Finding Areas of Two-Dimensional Figures (6.8.D)

Transcription

1 ircumference and rea ircumference and rc Length reas of ircles and Sectors reas of Polygons and omposite Figures Effects of hanging Dimensions P t ((p. 68) Posters asaltic olumns (p. (p 61) SEE the ig Idea (p 618) Table Top (p. Population Density (p (p. 607) London Eye (p. 603) Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

2 Maintaining Mathematical Proficiency Finding reas of Two-Dimensional Figures (6.8.D) Example 1 Find the area of each figure. a m b in. 8. m 1. in m = 1 h(b 1 + b ) Write area formula. = bh = 1 (8.)( ) Substitute. = (3.35)(1.) = Simplify. =.69 The area is square meters. The area is.69 square inches. Find the area of the figure ft 36 cm 8.8 cm 5 1 yd ft 1.6 cm 5 1 yd 8 Finding a Missing Dimension (.5.) Example rectangle has a perimeter of 10 meters and a length of 3 meters. What is the width of the rectangle? P = + w Write formula for perimeter of a rectangle. 10 = (3) + w Substitute 10 for P and 3 for. 10 = 6 + w Multiply and 3. = w Subtract 6 from each side. = w Divide each side by. The width is meters. Find the missing dimension.. rectangle has a perimeter of 8 inches and a width of 5 inches. What is the length of the rectangle? 5. triangle has an area of 1 square centimeters and a height of 1 centimeters. What is the base of the triangle? 6. rectangle has an area of 8 square feet and a width of 7 feet. What is the length of the rectangle? 7. STRT RESONING How is the area formula for a parallelogram derived from the area formula for a rectangle? 595

3 Mathematical Thinking Solving a Simpler Form of a Problem ore oncept omposite Figures and rea Mathematically profi cient students analyze mathematical relationships to connect and communicate mathematical ideas. (G.1.F) composite figure is a figure that consists of triangles, squares, rectangles, and other two-dimensional figures. To find the area of a composite figure, separate it into figures with areas you know how to find. Then find the sum of the areas of those figures. triangle trapezoid Find the area of the composite figure. Finding the rea of a omposite Figure SOLUTION The figure consists of a parallelogram and a rectangle. rea of parallelogram rea of rectangle = bh = w = (3.1)(10.) = (3.1)(11.7) = 35.6 = cm 3.1 cm 11.7 cm The area of the figure is = square centimeters. Monitoring Progress Find the area of the composite figure. 1. m. 6 in ft 10. m 8 in. 18 in. 5. ft 10. m 6 in. 9.8 ft. ft 596 hapter 11 ircumference and rea

4 11.1 ircumference and rc Length TEXS ESSENTIL KNOWLEDGE ND SKILLS G.1. G.1.D Essential Question How can you find the length of a circular arc? Finding the Length of a ircular rc Work with a partner. Find the length of each red circular arc. a. entire circle b. one-fourth of a circle 5 y 5 y x x c. one-third of a circle d. five-eighths of a circle y y x x Writing a onjecture NLYZING MTHEMTIL RELTIONSHIPS To be proficient in math, you need to notice if calculations are repeated and look both for general methods and for shortcuts. Work with a partner. The rider is attempting to stop with the front tire of the motorcycle in the painted rectangular box for a skills test. The front tire makes exactly one-half additional revolution before stopping. The diameter of the tire is 5 inches. Is the front tire still in contact with the painted box? Explain. ommunicate Your nswer 3. How can you find the length of a circular arc?. motorcycle tire has a diameter of inches. pproximately how many inches does the motorcycle travel when its front tire makes three-fourths of a revolution? 3 ft Section 11.1 ircumference and rc Length 597

5 11.1 Lesson What You Will Learn ore Vocabulary circumference, p. 598 arc length, p. 599 radian, p. 601 Previous circle diameter radius Use the formula for circumference. Use arc lengths to find measures. Solve real-life problems. Measure angles in radians. Using the Formula for ircumference The circumference of a circle is the distance around the circle. onsider a regular polygon inscribed in a circle. s the number of sides increases, the polygon approximates the circle and the ratio of the perimeter of the polygon to the diameter of the circle approaches π For all circles, the ratio of the circumference to the diameter d is the same. This ratio is = π. Solving for yields the formula for the circumference of a circle, d = πd. ecause d = r, you can also write the formula as = π(r) = πr. ore oncept ircumference of a ircle The circumference of a circle is = πd or = πr, where d is the diameter of the circle and r is the radius of the circle. d r = πd = πr USING PREISE MTHEMTIL LNGUGE You have sometimes used 3.1 to approximate the value of π. Throughout this book, you should use the π key on a calculator, then round to the hundredths place unless instructed otherwise. Using the Formula for ircumference Find each indicated measure. a. circumference of a circle with a radius of 9 centimeters b. radius of a circle with a circumference of 6 meters SOLUTION a. = πr = π 9 = 18π The circumference is about centimeters. b. = πr 6 = πr 6 π = r.1 r The radius is about.1 meters. Monitoring Progress Help in English and Spanish at igideasmath.com 1. Find the circumference of a circle with a diameter of 5 inches.. Find the diameter of a circle with a circumference of 17 feet. 598 hapter 11 ircumference and rea

6 Using rc Lengths to Find Measures n arc length is a portion of the circumference of a circle. You can use the measure of the arc (in degrees) to find its length (in linear units). ore oncept rc Length In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360. rc length of = m πr 360, or rc length of = m 360 πr P r Using rc Lengths to Find Measures Find each indicated measure. a. arc length of b. circumference of Z c. m RS 8 cm 60 P Z 0 X.19 in. Y 15.8 m S T R m SOLUTION a. rc length of = 360 π(8) 8.38 cm b. rc length of XY = m XY c. rc length of RS = m RS 360 πr = π(15.8) = m RS = π(15.8) = m RS in. = 165 m RS Monitoring Progress Help in English and Spanish at igideasmath.com Find the indicated measure. 3. arc length of PQ. circumference of N 5. radius of G Q P 9 yd 75 R S L 70 N 61.6 m M E G ft F Section 11.1 ircumference and rc Length 599

7 Solving Real-Life Problems Using ircumference to Find Distance Traveled The dimensions of a car tire are shown. To the nearest foot, how far does the tire travel when it makes 15 revolutions? 5.5 in. OMMON ERROR lways pay attention to units. In Example 3, you need to convert units to get a correct answer. SOLUTION Step 1 Find the diameter of the tire. d = 15 + (5.5) = 6 in. 5.5 in. Step Find the circumference of the tire. = πd = π 6 = 6π in. Step 3 Find the distance the tire travels in 15 revolutions. In one revolution, the tire travels a distance equal to its circumference. In 15 revolutions, the tire travels a distance equal to 15 times its circumference. Distance traveled = Number of revolutions ircumference = 15 6π 15. in. Step Use unit analysis. hange 15. inches to feet. 15. in. 1 ft = 10.1 ft 1 in. The tire travels approximately 10 feet. 15 in. Using rc Length to Find Distances The curves at the ends of the track shown are 180 arcs of circles. The radius of the arc for a runner on the red path shown is 36.8 meters. bout how far does this runner travel to go once around the track? Round to the nearest tenth of a meter m.0 m SOLUTION The path of the runner on the red path is made of two straight sections and two semicircles. To find the total distance, find the sum of the lengths of each part. Distance = Length of each straight section + = (8.39) + ( 1 π 36.8 ) 00.0 Length of each semicircle The runner on the red path travels about 00.0 meters. Monitoring Progress 8.39 m Help in English and Spanish at igideasmath.com 6. car tire has a diameter of 8 inches. How many revolutions does the tire make while traveling 500 feet? 7. In Example, the radius of the arc for a runner on the blue path is.0 meters, as shown in the diagram. bout how far does this runner travel to go once around the track? Round to the nearest tenth of a meter. 600 hapter 11 ircumference and rea

