a width of 5 inches, and a height of 4 inches? Justify your selection. 5. TRUCKS Use the table at the right. Inside Dimensions of Moving Trucks a. What is the approximate volume of the small truck? b. The Davis family is moving, and they estimate that they will need a truck with about 1,50 cubic feet. Which truck would be best for them to rent? c. About how many cubic feet greater is the volume of the Mega Moving Truck than the -bedroom moving truck? Truck Length (ft) Width (ft) Van _ Small Truck 11 1_ 13 -Bedroom Moving Truck 3-Bedroom Moving Truck Mega Moving Truck 14 1_ 0 5_ 6 1_ 4 7 5_ 1 7 7_ 1 7 1_ 7 7_ 1 Height (ft) 6 6 3_ 4 7 1_ 6 8 1_ 1 8 5_ 1 6. ESTIMATION Delsin estimates that the volume of a rectangular prism with a length of 5.8 centimeters, a width of 3 centimeters, and a height of 1. centimeters is less than 180 cubic centimeters. Is he correct? Explain. Lesson 4B Volume and Surface Area of Rectangular Prisms 573

57 C 7. FIND THE ERROR Amanda is finding the volume of the rectangular prism. Find her mistake and correct it cm 1.4 cm 6.8 cm V = l + w + h V = V = 41.9 cm 3 8. CHALLENGE Refer to the prism at the right. If all the dimensions of the prism doubled, would the volume double? Explain your reasoning. 9. E WRITE MATH Explain why cubic units are used to measure volume instead of linear units or square units. Test Practice 30. Don used the shoebox to create a home for the toad he caught. 18 in. 9 in. 10 in. Find the volume of the shoebox. A. in 3 C. 1,60 in 3 B. 864 in 3 D. 1,710 in A cereal company is creating a new size box in which to package cereal. The box has a width of 7 centimeters, a base of 7 centimeters, and a volume of 6,46 cubic centimeters. Find the height of the cereal box. F. 34 centimeters G. 38 centimeters H. 4 centimeters I. 46 centimeters 3. TOYS Tiffany is using wooden cubes to make rectangular prisms. If she has exactly 8 wooden cubes, make a model to find the length, width, and height of two possible rectangular prisms. (Lesson 3D) 33. Find the area of the figure 40 mm at the right. (Lesson 3C) 10 mm 50 mm 5 mm 0 mm 574 Perimeter, Area, and Volume

58 Multi-Part Lesson 4 Volume and Surface Area of Rectangular Prisms PART A B C D E Main Idea Find the surface area of rectangular prisms using models and nets. Vocabulary nets Surface Area of Rectangular Prisms CEREAL If you want to know the amount of cereal that can fit in the box, you would find the volume. But if you want to know how much cardboard is needed to make the box, you would find the surface area. Get ConnectED GLE Develop and use formulas for surface area and volume of 3-dimensional figures. SPI Determine the surface area and volume of prisms, pyramids and cylinders. One way to find the surface area is to use a net. Nets are two-dimensional patterns of threedimensional figures. When you construct a net, you are decomposing the three-dimensional figure into separate shapes. Make a Net of a Prism What do you need to find? the surface area of the cardboard box Use a cereal box in the shape of a rectangular prism. Measure and record the length, width, and height of the box. Using a marker, label the top, bottom, front, back, and side faces of the box. Using scissors, carefully cut along three edges of the top face and then cut down each vertical edge. top back side front side the Results 1. What do you notice about the top and bottom faces, the left and right faces, and the front and back faces?. Find the area of each face. Then find the sum of these figures. Lesson 4C Volume and Surface Area of Rectangular Prisms 575

