-7 What You ll Learn To find the areas of circles, sectors, and segments of circles... nd Why To compare the area of different-size pizzas, as in Example reas of ircles and Sectors heck Skills You ll Need G for Help Lesson -6. What is the radius of a circle with diameter 9 cm? 4.5 cm. What is the diameter of a circle with radius? 6 ft. Find the circumference of a circle with diameter in. π or about 7.7 in. 4. Find the circumference of a circle with radius m. 6π or about 8.8 m New Vocabulary sector of a circle segment of a circle -7. Plan bjectives To find the areas of circles, sectors, and segments of circles Examples Real-World onnection Finding the rea of a Sector of a ircle Finding the rea of a Segment of a ircle Finding reas of ircles and Parts of ircles Hands-n ctivity: Exploring the rea of a ircle Use a compass to draw a large circle. Fold the circle horizontally and vertically. ut the circle into four wedges on the fold lines. Fold each wedge into quarters. ut each wedge on the fold lines. You will have 6 wedges. Math ackground reas of circles and sectors can be estimated without using the constant p. ne estimation technique measures the areas of inscribed and circumscribed polygons much as lower and upper sums estimate Riemann integrals. alculating geometric probability may require calculating the areas of sectors and circles. More Math ackground: p. 5D Lesson Planning and Tape the wedges to a piece of paper to form the figure shown here. r See p. 5E for a list of the resources that support this lesson.. How does the area of the figure compare with area of the circle? They are equal.. The base of the figure is formed by arcs of the circle. Explain how the length b relates to the circumference of the circle. b N. Explain how the length b relates to the radius r of the circle. b N πr 4. If you increase the number of wedges, the figure you create becomes more and more like a rectangle with base b and height r. Write an expression for the area of the rectangle in terms of r. πr b PowerPoint ell Ringer Practice heck Skills You ll Need For intervention, direct students to: Finding ircumference Lesson -6: Example 4 Extra Skills, Word Problems, Proof Practice, h. Lesson -7 reas of ircles and Sectors 575 Special Needs L fter Example, have students contact a local pizza parlor and find out the diameters and prices of a small and large pizza. Have them determine which is a better buy. learning style: tactile elow Level L Encourage students to label parts when finding areas within circles. For example, label the segment I and the triangle II in Example. learning style: visual 575
. Teach Guided Instruction Hands-n ctivity Discuss as a class why base b is approximately but not exactly equal to pr. Students should mention that a segment is shorter than a curved path between the same two points. onnection to alculus The ctivity s method of finding the area of a curved region is further developed in the study of calculus. Math Tip Discuss the fact that the medium pizza has a percent greater radius but a 44 percent greater area than the small pizza. Have students verify these figures and explain the difference in the percents of increase. Teaching Tip sk: How is finding the area of a sector like finding arc length? oth have the ratio measure of the arc 6 as a factor. EXMPLE EXMPLE EXMPLE Tactile Learners Students may cut apart segments of a circle and relate the areas to the area of the circle. X Y Quick heck In the diagrams on the preceding page, area of a circle area of a parallelogram < b? r <? r pr, and the approximations improve as the circle is cut into more pieces. Key oncepts Theorem - rea of a ircle The area of a circle is the product of p and the square of the radius. pr EXMPLE Real-World onnection Food How much more pizza is in a -in.-diameter pizza than in a -in. pizza? radius of small pizza 5 Find the radii. radius of medium pizza 6 area of small pizza p(5) 5p Use the formula for area of a circle. area of medium pizza p(6) 6p difference in area 6p - 5p p < 4.55759 Use a calculator. There is about 5 in. more pizza in the medium pizza. How much more pizza is in a 4-in.-diameter pizza than in a -in. pizza? about 4 in. sector of a circle is a region bounded by an arc of the circle and the two radii to the arc s endpoints.you name a sector using one arc endpoint, the center of the circle, and the other arc endpoint. The slice of pizza at the left is sector XY of a circle. The area of a sector is a fractional part of the area of a circle. The ratio of a sector s measure of the arc area to a circle s area is 6. Key oncepts Theorem - rea of a Sector of a ircle The area of a sector of a circle is the product of r measure of the arc the ratio 6 and the area of the circle. rea of sector m 6? pr r EXMPLE Finding the rea of a Sector of a ircle Quick heck 576 hapter rea Find the area of sector ZM. Leave your answer in terms of p. area of sector ZM mzm? pr Z 6 7 7 6? p() 8p The area of sector ZM is 8p cm. cm M ritical Thinking circle has a diameter of cm. What is the area of a sector bounded by a 88 major arc? Round your answer to the nearest tenth. 8.5 cm 576 dvanced Learners L4 Following Example, have students investigate how to make a pizza with twice the area of another pizza. learning style: verbal English Language Learners ELL Help students distinguish between segment of a circle and line segments. Then have students compare finding the length of an arc of a circle with finding the area of a sector of a circle. learning style: verbal