8 Measuring ngles in Radians Recall that in a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360. To see why, consider the diagram. r circle of radius 1 has circumference π, so the arc length of D is m D 360 π. 1 D Recall that all circles are similar and corresponding lengths of similar figures are proportional. ecause m = m D, and D are corresponding arcs. So, you can write the following proportion. rc length of = rc length of r D 1 rc length of = r rc length of D rc length of = r m D 360 π This form of the equation shows that the arc length associated with a central angle is proportional to the radius of the circle. The constant of proportionality, m D 360 π, is defined to be the radian measure of the central angle associated with the arc. In a circle of radius 1, the radian measure of a given central angle can be thought of as the length of the arc associated with the angle. The radian measure of a complete circle (360 ) is exactly π radians, because the circumference of a circle of radius 1 is exactly π. You can use this fact to convert from degree measure to radian measure and vice versa. ore oncept onverting between Degrees and Radians Degrees to radians Multiply degree measure by π radians, or π radians Radians to degrees Multiply radian measure by 360 π radians, or 180 π radians. onverting between Degree and Radian Measure a. onvert 5 to radians. b. onvert 3π SOLUTION a. 5 π radians 180 = π radian b. 3π radians to degrees. radians 180 π radians = 70 So, 5 = π radian. So, 3π radians = 70. Monitoring Progress Help in English and Spanish at igideasmath.com 8. onvert 15 to radians. 9. onvert π 3 radians to degrees. Section 11.1 ircumference and rc Length 601

9 11.1 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and ore oncept heck 1. WRITING Describe the difference between an arc measure and an arc length.. WHIH ONE DOESN T ELONG? Which phrase does not belong with the other three? Explain your reasoning. the distance around a circle π times twice the radius π times the diameter the distance from the center to any point on the circle Monitoring Progress and Modeling with Mathematics In Exercises 3 10, find the indicated measure. (See Examples 1 and.) 3. circumference of a circle with a radius of 6 inches. diameter of a circle with a circumference of 63 feet 5. radius of a circle with a circumference of 8π 1. ERROR NLYSIS Describe and correct the error in finding the length of GH. G 75 5 cm H rc length of GH = m GH πr = 75 π(5) = 750π cm 6. exact circumference of a circle with a diameter of 5 inches 7. arc length of 8 ft P 5 8. m DE D 8.73 in. 9. circumference of 10. radius of R 7.5 m F G cm E Q 10 in. 60 R L M 13. PROLEM SOLVING measuring wheel is used to calculate the length of a path. The diameter of the wheel is 8 inches. The wheel makes 87 complete revolutions along the length of the path. To the nearest foot, how long is the path? (See Example 3.) 1. PROLEM SOLVING The radius of the front wheel of your bicycle is 3.5 centimeters. You ride 0 meters. How many complete revolutions does the front wheel make? In Exercises 15 18, find the perimeter of the shaded region. (See Example.) ERROR NLYSIS Describe and correct the error in finding the circumference of in. = πr = π(9) =18π in hapter 11 ircumference and rea

10 x + y = In Exercises 5 and 6, find the circumference of the circle with the given equation. Write the circumference in terms of π (x + ) + (y 3) = 9 7. USING STRUTURE semicircle has endpoints In Exercises 19, convert the angle measure. (See Example 5). (, 5) and (, 8). Find the arc length of the semicircle. 8. RESONING EF is an arc on a circle with radius r. 19. onvert 70 to radians. Let x be the measure of EF. Describe the effect on the length of EF if you (a) double the radius of the circle, and (b) double the measure of EF. 0. onvert 300 to radians. 11π 1 1. onvert radians to degrees. 9. MKING N RGUMENT Your friend claims that it is π. onvert radian to degrees. 8 possible for two arcs with the same measure to have different arc lengths. Is your friend correct? Explain your reasoning. 3. PROLEM SOLVING The London Eye is a Ferris wheel in London, England, that travels at a speed of 0.6 meter per second. How many minutes does it take the London Eye to complete one full revolution? 67.5 m 30. PROLEM SOLVING Over 000 years ago, the Greek scholar Eratosthenes estimated Earth s circumference by assuming that the Sun s rays were parallel. He chose a day when the Sun shone straight down into a well in the city of Syene. t noon, he measured the angle the Sun s rays made with a vertical stick in the city of lexandria. Eratosthenes assumed that the distance from Syene to lexandria was equal to about 575 miles. Explain how Eratosthenes was able to use this information to estimate Earth s circumference. Then estimate Earth s circumference. t ligh m = 7. lexandria sun stick t ligh sun well. PROLEM SOLVING You are planning to plant a circular garden adjacent to one of the corners of a building, as shown. You can use up to 38 feet of fence to make a border around the garden. What radius can the garden have? hoose all that apply. Explain your reasoning. 1 1 Syene center of Earth Not drawn to scale 31. NLYZING RELTIONSHIPS In, the ratio of the length of PQ to the length of RS is to 1. What is the ratio of PQ to RS? to 1 to 1 1 to D 1 to 3. NLYZING RELTIONSHIPS 5 arc in and a D arc in P have the same length. What is the ratio of the radius r1 of to the radius r of P? Explain your reasoning. Section 11.1 ircumference and rc Length 603