59 Orthogonal drawings consist of separate views of an object taken from different angles. The drawing can be used to create the actual figure. Make a Net from a Drawing Create a rectangular prism from the orthogonal drawing. Use grid paper to draw a net from the orthogonal drawing. Orthogonal Drawing View Drawing Front and Back Sides Top and Bottom top side front side back bottom Fold the net into a threedimensional figure. the Results 3. A rectangular prism has 6 faces. Why does the orthogonal view show only 3 faces? 4. MAKE A CONJECTURE Write a formula for finding the surface area of a rectangular prism. Use your formula to find the surface area of the rectangular prism in Activity. and Apply Draw a net for each figure. Then find the area of the net mm 6 mm 3 in. 6 mm in. 4 in. 3 ft 6 ft ft 8. Create a net and a rectangular prism from the orthagonal drawing. Then find the surface area of the prism. 9. A rectangular prism has a length of 8 centimeters, a width of centimeters, and a height of 3 centimeters. What is the surface area? Orthogonal Drawing View Drawing Front and Back Sides Top and Bottom 576 Perimeter, Area, and Volume

60 Multi-Part Lesson 4 Volume and Surface Area of Rectangular Prisms PART A B C D E Main Idea Find the surface areas of rectangular prisms. Vocabulary surface area Get ConnectED GLE Develop and use formulas for surface area and volume of 3-dimensional figures. SPI Determine the surface area and volume of prisms, pyramids and cylinders. Surface Area of Rectangular Prisms Use 10 centimeter cubes to build four different rectangular prisms. Count the number of squares on the outside of each prism. The total number of squares is the surface area. 1. Copy and complete the table for each prism. Prism Length (cm) Width (cm) Height (cm) Volume (cm 3 ) Surface Area (cm ) Describe the prisms with the greatest and least surface areas. The surface area of a prism is the sum of the areas of its faces. l h back h w l w side bottom front side h top l front and back: top and bottom: two sides: lh + lh = lh lw + lw = lw hw + hw = hw lh + lw + hw Surface Area of a Rectangular Prism Words Symbols The surface area S.A. of a rectangular prism with length l, width w, and height h is the sum of the areas of the faces. S.A. = lh + lw + hw h w Model l Lesson 4D Volume and Surface Area of Rectangular Prisms 577

61 Find the surface area of the rectangular prism. Find the area of each face. front and back: lh = (7)(4) or 56 top and bottom: lw = (7)(5) or 70 left and right sides: hw = (4)(5) or 40 Add to find the surface area. The surface area is or 166 square feet. Find the Surface Area of a Rectangular Prism 7 ft 5 ft top 5 ft bottom 7 ft 7 ft 4 ft 4 ft back side front side 4 ft 5 ft 7 ft 7 ft 5 ft a. Find the surface area of the 4 m 3 m rectangular prism. m GEOLOGY A geode is being sent as a gift. It is packed in a box that measures 7 inches long, 3 inches wide, and 16 inches tall. What is the surface area of the box? S.A. = lh + lw + hw Surface area of a prism Real-World Link A geode is a hollow rock that is lined on the inside with crystal. The largest geode ever found is 30 feet deep and large enough for people to walk through. S.A. = (7)(16) + (7)(3) + (16)(3) l = 7, w = 3, h = 16 S.A. = 14(16) + 14(3) + 3(3) S.A. = S.A. = 36 Multiply. Multiply. Add. The surface area of the box is 36 square inches. b. PAINTING Nadine is going to paint her younger sister s toy chest, including the bottom. What is the approximate surface area that she will paint? 38 in. 19 in. 19 in. 578 Perimeter, Area, and Volume

