Areas of Regular Polygons and Circles

Transcription

1 reas of Regular Polygons and ircles Find areas of regular polygons. Find areas of circles. Vocabulary apotem can you find te area of a polygon? Te foundations of most gazebos are saped like regular exagons. Suppose te owners of tis gazebo would like to install tile on te floor. If tiles are sold in square feet, ow can tey find out te actual area of tiles needed to cover te floor? Look Back To review apotem, see Lesson 1-4. RS OF RULR POLYONS In regular exagon BF inscribed in circle, and F are radii from te center of te circle to two vertices of te exagon. H is drawn from te center of te regular polygon perpendicular to a side of te polygon. Tis segment is called an apotem. Triangle F is an isosceles triangle, since te radii are congruent. If all of te radii were drawn, tey would separate te exagon into 6 nonoverlapping congruent isosceles triangles. B H F Te area of te exagon can be determined by adding te areas of te triangles. Since H is perpendicular to F, it is an altitude of F. Let a represent te lengt of H and let s represent te lengt of a side of te exagon. rea of F 1 b s B H a F 1 sa Te area of one triangle is 1 sa square units. So te area of te exagon is 6 1 sa square units. Notice tat te perimeter P of te exagon is 6s units. We can substitute P for 6s in te area formula. So, 6 1 sa becomes 1 Pa. Tis formula can be used for te area of any regular polygon. rea of a Regular Polygon If a regular polygon as an area of square units, a perimeter of P units, and an apotem of a units, ten 1 Pa. 610 apter 11 reas of Polygons and ircles R. ilbert/h. rmstrong Roberts

2 xample 1 rea of a Regular Polygon Find te area of a regular pentagon wit a perimeter of 40 centimeters. K potem: Te central angles of a regular pentagon are all congruent. Terefore, te measure of eac angle is 36 0 or 7. P Q is an apotem of pentagon JKLMN. 5 It bisects NPM and is a perpendicular bisector J P N Q M L Problem Solving Tere is anoter metod for finding te apotem of a regular polygon. You can use te Interior ngle Sum Teorem to find m PMQ and ten write a trigonometric ratio to find PQ. of N M. So, m MPQ 1 (7) or 36. Since te perimeter is 40 centimeters, eac side is 8 centimeters and QM 4 centimeters. Write a trigonometric ratio to find te lengt of P Q. tan MPQ Q M PQ lengt of opposite side tan lengt of adjacent side tan 36 P 4Q m MPQ 36, QM 4 (PQ) tan 36 4 Multiply eac side by PQ. 4 PQ ivide eac side by tan 36. tan 36 PQ 5.5 Use a calculator. rea: 1 Pa rea of a regular polygon 1 (40)(5.5) 110 P 40, a 5.5 Simplify. So, te area of te pentagon is about 110 square centimeters. RS OF IRLS You can use a calculator to elp derive te formula for te area of a circle from te areas of regular polygons. rea of a ircle ollect ata Suppose eac regular polygon is inscribed in a circle of radius r. 1. opy and complete te following table. Round to te nearest undredt. Inscribed Polygon Number of Sides Measure of a Side Measure of potem rea r 1.18r 0.77r 0.6r 0.31r 0.16r 0.5r 0.81r 0.9r 0.95r 0.99r 0.998r nalyze te ata. Wat appens to te appearance of te polygon as te number of sides increases? 3. Wat appens to te areas as te number of sides increases? 4. Make a conjecture about te formula for te area of a circle. Lesson 11-3 reas of Regular Polygons and ircles 611