11 33. PROLEM SOLVING How many revolutions does the smaller gear complete during a single revolution of the larger gear? 7 3. USING STRUTURE Find the circumference of each circle. a. a circle circumscribed about a right triangle whose legs are 1 inches and 16 inches long b. a circle circumscribed about a square with a side length of 6 centimeters c. a circle inscribed in an equilateral triangle with a side length of 9 inches 35. REWRITING FORMUL Write a formula in terms of the measure θ (theta) of the central angle (in radians) that can be used to find the length of an arc of a circle. Then use this formula to find the length of an arc of a circle with a radius of inches and a central angle of 3π radians. 36. HOW DO YOU SEE IT? ompare the circumference of P to the length of DE. Explain your reasoning. 37. MKING N RGUMENT In the diagram, the measure of the red shaded angle is 30. The arc length a is. Your classmate claims that it is possible to find the circumference of the blue circle without finding the radius of either circle. Is your classmate correct? Explain your reasoning. r r a D 3 P E 38. MODELING WITH MTHEMTIS What is the measure (in radians) of the angle formed by the hands of a clock at each time? Explain your reasoning. a. 1:30 p.m. b. 3:15 p.m. 39. MTHEMTIL ONNETIONS The sum of the circumferences of circles,, and is 63π. Find. 3x x 0. THOUGHT PROVOKING Is π a rational number? ompare the rational number 355 to π. Find a 113 different rational number that is even closer to π. 1. PROOF The circles in the diagram are concentric and FG GH. Prove that JK and NG have the same length. L K M N F J 5x G H. REPETED RESONING is divided into four congruent segments, and semicircles with radius r are drawn. r a. What is the sum of the four arc lengths? b. What would the sum of the arc lengths be if was divided into 8 congruent segments? 16 congruent segments? n congruent segments? Explain your reasoning. Maintaining Mathematical Proficiency Find the area of the polygon with the given vertices. (Section 1.) Reviewing what you learned in previous grades and lessons 3. X(, ), Y(8, 1), Z(, 1). L( 3, 1), M(, 1), N(, 5), P( 3, 5) 60 hapter 11 ircumference and rea

12 11. reas of ircles and Sectors TEXS ESSENTIL KNOWLEDGE ND SKILLS G.1. Essential Question How can you find the area of a sector of a circle? Finding the rea of a Sector of a ircle Work with a partner. sector of a circle is the region bounded by two radii of the circle and their intercepted arc. Find the area of each shaded circle or sector of a circle. a. entire circle b. one-fourth of a circle 8 y 8 y 8 8 x 8 8 x 8 8 c. seven-eighths of a circle d. two-thirds of a circle y y x x RESONING To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution. Finding the rea of a ircular Sector Work with a partner. center pivot irrigation system consists of 00 meters of sprinkler equipment that rotates around a central pivot point at a rate of once every 3 days to irrigate a circular region with a diameter of 800 meters. Find the area of the sector that is irrigated by this system in one day. ommunicate Your nswer 3. How can you find the area of a sector of a circle?. In Exploration, find the area of the sector that is irrigated in hours. Section 11. reas of ircles and Sectors 605

13 11. Lesson What You Will Learn ore Vocabulary population density, p. 607 sector of a circle, p. 608 Previous circle radius diameter intercepted arc Use the formula for the area of a circle. Use the formula for population density. Find areas of sectors. Use areas of sectors. Using the Formula for the rea of a ircle You can divide a circle into congruent sections and rearrange the sections to form a figure that approximates a parallelogram. Increasing the number of congruent sections increases the figure s resemblance to a parallelogram. The base of the parallelogram that the figure approaches is half of the circumference, so b = 1 = 1 (πr) = πr. The height is the radius, so h = r. So, the area of the parallelogram is = bh = (πr)(r) = πr. ore oncept rea of a ircle The area of a circle is = πr where r is the radius of the circle. r r r 1 = πr = πr 606 hapter 11 ircumference and rea Using the Formula for the rea of a ircle Find each indicated measure. a. area of a circle with a radius of.5 centimeters b. diameter of a circle with an area of square centimeters SOLUTION a. = πr Formula for area of a circle = π (.5) Substitute.5 for r. = 6.5π Simplify Use a calculator. The area of the circle is about square centimeters. b. = πr Formula for area of a circle = πr Substitute for π = r Divide each side by π. 6 r Find the positive square root of each side. The radius is about 6 centimeters, so the diameter is about 1 centimeters. Monitoring Progress 1. Find the area of a circle with a radius of.5 meters. Help in English and Spanish at igideasmath.com. Find the radius of a circle with an area of square feet.

14 Using the Formula for Population Density The population density of a city, county, or state is a measure of how many people live within a given area. number of people Population density = area of land Population density is usually given in terms of square miles but can be expressed using other units, such as city blocks. Using the Formula for Population Density a. bout 30,000 people live in a 5-mile radius of a city s town hall. Find the population density in people per square mile. b. region with a 3-mile radius has a population density of about 6195 people per square mile. Find the number of people who live in the region. SOLUTION a. Step 1 Find the area of the region. = πr = π 5 = 5π The area of the region is 5π 78.5 square miles. Step Find the population density. Population density = number of people area of land = 30,000 5π 575 Formula for population density Substitute. Use a calculator. The population density is about 575 people per square mile. b. Step 1 Find the area of the region. = πr = π 3 = 9π The area of the region is 9π 8.7 square miles. Step Let x represent the number of people who live in the region. Find the value of x. number of people Population density = Formula for population density area of land 6195 x 9π Substitute. 175,159 x Multiply and use a calculator. The number of people who live in the region is about 175,159. Monitoring Progress Help in English and Spanish at igideasmath.com 3. bout 58,000 people live in a region with a -mile radius. Find the population density in people per square mile.. region with a 3-mile radius has a population density of about 1000 people per square mile. Find the number of people who live in the region. Section 11. reas of ircles and Sectors 607

15 NLYZING MTHEMTIL RELTIONSHIPS The area of a sector is a fractional part of the area of a circle. The area of a sector formed by a 5 arc is 5 360, or 1 of the area of 8 the circle. Finding reas of Sectors sector of a circle is the region bounded by two radii of the circle and their intercepted arc. In the diagram below, sector P is bounded by P, P, and. ore oncept rea of a Sector The ratio of the area of a sector of a circle to the area of the whole circle (πr ) is equal to the ratio of the measure of the intercepted arc to 360. rea of sector P πr = m 360, or rea of sector P = m 360 πr P r Finding reas of Sectors Find the areas of the sectors formed by UTV. SOLUTION Step 1 Find the measures of the minor and major arcs. ecause m UTV = 70, m UV = 70 and m USV = = 90. Step Find the areas of the small and large sectors. rea of small sector = m UV 360 πr Formula for area of a sector = π 8 Substitute Use a calculator. rea of large sector = m USV 360 πr Formula for area of a sector S = π 8 Substitute. U T 70 8 in. V Use a calculator. The areas of the small and large sectors are about square inches and about square inches, respectively. Monitoring Progress Find the indicated measure. 5. area of red sector 6. area of blue sector Help in English and Spanish at igideasmath.com F 1 ft 10 G D E 608 hapter 11 ircumference and rea

16 Using reas of Sectors Find the area of V. Using the rea of a Sector V T 0 = 35 m U SOLUTION rea of sector TVU = m TU 360 rea of V Formula for area of a sector 35 = rea of V Substitute. 315 = rea of V Solve for area of V. The area of V is 315 square meters. Finding the rea of a Region rectangular wall has an entrance cut into it. You want to paint the wall. To the nearest square foot, what is the area of the region you need to paint? 16 ft 10 ft 16 ft OMMON ERROR Use the radius (8 feet), not the diameter (16 feet), when you calculate the area of the semicircle. SOLUTION 36 ft The area you need to paint is the area of the rectangle minus the area of the entrance. The entrance can be divided into a semicircle and a square. rea of wall = rea of rectangle (rea of semicircle + rea of square) = 36(6) [ 180 = 936 (3π + 56) (π 8 ) + 16 ] The area you need to paint is about 579 square feet. Monitoring Progress Help in English and Spanish at igideasmath.com 7. Find the area of H. 8. Find the area of the figure. F = 1.37 cm 85 H G 7 m 7 m 9. If you know the area and radius of a sector of a circle, can you find the measure of the intercepted arc? Explain. Section 11. reas of ircles and Sectors 609