62 Example 1 Find the surface area of each rectangular prism m ft 5 ft 3. 7 m 6 m 6.5 ft 15 cm 7 cm cm Example 4. VIDEO GAMES A game box for video games is shaped like a rectangular prism. What is the surface area of the game box? 15 cm 11 cm 16 cm = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP. Example 1 Find the surface area of each rectangular prism in. 1 in. 5 in ft 7 ft 3 ft cm 4.3 cm 8.1 cm ft 4 ft 0 ft m 15.1 m 5.5 m mm 5 mm 5 mm Example 11. DISPLAYS Tomás keeps his diecast car in a glass display case as shown. What is the surface area of the glass, including the bottom? 5 in. 1. CAKES A full sheet cake is typically 18 inches by 4 inches by inches. What is the minimum surface area of a rectangular box that will contain the cake? 15 in. 6 in. B 13. ESTIMATION Martina estimates that the surface area of a rectangular prism with a length of 13. feet, a width of 6 feet, and a height of 8 feet is about 460 square feet. Is her estimate reasonable? Explain your reasoning. Lesson 4D Volume and Surface Area of Rectangular Prisms 579

63 Classify each measure as length, area, surface area, or volume. Explain your reasoning. Include an appropriate unit of measure. 14. the amount of water in a lake 15. the amount of land available to build a house 16. the amount of wrapping paper needed to cover a box 17. the height of a tree 18. BIRDS Chrissy is making a bird nesting box for her backyard. a. What is the surface area of the nesting box, including the hole? b. What is the surface area if the width is doubled? c. What is the surface area if the width is half as great? 5.5 in. 9 in. 7.5 in. 19 SHIPPING Find the surface area of each shipping package. Which package has the greater surface area? Does the same package have a greater volume? Explain. 3 in. 8 in. 1 in. 11 in. 6 in. 14 in. C 0. REASONING Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. If all the dimensions of a cube are doubled, the surface area is four times greater. 1. CHALLENGE All of the triangular faces of the figure are congruent. a. What is the area of one of the triangular faces? the square face? b. Use what you know about finding the surface area of a rectangular prism to find the surface area of the square pyramid. 1 in. 8 in. 1 in.. E WRITE MATH What is the relationship between area and surface area? 580 Perimeter, Area, and Volume

64 Test Practice 3. Which net can be used to make and find the surface area of a cube? A. B. 4. Dilip is going to paint a shoebox to use for storage of his trading cards. The shoebox is 3 inches long, 10 inches wide, and 8 inches high. Find the surface area of the shoebox. F. 46 square inches G. 88 square inches H. 988 square inches I. 1,840 square inches C. D. 5. SHORT RESPONSE A company is experimenting with two new boxes for packaging merchandise. Each box is a cube with the side lengths shown. What is the ratio of the surface area of the smaller box to the surface area of the larger box? 1 in. 18 in. Find the volume of each prism. (Lesson 4B) m 5 m 4 m 11 in. 9. POOL Mrs. Norway wants to build a pathway around a circular pool. The pool has a radius of 10 feet. The distance from the center of the pool to the outside edge of the pathway will be 14 feet. Find the area of the pathway. Use 3.14 for π. Round to the nearest tenth. (Lesson 3D) 7 in. 3 in. 13 mm 17.5 mm 9.7 mm 30. Find the area of the composite figure shown. (Lesson 3C) 19 m 5 m 7 m 15 m Lesson 4D Volume and Surface Area of Rectangular Prisms 581

65 in Landscape Architecture Planting in C ircles Do you have an artistic side, and do you enjoy being outdoors? If so, a career in landscape design might be a perfect fit for you. Landscape architects design outside areas such as yards, parks, playgrounds, campuses, shopping centers, and golf courses. Their designed areas are not only meant to be beautiful, but also functional and compatible with the natural environment. A landscape architect must be proficient in mathematics, science, and the use of computer-aided design. Are you interested in a career as a landscape architect? Take some of the following courses in high school. Algebra Architectural Design Botany Get ConnectED 58 Perimeter, Area, and Volume Drafting/Illustrative Design Technology Geometry