part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle. To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed. Segment of a circle PowerPoint dditional Examples circular archery target has a -ft diameter. It is yellow except for a red bull s-eye at the center with a 6-in. diameter. Find the area of the yellow region. Round to the nearest whole number. 44 in. This pizza is cut into two segments. rea of sector rea of triangle rea of segment Find the area of sector. Leave your answer in terms of p. Quick heck Example (page 576) EXMPLE Finding the rea of a Segment of a ircle Find the area of the shaded segment. Round your answer to the nearest tenth. area of sector m? pr Use the formula 6 for area of a sector. 9 6? p() Substitute. in. 4? p 5p area of bh Use the formula for area of a triangle. ()() Substitute. 5 area of segment 5p - 5 < 8.5986 Use a calculator. The area of the segment is about 8.5 in. circle has a radius of cm. Find the area of the smaller segment of the circle determined by a 68 arc. Round your answer to the nearest tenth.. cm EXERISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving G Practice by Example for Help Find the area of each circle. Leave your answer in terms of π.... 4. 6 m cm.7 ft 9π m.5π cm.75π ft π 9 in. 5. griculture Some farmers use a circular irrigation method. n irrigation arm acts as the radius of an irrigation circle. How much more land is covered with an irrigation arm of ft than by an irrigation arm of 5 ft? about 86,94 ft in. 6. What is the difference in the areas of a circular table with diameter 6 ft and a circular table with diameter? about ft π m Find the area of the shaded segment. Round your answer to the nearest tenth. 6 m 5. Daily Notetaking Guide -7 L Daily Notetaking Guide -7 dapted Instruction L losure circle with a diameter of in. contains an inscribed square as shown below. Find the area of the shaded segments. Leave your answer in terms of p. (5π 5) in. 4 ft Lesson -7 reas of ircles and Sectors 577 577
. Practice ssignment Guide -4 hallenge 5-4 Test Prep 4-4 Mixed Review 44-48 Homework Quick heck To check students understanding of key skills and concepts, go over Exercises 8, 8, 6, 9,. Error Prevention! Exercises 4 Some students may substitute d for r in the formula pr. Use these exercises to identify and correct the error. Diversity Exercise 4 Point out that the bow of a canoe is its front end. GPS Guided Problem Solving Enrichment Reteaching dapted Practice Practice Name lass Date Practice -7 Find the surface area of each sphere. Round your answers to the nearest tenth. Surface reas and Volumes of Spheres L L4 L L L Example (page 576) Example (page 577) pply Your Skills. (4π ± 6) ft. (54π ±.5 ") cm 4. (π ± 6 ") m 5. (4 π) ft 6. (64 6π) ft 7. (784 96π) in. Find the area of each shaded sector of a circle. Leave your answer in terms of π. 7. 8. 9. 45 6 cm 8 yd 4.5π yd 64π cm 69π 6 m... 6 cm in. π in. π ft 56π cm 45 4 ft Find the area of sector TP in ( using the given information. Leave your answer in terms of π. 5π. r 5 m, m 98 m π TP 4. r 6 ft, m 5 4 TP 5. d 6 in., m 58 4π in. PT 6. d 5 cm, m PT 88 8.5π cm Find the area of each shaded segment. Round your answer to the nearest tenth. 7. 8. 9. 6 m 6 cm. cm 8. ft. m circle has the given radius. Find the area of the smaller segment of the circle determined by an arc with the given measure. Round to the nearest tenth.. radius 5 m, arc 68.4 m. radius 4 cm, arc 8.4 cm Find the area of the shaded region. Leave your answer in terms of π and in simplest radical form. 7. See left... 4. 5. 6. 7. ft 9 cm 6 m 6 6 m 4 in.... 7 m 4 in. 4. 5. 6. ft m Find the volume of each sphere. Round your answers to the nearest tenth. 7. 8. 9. 4 mi 4 cm... cm S.. 9,79 cm The volume of each sphere is given. Find the surface area. Round your answers to the nearest whole number. cm 7 m m S.. 45,4 yd Pearson Education, Inc. ll rights reserved. G nline Homework Help Visit: PHSchool.com Web ode: aue-7 8. Multiple hoice The diver at the right is collecting samples from the ocean floor. The line to the diver is ft long, and the diver is working at a depth of. What is the approximate area of the circle that the diver can cover?, ft 5,4 ft,4 ft 5,4 ft ft. V ft 4. V 75 m 5. V 45 cm Use the given circumference to approximate the volume of each object. Round your answers to the nearest whole number. 6. a baseball with 4 cm 7. a basketball with 75 cm 8. a volleyball with 69 cm 9. a golf ball with.5 cm 578 hapter rea 578 9. Lower outside; the lower inside and top pieces have base areas 8π in., but the lower outside pieces have base areas 8.75π in..