17 11. Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and ore oncept heck 1. VOULRY (n) of a circle is the region bounded by two radii of the circle and their intercepted arc.. WRITING The arc measure of a sector in a given circle is doubled. Will the area of the sector also be doubled? Explain your reasoning. Monitoring Progress and Modeling with Mathematics In Exercises 3 10, find the indicated measure. (See Example 1.) In Exercises 15 18, find the areas of the sectors formed by DFE. (See Example 3.) 3. area of. area of 0 in. 0. cm 15. G E 10 in. 60 F D 16. E D F 56 1 cm G 5. area of a circle with a radius of 5 inches 6. area of a circle with a diameter of 16 feet 17. D G 137 F 8 m E 18. G F 75 ft D E 7. radius of a circle with an area of 89 square feet 8. radius of a circle with an area of 380 square inches 9. diameter of a circle with an area of 1.6 square inches 10. diameter of a circle with an area of 676π square centimeters In Exercises 11 1, find the indicated measure. (See Example.) 11. bout 10,000 people live in a region with a 1-mile radius. Find the population density in people per square mile. 1. bout 650,000 people live in a region with a 6-mile radius. Find the population density in people per square mile. 13. region with a -mile radius has a population density of about 6366 people per square mile. Find the number of people who live in the region. 1. bout 79,000 people live in a circular region with a population density of about 513 people per square mile. Find the radius of the region. 19. ERROR NLYSIS Describe and correct the error in finding the area of the circle. = π(1) = 1π 5.39 ft 0. ERROR NLYSIS Describe and correct the error in finding the area of sector XZY when the area of Z is 55 square feet. W Z 1 ft X 115 Y Let n be the area of sector XZY. n 360 = n ft 610 hapter 11 ircumference and rea

18 In Exercises 1 and, the area of the shaded sector is shown. Find the indicated measure. (See Example.) 1. area of M 30. MKING N RGUMENT Your friend claims that if the radius of a circle is doubled, then its area doubles. Is your friend correct? Explain your reasoning. 31. MODELING WITH MTHEMTIS The diagram shows = cm K 50 the area of a lawn covered by a water sprinkler. M J L. radius of M J = 1.36 m M 15 ft 89 L 15 K In Exercises 3 8, find the area of the shaded region. (See Example 5.) 3.. 6m a. What is the area of the lawn that is covered by the sprinkler? b. The water pressure is weakened so that the radius is 1 feet. What is the area of the lawn that will be covered? 0 in. 3. MODELING WITH MTHEMTIS The diagram shows a projected beam of light from a lighthouse. m 0 in ft mi 8 cm 5 in. 7. lighthouse 8. 3m m a. What is the area of water that can be covered by the light from the lighthouse? b. What is the area of land that can be covered by the light from the lighthouse? 9. PROLEM SOLVING The diagram shows the shape of a putting green at a miniature golf course. One part of the green is a sector of a circle. Find the area of the putting green. (3x ) ft 5x ft 33. NLYZING RELTIONSHIPS Look back at the Perimeters of Similar Polygons Theorem (Theorem 8.1) and the reas of Similar Polygons Theorem (Theorem 8.) in Section 8.1. How would you rewrite these theorems to apply to circles? Explain your reasoning. 3. NLYZING RELTIONSHIPS square is inscribed in (x + 1) ft a circle. The same square is also circumscribed about a smaller circle. Draw a diagram that represents this situation. Then find the ratio of the area of the larger circle to the area of the smaller circle. Section 11. reas of ircles and Sectors 611

19 35. ONSTRUTION The table shows how students get to school. Method Percent of students bus 65% walk 5% other 10% a. Explain why a circle graph is appropriate for the data. b. You will represent each method by a sector of a circle graph. Find the central angle to use for each sector. Then construct the graph using a radius of inches. c. Find the area of each sector in your graph. 36. HOW DO YOU SEE IT? The outermost edges of the pattern shown form a square. If you know the dimensions of the outer square, is it possible to compute the total colored area? Explain. 37. STRT RESONING circular pizza with a 1-inch diameter is enough for you and friends. You want to buy pizzas for yourself and 7 friends. 10-inch diameter pizza with one topping costs $6.99 and a 1-inch diameter pizza with one topping costs $1.99. How many 10-inch and 1-inch pizzas should you buy in each situation? Explain. a. You want to spend as little money as possible. b. You want to have three pizzas, each with a different topping, and spend as little money as possible. c. You want to have as much of the thick outer crust as possible. Maintaining Mathematical Proficiency Find the area of the figure. (Skills Review Handbook) in. 18 in. ft. 7 ft 10 ft 38. THOUGHT PROVOKING You know that the area of a circle is πr. Find the formula for the area of an ellipse, shown below. a 39. MULTIPLE REPRESENTTIONS onsider a circle with a radius of 3 inches. a. omplete the table, where x is the measure of the arc and y is the area of the corresponding sector. Round your answers to the nearest tenth. b b x y b. Graph the data in the table. c. Is the relationship between x and y linear? Explain. d. If parts (a) (c) were repeated using a circle with a radius of 5 inches, would the areas in the table change? Would your answer to part (c) change? Explain your reasoning. 0. RITIL THINKING Find the area between the three congruent tangent circles. The radius of each circle is 6 inches. 1. PROOF Semicircles with diameters equal to three sides of a right triangle are drawn, as shown. Prove that the sum of the areas of the two shaded crescents equals the area of the triangle. Reviewing what you learned in previous grades and lessons 13 in in. 3 ft 5 ft a 61 hapter 11 ircumference and rea

20 What Did You Learn? ore Vocabulary circumference, p. 598 arc length, p. 599 radian, p. 601 population density, p. 607 sector of a circle, p. 608 ore oncepts Section 11.1 ircumference of a ircle, p. 598 rc Length, p. 599 onverting between Degrees and Radians, p. 601 Section 11. rea of a ircle, p. 606 Population Density, p. 607 rea of a Sector, p. 608 Mathematical Thinking 1. In Exercise 13 on page 60, why does it matter how many revolutions the wheel makes?. Your friend is confused with Exercise 19 on page 610. What question(s) could you ask your friend to help them figure it out? 3. In Exercise 0 on page 61, explain how you started solving the problem and why you started that way. Study Skills Kinesthetic Learners Incorporate physical activity. ct out a word problem as much as possible. Use props when you can. Solve a word problem on a large whiteboard. The physical action of writing is more kinesthetic when the writing is larger and you can move around while doing it. Make a review card. 613