66 4m 10 m 3 ft %FTJHO %FTJHO GLE For each problem, use the information in the designs. 1. In Design, what is the radius of the larger grassy area?. The small circular fountain in Design 1 is surrounded by a stone wall. Find the circumference of the wall. Use _ for π Find the circumference of the smaller grassy area in Design. Use 3.14 for π. 4. In Design, how much greater is the lawn area in the larger circle than in the smaller circle? Use 3.14 for π. 5. In Design, the smaller circle is surrounded by a path 1 meter wide. What is the circumference of the path? Use the π key on a calculator and round to the nearest tenth. 6. In Design 1, the area of the large circular patio is about 01.1 square feet. What is the radius of the patio? Round to the nearest foot. Problem Solving in Landscape Architecture 583

67 Chapter Study Guide and Review Be sure the following Key Concepts are noted in your Foldable. Perimeter Area Volume Key Vocabulary base perimeter center pi circle radius circumference rectangular prism composite figure semicircle cubic units surface area diameter three-dimensional figure height volume nets Key Concepts Area (Lesson 1) h b A = bh h b 1 b h b A = _ bh or A = _ 1 bh A = 1 _ h(b 1 + b ) A = πr Circumference and Perimeter (Lessons and 3) r d = r C = πd C = πr h b b P = b + h Volume and Surface Area (Lesson 4) l w h V = lwh V = Bh r h SA = lh + lw + hw Vocabulary Check Choose the correct term or number to complete each sentence. 1. The amount of space that a three-dimensional figure contains is called its (area, volume).. The shortest distance from the base to the opposite side of a parallelogram is called the (height, center). 3. The distance around any closed figure is called its (area, perimeter). 4. In estimating the circumference of a circle, round the value of π to (3, 4). 5. Cubic units are used when calculating (area, volume). 6. The distance around a circle is called the (diameter, circumference). 584 Perimeter, Area, and Volume

68 Multi-Part Lesson Review Lesson 1 Area Area of Parallelograms (Lesson 1A) Find the area of each parallelogram ft m.5 m 3 m 31 ft in. 5 in. 11. DECKS Find the height of a deck if it is a parallelogram with base 8 _ 1 4 feet and an area of 49 _ 1 square feet. 7 in. EXAMPLE 1 Find the area of the parallelogram. A = bh A = 6 5 A = 30 in EXAMPLE 5 in. 6 in. Find the area of a parallelogram with base 4.3 meters and height 11. meters. A = bh A = A = The area is square meters. Area of Triangles (Lesson 1C) Find the area of each triangle m 3 m m 11 in. 18 in. 3 m 16. FLAGS How much material is needed to make a triangular flag with base 1_ 4 feet and height 8 _ 1 feet? EXAMPLE 3 Find the area of the triangle. A = _ 1 bh A = _ 1 (75 50) 50 m A = 1,875 m 75 m EXAMPLE 4 Find the area of a triangular garden with base 8 feet and height 7 feet. A = _ 1 bh A = _ 1 (8)(7) A = _ 1 (56) or 8 The area is 8 square feet. Chapter Study Guide and Review 585

69 Chapter Study Guide and Review Lesson 1 Area (continued) Area of Trapezoids (Lesson 1D) Find the area of each trapezoid m 13 m 1 m in. 0.9 in. 1.5 in. 19. PAINT One wall in Franco s room is in the shape of a trapezoid. The base of the wall at the floor is 11 feet long, and the base at the ceiling is 8 feet. The height of the wall is 7 feet. Find the area of the wall to be painted. EXAMPLE 5 Find the area of the trapezoid. A = _ 1 h(b 1 + b ) 14 in. A = _ 1 15( ) A = _ in. A = 47.5 The area of the trapezoid is 47.5 square inches. 15 in. 0. WINDOWS A window is shaped like a trapezoid. The bases are 30 inches and 40 inches. The height of the window is 4 inches. Find the area of the window. Lesson Circles Circumference (Lesson B) Find the radius or diameter of each circle with the given dimensions. 1. d = 58 cm. r = 7 in. 3. r = 9 ft 4. d = 3 yd Find the circumference _ of each circle. Use 3.14 or for π. Round to the 7 nearest tenth cm 6 m 7. RIDES The circular floor of a carousel has a diameter of 40 feet. Find the circumference of the floor. EXAMPLE 6 Find the radius of a circle with diameter 68 yards. The radius of a circle is half its diameter. So, the radius of a circle with a diameter of 68 yards is _ 1 of 68 yards, or 34 yards. EXAMPLE 7 Find the circumference of the circle at the right. Use 3.14 for π. Round to the nearest tenth. C = πd Circumference of a circle C Replace d with 5. C 15.7 Multiply. The circumference is about 15.7 centimeters. 5 cm 586 Perimeter, Area, and Volume