Real-World onnection areers Training as a pastry chef is part of the background of many executive chefs. hallenge 8. lue region; let x. rea of blue 4x πx ; area of yellow πx x. π 9. 5" units 4a. Problem Solving Hint For Exercise 9, where must T and U lie in a diagram of (T and (U? oat path ft ft ft ft 4b. Find the area of 4 of a circle of radius and add 4 of a circle of radius. lesson quiz, PHSchool.com, Web ode: aua-7 9. Writing The merican Institute of aking GPS suggests a technique for cutting and serving a tiered cake. The tiers of a cake have the same height and have radii 8 in. and in. The top tier and the cake directly under it are each cut into 8 wedges as shown. The outer ring of the -inch tier is cut into pieces. Which would be larger, a piece from the top or a piece from the outer ring? Explain. See margin, p. 578.. How many circles with radius 4 in. will have the same total area as a circle with radius in.? 9 circles. Games dart board has diameter in. and is divided into congruent sectors. Find the area of one sector. Round your answer to the nearest tenth. 5.7 in.. In a circle, a 98 sector has area 6p in.. What is the circle s radius? in.. pen-ended Draw a circle and a sector so that the area of the sector is 6p cm. Give the radius of the circle and the measure of the arc of the sector. See margin. 4. method for finding the area of a segment determined by a minor arc is described on page 577. 4a. See margin. a. Describe two ways to find the area of a segment determined by a major arc. b. If m 9 in a circle of radius, find the areas of the two segments determined by. 5π 5; 75π ± 5 Find the area of the shaded region. Leave your answer in terms of π. 5. 6. 7. 7 m m (49π 7.5 ") m ( 5π) m 4π m 8. ircle at the right is inscribed in square D and circumscribed about square PQRS. Which is smaller, P Q the blue region or the yellow region? Explain. 9. ircles T and U each have a radius and TU. Find S R the area of the region that is contained inside both circles. D 4. Recreation n -by- ft floating dock is anchored in the middle of a pond. The bow of a canoe is tied to a corner of the dock with a -ft rope as shown in the picture below. a. Sketch a diagram of the region in which the bow of the canoe can travel. b. Write a plan for finding the area. 4a b. See left. c. Find the area. Round your answer to the nearest square foot. 9 ft 4 m Lesson -7 reas of ircles and Sectors 579 4. ssess & Reteach Lesson Quiz Solve.. park contains two circular playgrounds. ne has a diameter of 6 m, and the other has a diameter of 4 m. How much greater is the area of the larger playground? Round to the nearest whole number. 57 m. circle has an 8-in. radius. Find the area of a sector whose arc measures 5. Leave your answer in terms of p. 4π in. For Exercises and 4, find the area of the shaded segment. Round to the nearest whole unit.. 4. PowerPoint 5 cm 8 in. 6 cm in. lternative ssessment Have student partners plan how to sell pizzas. Using the area formulas from this lesson, they should select two different sizes and price them proportionately. Have students write a paragraph explaining how the size of each pizza justifies its price.. nswers may vary. Sample: 8 cm radius; 98 arc 8 cm 4. a. nswers may vary. Sample: Subtract the minor arc segment area from the area of the circle, or add the areas of the major sector and k formed. 579
Test Prep For additional practice with a variety of test item formats: Standardized Test Prep, p. 59 Test-Taking Strategies, p. 588 Test-Taking Strategies with Transparencies Test Prep Multiple hoice Short Response 4. circle has area 7p yd. What is the area of a sector of the circle?. p yd. p yd. 4p yd D. 6p yd 4. sector of 9 has area p mm. What is the area of the circle? G F. p mm G. 4p mm H. 8p mm J. 4 mm 4. Each of three water sprinklers covers a m semicircle of radius m. The shaded m region remains dry. Find the area of the shaded region to the nearest square meter. Show your work or explain how you found the area. See margin. Use this heckpoint Quiz to check students understanding of the skills and concepts of Lessons -4 through -7. Grab & Go heckpoint Quiz G Mixed Review for Help Lesson -6 Find the length of in each circle. Leave your answers in terms of π. 44. 45. 46. 6 cm 7 5 m 6 in. 4 Lesson 6-5 Lesson 4-7 π cm π m 8π in. 47. Three sides of a trapezoid are congruent. The fourth side is 4 in. longer than each of the other three. The perimeter is 49 in. Find the length of each side. 4 in., 4 in., 4 in., 54 in. 48. Write a proof. (Hint: First prove D overlapping triangles are congruent.) F Given: >, & > & See margin. Prove: #DF > #EF E heckpoint Quiz Lessons -4 through -7 4. [] area of rectangle? 4 area of semicircles?( π? ) 6π 4 6π N 5. m [] no work shown 48. Show that ke kd by S. Then use E D and segment subtraction to show that D E. Now with vertical lfd lfe, kdf kef by S. 6. π in. 7. 7π m 8. (6π ) cm 9..5π ft 58 hapter rea The similarity ratio of k to kdef is : 5. Fill in the missing information.. The perimeter of # is 6 in.. The area of # is 9. 45 in. The perimeter of #DEF is 9. 6 in. The area of #DEF is 5 in... The areas of two similar triangles are.44 and.. Find their similarity ratio. 4. Find the area of # if a 4 ft, b 7 ft and m&. 4. 6 : 5 5. regular -gon has perimeter 4 cm. Find its area to the nearest tenth. 44.8 cm Find the area of each shaded region. Leave answers in terms of π. 6. 7. 8. 9. in. 9 m 8 cm 9π. In a circle of radius 8 mm, m 45. Find the length of in terms of p. mm 6 ft 5 58