21 Quiz 1. Find the circumference of a circle with a radius of 7 5 inches. (Section 11.1) 8. Find the radius of a circle with a circumference of 30 meters. (Section 11.1) Find the indicated measure. (Section 11.1) 3. m EF. arc length of QS 5. circumference of N 13.7 m E 7 m G F Q S cm 83 R N L 8 in. M 8 6. onvert 6 to radians and 5π 9 Find the indicated measure. (Section 11.) radians to degrees. (Section 11.1) 7. area of a circle with a diameter of 10 yards 8. radius of a circle with an area of 38.5 square kilometers Use the figure to find the indicated measure. (Section 11.) H 9. area of red sector 10. area of blue sector L 100 J 1 yd K 11. Find the area of. (Section 11.) = 50 mm ft 1 ft 1. You are using one of your school s colors to paint around the shaded region of the basketball court shown. Find the perimeter of the shaded region. (Section 11.1) 3 m 13. The two white congruent circles just fit into the blue circle. What is the area of the blue region? (Section 11.) 1. bout 750,000 people live in a region with a 10-mile radius. (Section 11.) a. Find the population density in people per square mile. b. The same number of people live in a region with a 0-mile radius. Is this population density one-half of the population density you found in part (a)? Explain. 61 hapter 11 ircumference and rea

22 11.3 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.11. G.11. reas of Polygons and omposite Figures Essential Question How can you find the area of a regular polygon? The center of a regular polygon is the center of its circumscribed circle. The distance from the center to any side of a regular polygon is called the apothem of a regular polygon. P apothem P center Finding the rea of a Regular Polygon Work with a partner. Use dynamic geometry software to construct each regular polygon with side lengths of, as shown. Find the apothem and use it to find the area of the polygon. Describe the steps that you used. a. b. 7 6 D 3 E c. E 8 7 D d. F 10 9 E 6 G 8 7 D 5 6 F 3 H Writing a Formula for rea RESONING To be proficient in math, you need to know and flexibly use different properties of operations and objects. Work with a partner. Generalize the steps you used in Exploration 1 to develop a formula for the area of a regular polygon. ommunicate Your nswer 3. How can you find the area of a regular polygon?. Regular pentagon DE has side lengths of 6 meters and an apothem of approximately.13 meters. Find the area of DE. Section 11.3 reas of Polygons and omposite Figures 615

23 11.3 Lesson What You Will Learn ore Vocabulary center of a regular polygon, p. 617 radius of a regular polygon, p. 617 apothem of a regular polygon, p. 617 central angle of a regular polygon, p. 617 Previous rhombus kite Find areas of rhombuses and kites. Find angle measures and areas of regular polygons. Find areas of composite figures. Finding reas of Rhombuses and Kites You can divide a rhombus or kite with diagonals d 1 and d into two congruent triangles with base d 1, height 1 d, and area 1 d 1 ( 1 d ) = 1 d 1 d. So, the area of a rhombus or kite is ( 1 d 1 d ) = 1 d 1 d. d d 1 ore oncept 1 = d 1 d 1 d d 1 = d 1 d 1 d rea of a Rhombus or Kite The area of a rhombus or kite with diagonals d 1 and d is 1 d 1 d. d 1 d d d 1 d 1 Finding the rea of a Rhombus or Kite Find the area of each rhombus or kite. a. b. 8 m 7 cm 6 m 10 cm SOLUTION a. = 1 d 1 d b. = 1 d 1 d = 1 (6)(8) = 1 (10)(7) = = 35 So, the area is square meters. So, the area is 35 square centimeters. Monitoring Progress Help in English and Spanish at igideasmath.com 1. Find the area of a rhombus with diagonals d 1 = feet and d = 5 feet.. Find the area of a kite with diagonals d 1 = 1 inches and d = 9 inches. 616 hapter 11 ircumference and rea

24 center P M Q apothem PQ N radius PN MPN is a central angle. Finding ngle Measures and reas of Regular Polygons The diagram shows a regular polygon inscribed in a circle. The center of a regular polygon and the radius of a regular polygon are the center and the radius of its circumscribed circle. The distance from the center to any side of a regular polygon is called the apothem of a regular polygon. The apothem is the height to the base of an isosceles triangle that has two radii as legs. The word apothem refers to a segment as well as a length. For a given regular polygon, think of an apothem as a segment and the apothem as a length. central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices of the polygon. To find the measure of each central angle, divide 360 by the number of sides. Finding ngle Measures in a Regular Polygon NLYZING MTHEMTIL RELTIONSHIPS FG is an altitude of an isosceles triangle, so it is also a median and angle bisector of the isosceles triangle. In the diagram, DE is a regular pentagon inscribed in F. Find each angle measure. a. m F b. m FG c. m GF SOLUTION a. F is a central angle, so m F = = 7. E b. FG is an apothem, which makes it an altitude of isosceles F. So, FG bisects F and m FG = 1 m F = 36. c. y the Triangle Sum Theorem (Theorem 5.1), the sum of the angle measures of right GF is 180. So, m GF = = 5. G F D X Q Y Monitoring Progress In the diagram, WXYZ is a square inscribed in P. Help in English and Spanish at igideasmath.com P 3. Identify the center, a radius, an apothem, and a central angle of the polygon. W Z. Find m XPY, m XPQ, and m PXQ. REDING In this book, a point shown inside a regular polygon marks the center of the circle that can be circumscribed about the polygon. You can find the area of any regular n-gon by dividing it into congruent triangles. = rea of one triangle Number of triangles = ( 1 s a ) n ase of triangle is s and height of triangle is a. Number of triangles is n. = 1 a (n s) ommutative and ssociative Properties of Multiplication = 1 a P There are n congruent sides of length s, so perimeter P is n s. ore oncept rea of a Regular Polygon The area of a regular n-gon with side length s is one-half the product of the apothem a and the perimeter P. = 1 ap, or = 1 a ns a a s s Section 11.3 reas of Polygons and omposite Figures 617

25 Finding the rea of a Regular Polygon L M K J regular nonagon is inscribed in a circle with a radius of units. Find the area of the nonagon. SOLUTION The measure of central JLK is 360, or 0. pothem LM bisects the central angle, 9 so m KLM is 0. To find the lengths of the legs, use trigonometric ratios for right KLM. sin 0 = MK LK sin 0 = MK cos 0 = LM LK cos 0 = LM sin 0 = MK cos 0 = LM J L M 0 K The regular nonagon has side length s = (MK) = ( sin 0 ) = 8 sin 0, and apothem a = LM = cos 0. So, the area is = 1 a ns = 1 ( cos 0 ) (9)(8 sin 0 ) 6.3 square units. Finding the rea of a Regular Polygon You are decorating the top of a table by covering it with small ceramic tiles. The tabletop is a regular octagon with 15-inch sides and a radius of about 19.6 inches. What is the area you are covering? R 19.6 in. 15 in. SOLUTION P Q Step 1 Find the perimeter P of the tabletop. n octagon R has 8 sides, so P = 8(15) = 10 inches. Step Find the apothem a. The apothem is height RS of PQR in. ecause PQR is isosceles, altitude RS bisects QP. So, QS = 1 (QP) = 1 P S Q (15) = 7.5 inches. 7.5 in. To find RS, use the Pythagorean Theorem (Theorem 9.1) for RQS. a = RS = = Step 3 Find the area of the tabletop. = 1 ap = 1 ( ) (10) The area you are covering with tiles is about square inches. Monitoring Progress Find the area of the regular polygon. Help in English and Spanish at igideasmath.com hapter 11 ircumference and rea