70 Lesson Circles (continued) Area of Circles (Lesson D) _ Find the area of each circle. Use 3.14 or 7 for π. Round to the nearest tenth m 4 in. 30. WRESTLING A wrestling mat is a square with a circular ring inside. The ring has a radius of 4.5 meters. Find the area within the circle to the nearest tenth. Use 3.14 for π. Find _ the area of each semicircle. Use 3.14 or for π. Round to the nearest tenth yd 14 cm EXAMPLE 8 Find the area of the circle. Use 3.14 or _ for π. Round to the 7 nearest tenth. A = πr Area of a circle A Replace r with 9. A 54.3 Multiply. 9 m The area of the circle is 54.3 square meters. EXAMPLE 9 Find the area 8 cm of the semicircle. _ Use 3.14 or for π. Round to 7 the nearest tenth. A = _ 1 πr Area of a semicircle A _ Replace r with 8. A Multiply. The area of the circle is square centimeters. Lesson 3 Composite Figures Perimeter of Composite Figures (Lesson 3A) Find the perimeter of each figure. Use 3.14 for π. Round to the nearest tenth cm cm 4 cm 175 ft 80 ft 7 cm EXAMPLE 10 Find the 33 m perimeter of the figure. The figure is made up of three sides of a rectangle and a semicircle. Length of semicircle: _ 1 π d _ The perimeter of the figure is or 71.6 meters. 80 m Chapter Study Guide and Review 587

71 Chapter Study Guide and Review Lesson 3 Composite Figures (continued) Area of Composite Figures (Lesson 3C) Find the area of each figure. Use 3.14 for π. Round to the nearest tenth mm 15 mm m 10 mm 13.1 m 37. LANDSCAPING Find the area of the flower garden shown. MA.7.A.1. EXAMPLE 11 Find the area of the figure. Use 3.14 for π. Round to the nearest tenth. 13 in. 15 in. 30 in. The diameter of the semicircle is or 15 inches. 6 ft 38. CONSTRUCTION The side of a four season room is being constructed. Find the area of the side being built of glass windows. 5 ft ft 10 ft 4 ft 8 ft Area of rectangle A = lw Area of semicircle A = _ 1 πr MA.7.A.1. A = A _ A = 390 A 88.3 The area of the figure is or square inches. PSI: Make a Model (Lesson 3D) Solve. Use the make a model strategy. 39. CANS A grocer is stacking cans of tomato soup into a pyramid-shaped display. The bottom layer has 8 cans. There is one less can in each layer and there are 6 layers. How many cans are in the display? 40. BRICKS A bricklayer wants to arrange 16 bricks into a rectangular shape with the greatest perimeter possible. How many bricks will be in each row? EXAMPLE 1 A cheerleading squad formed a pyramid. There were 5 cheerleaders on the bottom and one less cheerleader in each row. How many rows were in the pyramid, if there are 1 cheerleaders? Using 1 cubes, place 5 cubes on the bottom and one less cube in each layer as shown. There are 3 rows. 588 Perimeter, Area, and Volume

72 Lesson 4 Volume and Surface Area of Rectangular Prisms Volume of Rectangular Prisms (Lesson 4B) Find the volume of each figure yd 3 yd 8 yd 4. ft 9 ft 6 ft Find the missing dimension of each rectangular prism in. h V = 5,080 in 3 in. w 4.8 m V = m m 45. BUILDINGS What is the volume of an office building with a length 168 yards, width 115 yards, and height 96 yards? EXAMPLE 13 Find the volume of the figure. V = lwh V = V = in. 4 in. 8 in. The volume is 160 cubic inches. EXAMPLE 14 Find the missing dimension of the rectangular prism. V = lwh 3,40 = l ,40 = l 16 _ 3,40 16 = _ l = l The length is 15 feet. l = 3,40 ft 3 18 ft 1 ft Surface Area of Rectangular Prisms (Lesson 4D) Find the surface area of each rectangular prism. Round to the nearest tenth if necessary cm 6 cm 6 cm 4 yd 6 yd 48. STORAGE A storage trunk is 5 feet long, feet wide, and 4 feet tall. How much wood is needed to make the trunk? 49. CAKE A baker needs to put icing on a rectangular cake. The cake is 14 inches long, 1 inches wide, and 4 inches tall. What is the surface area of the cake, not including the bottom? 8 yd EXAMPLE 15 Find the surface area of the rectangular prism. 5 ft 9 ft ft S.A. = lh + lw + hw S.A. = ()(5) + ()(9) + (5)(9) S.A. = S.A. = 146 The surface area is 146 square feet. Chapter Study Guide and Review 589

73 Practice Chapter Test Find the area of each figure in. 11 in. 6. CRATERS The Meteor Crater is located in Arizona. This crater is circular and has a radius of about 0.4 mile. About how many square miles does this crater cover? Use 3.14 for π. 3. REASONING Find the height of a triangle with a base of 4 millimeters and an area of 66 square millimeters. 4. GARDENING The garden shown is being covered with fertilizer. If one bag of fertilizer covers 5 square meters, how many bags of fertilizer are needed to fertilize the garden? 7. STORAGE A rectangular storage room is 1 feet long by 18 feet wide. Find the number of cubic feet of space the room occupies if it is 9 feet high. 8. GEOMETRY A rectangular prism is made using exactly 1 cubes. Find a possible length, width, and height of the prism. Find the volume and surface area of each figure in cm cm 4 m 15 in. 4 cm 6 m 7 in. 5. MULTIPLE CHOICE The drawing shows two circles that have the same center. 1 in. 4 in. Which expression can be used to find the approximate circumference of the outer circle in inches? A. π(4 + 1) C. π(4 + 1) B. _ 1 (4 + 1) D. (4 + 1) 11. EXTENDED RESPONSE As a treat, Louise fills and freezes orange juice in the tray shown. Each compartment is a rectangular prism with the dimensions shown and labeled below. 4 cm 3 cm 3 cm Part A What is the volume of one compartment? Part B What is the total volume of the tray? 590 Perimeter, Area, and Volume

74 Preparing for Standardized Tests Extended Response: Scoring Points To receive all possible points for an extended-response question, your answer must be correct, and either all work must be shown or an explanation of how you found your answer must be given. A rectangular container of sandwiches is placed in a rectangular cooler. The extra space around the container is filled with ice. Part A What is the volume of ice that will fit in the cooler? Part B Explain in words how you determined the volume. Full Credit Answer 4 in. 5 in. 0 in. 15 in. 14 in. 1 in. Part A V = l w h V 1 = = 3,360 cubic inches V = = 300 cubic inches V 1 - V = 3,360 cubic inches cubic inches = 3,060 cubic inches Part B Volume is found by multiplying length times width times height. Volume of the cooler is 3,360 cubic inches. Volume of the sandwich container is 300 cubic inches. The volume of the ice is found by subtracting the container from the cooler. 3,360 cubic inches cubic inches = 3,060 cubic inches of ice The answer given above would earn full credit. If not all the steps were shown, or if there were errors, partial credit would have been awarded. A rectangular gift box is 18 inches long, 16 inches wide, and 4 inches high. Another gift box has dimensions that are half as long, half as wide, and half as high as the first. Part A How much greater is the volume of the larger gift box than that of the smaller gift box? Part B Explain your answer. If you find that you cannot answer every ery part of an open-ended d question, do as much as you can. You may earn partial credit. Preparing for Standardized Tests 591