26 Finding reas of omposite Figures Finding reas of omposite Figures Find the area of each composite figure. Round your answers to the nearest hundredth, if necessary. a. 1.6 in. 1 in. 1.5 in. b. 7 cm 60 cm 7 cm c. 6 ft 7 ft 3 ft 0.8 in. in. ft SOLUTION a. The composite figure consists of a regular hexagon and a trapezoid. The length of one side of the hexagon is (0.8) = 1.6 inches. The apothem is (1.6) (0.8) = inches. rea of composite figure = rea of regular hexagon + rea of trapezoid = 1 ( 1.9 )(6)(1.6) The area is about 8.90 square inches. b. The composite figure consists of a sector and a triangle. (1.5)(1 + ) rea of composite figure = rea of sector + rea of triangle = 360 π (7)() The area is about square centimeters. c. The composite figure consists of a kite and a parallelogram. rea of composite figure = rea of kite + rea of parallelogram = 33 The area is 33 square feet. = 1 (7)(6) + 3() Monitoring Progress 7. Find the area of the composite figure. Round your answer to the nearest hundredth, if necessary. Help in English and Spanish at igideasmath.com 3 ft 5 6 ft 7 ft Section 11.3 reas of Polygons and omposite Figures 619

27 11.3 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and ore oncept heck 1. WRITING Explain how to find the measure of a central angle of a regular polygon.. DIFFERENT WORDS, SME QUESTION Which is different? Find both answers. Find the radius of F. Find the apothem of polygon DE. Find F. Find the radius of polygon DE. G E F 6.8 D Monitoring Progress and Modeling with Mathematics In Exercises 3 6, find the area of the kite or rhombus. (See Example 1.) In Exercises 15 18, find the given angle measure for regular octagon DEFGH. (See Example.) 15. m GJH 16. m GJK 17. m KGJ 18. m EJH H K G F J E D In Exercises 7 10, use the diagram. 7. Identify the center of polygon JKLMN. 8. Identify a central angle of polygon JKLMN. Q N M 5 J P 6 K In Exercises 19, find the area of the regular polygon. (See Examples 3 and.) What is the radius of polygon JKLMN? L 3. an octagon with a radius of 11 units 10. What is the apothem of polygon JKLMN? In Exercises 11 1, find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree, if necessary sides sides 13. sides 1. 7 sides. a pentagon with an apothem of 5 units 5. ERROR NLYSIS Describe and correct the error in finding the area of the kite = 1 (3.6)(5.) = 9.7 So, the area of the kite is 9.7 square units. 60 hapter 11 ircumference and rea

28 6. ERROR NLYSIS Describe and correct the error in finding the area of the regular hexagon s = = 1 ns 1 = 9.5 So, the area of the hexagon is about 9.5 square units. In Exercises 7 30, find the area of the composite figure. (See Example 5.) MODELING WITH MTHEMTIS watch has a circular surface on a background that is a regular octagon. Find the area of the octagon. Then find the area of the silver border around the circular face. 0. cm 1 cm RITIL THINKING In Exercises 33 35, tell whether the statement is true or false. Explain your reasoning. 33. The area of a regular n-gon of a fixed radius r increases as n increases. 9 ft ft 3. The apothem of a regular polygon is always less than the radius ft 35. The radius of a regular polygon is always less than the side length m 36. RESONING Predict which figure has the greatest area and which has the least area. Explain your reasoning. heck by finding the area of each figure in. 9 in. 15 in. 13 in. 9 in cm cm 10 cm 5.1 cm 6.6 cm 15 in. 18 in. 31. MODELING WITH MTHEMTIS asaltic columns are geological formations that result from rapidly cooling lava. Giant s auseway in Ireland contains many hexagonal basaltic columns. Suppose the top of one of the columns is in the shape of a regular hexagon with a radius of 8 inches. Find the area of the top of the column to the nearest square inch. MTHEMTIL ONNETIONS In Exercises 37 and 38, write and solve an equation to find the indicated length(s). Round decimal answers to the nearest tenth. 37. The area of a kite is 3 square inches. One diagonal is twice as long as the other diagonal. Find the length of each diagonal. 38. One diagonal of a rhombus is four times the length of the other diagonal. The area of the rhombus is 98 square feet. Find the length of each diagonal. 39. USING EQUTIONS Find the area of a regular pentagon inscribed in a circle whose equation is given by (x ) + (y + ) = 5. Section 11.3 reas of Polygons and omposite Figures 61

29 0. HOW DO YOU SEE IT? Explain how to find the area of the regular hexagon by dividing the hexagon into equilateral triangles. Z U Y 1. RESONING The perimeter of a regular nonagon, or 9-gon, is 18 inches. Is this enough information to find the area? If so, find the area and explain your reasoning. If not, explain why not.. MKING N RGUMENT Your friend claims that it is possible to find the area of any rhombus if you only know the perimeter of the rhombus. Is your friend correct? Explain your reasoning. 3. PROOF Prove that the area of any quadrilateral with perpendicular diagonals is = 1 d 1 d, where d 1 and d are the lengths of the diagonals. P Q S T d. USING STRUTURE In the figure, an equilateral triangle lies inside a square inside a regular pentagon inside a regular hexagon. Find the approximate area of the entire shaded region to the nearest whole number. V X R W d 1 6. RITIL THINKING The area of a dodecagon, or 1-gon, is 10 square inches. Find the apothem of the polygon. 7. REWRITING FORMUL Rewrite the formula for the area of a rhombus for the special case of a square with side length s. Show that this is the same as the formula for the area of a square, = s. 8. REWRITING FORMUL Use the formula for the area of a regular polygon to show that the area of an equilateral triangle can be found by using the formula = 1 s 3, where s is the side length. 9. OMPRING METHODS Find the area of regular pentagon DE by using the formula = 1 ap, or = 1 a ns. Then find the area by adding the areas of smaller polygons. heck that both methods yield the same area. Which method do you prefer? Explain your reasoning. E 5 D 50. THOUGHT PROVOKING The area of a regular n-gon is given by = 1 ap. s n approaches infinity, what does the n-gon approach? What does P approach? What does a approach? What can you conclude from your three answers? Explain your reasoning. P 5. RITIL THINKING The area of a regular pentagon is 7 square centimeters. Find the length of one side USING STRUTURE Two regular polygons both have n sides. One of the polygons is inscribed in, and the other is circumscribed about, a circle of radius r. Find the area between the two polygons in terms of n and r. Maintaining Mathematical Proficiency Find the perimeter and the area of the figure m 1 m 53. Reviewing what you learned in previous grades and lessons (Skills Review Handbook) 9 cm 5. cm 8 yd 15 yd 10 yd 6 hapter 11 ircumference and rea

30 11. Effects of hanging Dimensions TEXS ESSENTIL KNOWLEDGE ND SKILLS G.10. Essential Question How does changing one or more dimensions of a rectangle affect its perimeter and area? hanging One Dimension MKING MTHEMTIL RGUMENTS To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. Work with a partner. a. Fold a piece of paper in half twice so that there are four layers. b. Draw a rectangle on the paper. Then use scissors to cut through the four layers so that you cut out four congruent rectangles. c. Place two rectangles side-by-side along either the length or the width so that you form a figure with double the length or double the width of a single rectangle. d. ompare the perimeter and the area of the figure formed by the two rectangles to the perimeter and the area of a single rectangle. e. Make a conjecture about how doubling the length or the width of a rectangle affects the perimeter and the area. f. Make a conjecture about how multiplying the length or the width of a rectangle by a positive number k affects the perimeter and the area. hanging Dimensions Proportionally Work with a partner. Use the rectangles from Exploration 1. a. rrange the four rectangles so that you form a rectangle with double the length and double the width of a single rectangle. b. ompare the perimeter and the area of the figure formed by the four rectangles to the perimeter and the area of a single rectangle. c. Make a conjecture about how doubling the length and the width of a rectangle affects the perimeter and the area. d. Make a conjecture about how multiplying the length and the width of a rectangle by a positive number k affects the perimeter and the area. ommunicate Your nswer 3. How does changing one or more dimensions of a rectangle affect its perimeter and area? Section 11. Effects of hanging Dimensions 63

31 11. Lesson What You Will Learn ore Vocabulary Previous perimeter area similar figures Describe the effects of non-proportional dimension changes. Describe the effects of proportional dimension changes. hanging Dimensions Non-Proportionally When you change one or more dimensions of a figure, you also change the perimeter and the area of the figure. hanging One Dimension Describe how the change affects the perimeter and the area of the figure. a. doubling the height b. multiplying the length by 1 3 cm 5 ft 3 cm 1 ft NLYZING MTHEMTIL RELTIONSHIPS Notice that when one dimension is multiplied by k, the area is multiplied by k. original = 1 bh new = 1 b(kh) = k ( 1 bh ) = k original SOLUTION a. b. efore change fter change Dimensions b = 3 cm, h = cm b = 3 cm, h = 8 cm Perimeter P = sum of side lengths = = 1 cm P = sum of side lengths = = cm rea = 1 bh = 1 (3)() = 6 cm = 1 bh = 1 (3)(8) = 1 cm Doubling the height increases the perimeter by = centimeters and increases the area by a factor of 1 6 =. efore change fter change Dimensions = 1 ft, w = 5 ft = ft, w = 5 ft Perimeter P = + w = (1) + (5) = 3 ft P = + w = () + (5) = 18 ft rea = w = 1(5) = 60 ft = w = (5) = 0 ft Multiplying the length by 1 decreases the perimeter by 3 18 = 16 feet and 3 decreases the area by a factor of 0 60 = 1 3. Monitoring Progress Help in English and Spanish at igideasmath.com 1. Describe how multiplying the width by 1 affects the perimeter and the area of the rectangle. 8 ft 10 ft 6 hapter 11 ircumference and rea

32 hanging Dimensions Non-Proportionally Describe how the change affects the perimeter and the area of the figure. a. doubling the length b. multiplying the base by 1 and tripling the width and doubling the height 6 m 8 in. NLYZING MTHEMTIL RELTIONSHIPS Notice that when one dimension is multiplied by j and another dimension is multiplied by k, the area is multiplied by jk. original = bh new = (jb)(kh) = jk(bh) = jk original 7 m SOLUTION a. b. efore change 6 in. fter change Dimensions = 7 m, w = 6 m = 1 m, w = 18 m Perimeter P = + w = (7) + (6) = 6 m P = + w = (1) + (18) = 6 m rea = w = 7(6) = m = w = 1(18) = 5 m Doubling the length and tripling the width increases the perimeter by 6 6 = 38 meters and increases the area by a factor of 5 = 6. efore change fter change Dimensions b = 6 in., h = 8 in. b = 3 in., h = 16 in. Perimeter P = sum of side lengths = = in. P = sum of side lengths = = in. rea = 1 bh = 1 (6)(8) = in. = 1 bh = 1 (3)(16) = in. Multiplying the base by 1 and doubling the height increases the perimeter by inches and does not change the area. Monitoring Progress Help in English and Spanish at igideasmath.com Describe how the change affects the perimeter and the area of the figure.. multiplying the length by 3. doubling the base and and the width by 1 tripling the height 7 m 5 ft 16 m 1 ft Section 11. Effects of hanging Dimensions 65

33 hanging Dimensions Proportionally When you change all the dimensions of a figure proportionally, the resulting figure is similar to the original figure. ore oncept hanging Dimensions Proportionally When you multiply all the linear dimensions of a figure by a positive number k, the perimeter and the area change as shown. efore multiplying all dimensions by k fter multiplying all dimensions by k Perimeter P kp rea k hanging Dimensions Proportionally Describe how doubling all the linear dimensions affects the perimeter and the area of the parallelogram. m 5 m 10 m SOLUTION efore change fter change Dimensions b = 10 m, h = m b = 0 m, h = 8 m Perimeter P = sum of side lengths = (10) + (5) kp = (30) = 60 m = 30 m rea = bh = 10() = 0 m k = (0) = (0) = 160 m Doubling all the linear dimensions of the parallelogram doubles the perimeter and increases the area by a factor of. Monitoring Progress Help in English and Spanish at igideasmath.com. Describe how multiplying all the linear dimensions by 1 affects the perimeter and the area of the triangle. 1 ft 1 ft 66 hapter 11 ircumference and rea

34 11. Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and ore oncept heck 1. VOULRY What is the difference between changing the linear dimensions of a figure non-proportionally and proportionally?. OMPLETE THE SENTENE When you change all the dimensions of a figure proportionally, the resulting figure is to the original figure. Monitoring Progress and Modeling with Mathematics In Exercises 3 10, describe how the change affects the perimeter and the area of the figure. (See Examples 1 3.) 3. doubling the base 7. doubling all the linear dimensions 8 yd 10 yd 18 yd 9 ft 1 ft 8. multiplying all the linear dimensions by 1 13 m. multiplying the width by 1 10 m 16 in. m 0 in. 5. multiplying the length by and tripling the width 9. tripling all the linear dimensions 8 ft 9 in. 1 in. 6. tripling the base and multiplying the height by multiplying all the linear dimensions by m cm 7 cm Section 11. Effects of hanging Dimensions 67