75 Test Practice Read each question. Then fill in the correct answer on the answer sheet provided by your teacher or on a sheet of paper. 1. The table below shows the areas of a triangle where the height of the triangle stays the same, but the base changes. Height (units) Area of Triangles Base (units) Area (square units) n Which expression can be used to find the area of a triangle that has a height of 4 units and a base of n units? A. n _ 4 B. 4n _ C. 4_ n D. 4n. GRIDDED RESPONSE José used a square baking pan to make a cake. The length of each side of the pan was 16 inches. Find the area of the pan in square inches. 4. In the spreadsheet below, a formula applied to the values in columns A and B results in the values in column C. What is the formula? A B C A. C = A- B C. C = A + B B. C = A - B D. C = A + B 5. SHORT RESPONSE In Mrs. Tucker s classroom library, there are 168 fiction and 4 nonfiction books. What is the ratio of fiction to nonfiction books in simplest form? 6. Which expression gives the surface area of a rectangular prism with length 5 units, width 8 units, and height 3 units? F. ()(5 ) + ()(8 ) + ()(3 ) G. (5)(8) + (5)(3) + (8)(3) H. (5)(8)(3) I. ()(5)(8 + 3) 7. Ted is making three picture frames like the one shown below. What length of wood does Ted need for all three picture frames? 3. Janet has a circular garden in her front yard with a diameter of 8 feet. How does the diameter d compare to the circumference C of the garden? F. d _ 1 3 C G. d _ 1 C H. d C I. d 3 C in in. A _ in. C. 7 1 _ 4 in. B _ 4 in. D _ 4 in. 59 Perimeter, Area, and Volume

76 8. SHORT RESPONSE Lynette is painting a 15-foot by 10-foot rectangular wall that has a 9-foot by 5-foot rectangular window at its center. How many square feet of wall will she paint? 15 ft 9 ft 5 ft 10 ft 11. GRIDDED RESPONSE The road sign shows the distances from the highway exit to certain businesses. What fraction of a mile is the restaurant from the exit? Restaurant Gas Station Hotel 0.65 mi 0.4 mi 1. mi 9. The cost of renting a car is shown in the advertisement. Which of the following equations can be used to find t, the cost in dollars of the rental for m miles? 1. For every $5 Marta earns mowing lawns, she puts $ in her savings account. How much money will she have to earn in order to deposit $30 into her savings account? F. $6 H. $15 G. $1 I. $ EXTENDED RESPONSE Leora is giftwrapping the box shown. F. t = 0.10m + 5 G. t = H. t = 50(m ) I. t = m 10. The circumference of a circle is centimeters. Find the length of the radius. Use 3.14 for π. A. 1 cm C. 5 cm B. 6 cm D. 3 cm 15 in. 9 in. 3 in. Part A Find the volume of the box. Part B If each dimension is doubled, explain what happens to the volume. Part C If only one dimension is doubled, what happens to the volume? Does it matter which dimension is doubled? Explain. NEED EXTRA HELP? If You Missed Question Go to Chapter-Lesson C 9-1A 9-B 7-1E 3-1B 9-4D -1D 9-3C 7-1E 9-B 4-1A 3-3C 9-4B For help with... GLE 3.4 GLE 3.4 SPI 4.4 SPI 3.6 SPI.6 SPI 4.5 SPI.1 GLE 4.3 SPI 3.6 SPI 4.4 SPI.5 SPI.6 SPI 4.5 Test Practice 593