35 11. ERROR NLYSIS Describe and correct the error in finding the perimeter and the area of the rectangle when the width is multiplied by 1. cm 8 cm P original = 0 cm P new = 1 (0) = 10 cm original = 16 cm new = ( 1 ) (16) = cm 1. ERROR NLYSIS Describe and correct the error in finding the perimeter and the area of the triangle when all the linear dimensions are doubled. 15 in. 8 in. P original = 0 in. P new = (0) = 80 in. original = 60 in. new = (60) = 10 in. 13. MKING N RGUMENT You and your friend are making posters for a school dance. Your friend claims that doubling the length and the width of the posters will double their areas. Is your friend correct? Explain your reasoning. 1. HOW DO YOU SEE IT? Describe the relationship between the areas of D and EFD. D F E 15. MODELING WITH MTHEMTIS Your patio is 6 feet long and feet wide. Explain two different ways you can change the dimensions of your patio to double its area. 16. NLYZING RELTIONSHIPS You multiply the height of a triangle by a positive number k, where k 1. Describe how you can change the base so that the area of the triangle is the same as the area of the original triangle. 17. NLYZING RELTIONSHIPS Describe how the change affects the circumference and the area of the circle. a. doubling the radius 9 ft b. multiplying the radius by 1 3 c. squaring the radius 18. THOUGHT PROVOKING The perimeter of rectangle EFGH is k times the perimeter of rectangle D and the area of EFGH is k times the area of D. an you be certain that D and EFGH are always similar? Explain. D H G Perimeter = P rea = E F Perimeter = kp rea = k Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Determine whether the figure has line symmetry, rotational symmetry, both, or neither. If the figure has line symmetry, determine the number of lines of symmetry. If the figure has rotational symmetry, describe any rotations that map the figure onto itself. (Section. and Section.3) hapter 11 ircumference and rea

36 What Did You Learn? ore Vocabulary center of a regular polygon, p. 617 radius of a regular polygon, p. 617 apothem of a regular polygon, p. 617 central angle of a regular polygon, p. 617 ore oncepts Section 11.3 rea of a Rhombus or Kite, p. 616 rea of a Regular Polygon, p. 617 rea of a omposite Figure, p. 619 Section 11. hanging Dimensions Non-Proportionally, p. 6 hanging Dimensions Proportionally, p. 66 Mathematical Thinking 1. In Exercise 50 on page 6, what conjecture did you make about the shape the n-gon approaches? What logical progression led you to determine whether your conjecture was correct?. What was the first step in the process you used to solve Exercise 15 on page 68? Why did you begin with this step? The art department at your school has decided to replace a broken window with an art project. Each color of glass has a different price. The principal asks your class to calculate the cost. an the school afford the window? To explore the answer to this question and more, go to igideasmath.com. Performance Task Window Design 69

37 11 hapter Review 11.1 ircumference and rc Length (pp ) The arc length of QR is 6.5 feet. Find the radius of P. rc length of QR = m QR Formula for arc length πr πr = Substitute. P 75 Q R 6.5 ft 6.5(360) = 75(πr) ross Products Property 5.00 r Solve for r. The radius of P is about 5 feet. Find the indicated measure. 1. diameter of P. circumference of F 3. arc length of P = 9. ft F 115 G in. 5.5 cm H. mountain bike tire has a diameter of 6 inches. To the nearest foot, how far does the tire travel when it makes 3 revolutions? 11. reas of ircles and Sectors (pp ) Find the area of sector D. rea of sector D = m 360 πr Formula for area of a sector = Substitute Use a calculator. The area of the small sector is about square meters. 10 m D 80 Find the area of the blue shaded region W V U T 9 in in. in. 7. T S 50 R = 7.93 ft Q 630 hapter 11 ircumference and rea

38 11.3 reas of Polygons and omposite Figures (pp ) regular hexagon is inscribed in H. Find (a) m EHG, and (b) the area of the hexagon. F H G 16 E a. FHE is a central angle, so m FHE = 360 = 60. pothem GH bisects FHE. 6 So, m EHG = 30. b. ecause EHG is a triangle, GE = 1 HE = 8 and GH = 3 GE = 8 3. So, s = (GE) = 16 and a = GH = 8 3. The area is = 1 a ns = 1 ( 8 3 ) (6)(16) square units. D Find the area of the kite or rhombus Find the area of the regular polygon Find the area of the composite figure in m 7 in. 1 in. 9 in. m 7 in. 8 m ft 7 ft 8 ft 17. platter is in the shape of a regular octagon with an apothem of 6 inches. Find the area of the platter. hapter 11 hapter Review 631

39 11. Effects of hanging Dimensions (pp ) Describe how the change affects the perimeter and the area of the figure. a. multiplying the base by and multiplying the height by 5 3 in. in. efore change fter change Dimensions b = in., h = 3 in. b = 8 in., h = 15 in. Perimeter P = sum of side lengths = = 1 in. P = sum of side lengths = = 0 in. rea = w = 1 (3)() = 6 in. = w = 1 (8)(15) = 60 in. Multiplying the base by 3 and multiplying the height by 5 increases the perimeter by 0 1 = 8 inches and increases the area by a factor of 60 6 = 10. b. multiplying all the linear dimensions by 1 6 ft 18 ft efore change fter change Dimensions = 18 ft, w = 6 ft = 9 ft, w = 3 ft Perimeter P = + w = (18) + (6) = 8 ft kp = 1 (8) = ft rea = w = 18(6) = 108 ft k = ( 1 ) (108) = 1 (108) = 7 ft Multiplying all the linear dimensions by 1 decreases the perimeter by a factor of 1 and decreases the area by a factor of 1. Describe how the change affects the perimeter and the area of the figure. 18. multiplying the base by multiplying the length by 0. multiplying all the 1 and the width by 6 linear dimensions by 5 1 cm 5 cm 9 m 3 5 in. 5 m in. 3 in. 63 hapter 11 ircumference and rea

40 11 hapter Test Find the area of the composite figure mm yd 13 mm 8 mm 5. cm 8 yd 1 yd 15 mm 9 cm.8 cm.8 cm Find the indicated measure.. circumference of F 5. m GH 6. area of shaded sector 6 in. E 10 F D J 7 ft G H 35 ft T Q S in. R 7. One diagonal of a rhombus is three times as long as the other diagonal. The area of the rhombus is 108 square inches. Find the length of each diagonal. Find the area of the regular polygon. 8. a hexagon with an apothem of 9 centimeters 9. a nonagon (9-gon) with a radius of 1 meter Describe how the change affects the perimeter and the area of the rectangle. 10. multiplying the width by 3 15 cm 11. multiplying the length by 1 and the width by 5 7 cm 1. The area of a circular pond is about 138,656 square feet. You are going to walk around the entire edge of the pond. bout how far will you walk? Round your answer to the nearest foot. 13. You want to make two wooden trivets, a large one and a small one. oth trivets will be shaped like regular pentagons. The perimeter of the small trivet is 15 inches, and the perimeter of the large trivet is 5 inches. Find the area of the small trivet. Then use this area to find the area of the large trivet. 1. In general, a cardboard fan with a greater area does a better job of moving air and cooling you. The fan shown is a sector of a cardboard circle. nother fan has a radius of 6 centimeters and an intercepted arc of 150. Which fan does a better job of cooling you? 10 9 cm hapter 11 hapter Test 633