3 You can see from te eometry ctivity tat te more sides a regular polygon as, te more closely it resembles a circle. If a circle as an area of square units and a radius of r units, ten r r rea of a ircle Square Yards square yard measures 36 inces by 36 inces or 196 square inces. xample Use rea of a ircle to Solve a Real-World Problem SWIN caterer as a 48-inc diameter table tat is 34 inces tall. Se wants a tableclot tat will touc te floor. Find te area of te tableclot in square yards. Te diameter of te table is 48 inces, and te tableclot must extend 34 inces in eac direction. So te diameter of te tableclot is or 116 inces. ivide by to find tat te radius is 58 inces. r (58) 10,568.3 rea of a circle Substitution Use a calculator. 48 in. 34 in. Te area of te tableclot is 10,568.3 square inces. To convert to square yards, divide by 196. Te area of te tableclot is 8. square yards to te nearest tent. Look Back To review inscribed and circumscribed polygons, see Lesson You can use te properties of circles and regular polygons to find te areas of inscribed and circumscribed polygons. xample 3 rea of an Inscribed Polygon Find te area of te saded region. ssume tat te triangle is equilateral. Te area of te saded region is te difference between te area of te circle and te area of te triangle. First, find te area of te circle. r (4) 50.3 rea of a circle Substitution Use a calculator. To find te area of te triangle, use properties of triangles. First, find te lengt of te base. Te ypotenuse of B is 4, so B is 3. Since (B), 4 3. Next, find te eigt of te triangle, B. Since m B is 60, B 3 3 or 6. Use te formula to find te area of te triangle. 1 b rea of a triangle (6) b 4 3, Use a calculator. 4 m B 4 B 3 Te area of te saded region is or 9.5 square meters to te nearest tent. 61 apter 11 reas of Polygons and ircles

4 oncept eck uided Practice 1. xplain ow to derive te formula for te area of a regular polygon.. OPN N escribe a metod for finding te base or eigt of a rigt triangle given one acute angle and te lengt of one side. Find te area of eac polygon. Round to te nearest tent. 3. a regular exagon wit a perimeter of 4 yards 4. a regular nonagon wit a perimeter of 108 meters Find te area of eac saded region. ssume tat all polygons tat appear to be regular are regular. Round to te nearest tent cm 3 in. pplication 7. UPHOLSTRY Tyra wants to cover te cusions of er papasan cair wit a different fabric. If tere are seven circular cusions tat are te same size wit a diameter of 1 inces, around a center cusion wit a diameter of 0 inces, find te area of fabric in square yards tat se will need to cover bot sides of te cusions. llow an extra 3 inces of fabric around eac cusion. Practice and pply For xercises 8 13, 6, , , 5, 8 31 See xamples xtra Practice See page Find te area of eac polygon. Round to te nearest tent. 8. a regular octagon wit a perimeter of 7 inces 9. a square wit a perimeter of 84 meters 10. a square wit apotem lengt of 1 centimeters 11. a regular exagon wit apotem lengt of 4 inces 1. a regular triangle wit side lengt of 15.5 inces 13. a regular octagon wit side lengt of 10 kilometers Find te area of eac saded region. ssume tat all polygons tat appear to be regular are regular. Round to te nearest tent in Lesson 11-3 reas of Regular Polygons and ircles 613 ristie's Images

5 3. LBR circle is inscribed in a square, wic is circumscribed by anoter circle. If te diagonal of te square is x, find te ratio of te area of te large circle to te area of te small circle. 4. K bakery sells single-layer mini-cakes tat are 3 inces in diameter for $4 eac. Tey also ave a 9-inc cake for $15. If bot cakes are te same tickness, wic option gives you more cake for te money, nine mini-cakes or one 9-inc cake? xplain. 5. PIZZ pizza sop sells 8-inc pizzas for $5 and 16-inc pizzas for $10. Wic would give you more pizza, two 8-inc pizzas or one 16-inc pizza? xplain. OORINT OMTRY Te coordinates of te vertices of a regular polygon are given. Find te area of eac polygon to te nearest tent. 6. T(0, 0), U( 7, 7), V(0, 14), W(7, 7) 7. ( 1, 0), H 0, 4 3, J 0, J(5, 0), K(.5,.5 ), L(0, 5), M(.5,.5 ), N( 5, 0), P(.5,.5 ), Q(0, 5), R(.5,.5 ) 9. (, ), B(0, 4), (, ), (4, 0), (, ), F(0, 4), (, ), H( 4, 0) Find te area of eac circle. Round to te nearest tent. 30. = = = = 91.4 SWIMMIN POOL For xercises 34 and 35, use te following information. Te area of a circular pool is approximately 7850 square feet. Te owner wants to replace te tiling at te edge of te pool. 34. Te edging is 6 inces wide, so se plans to use 6-inc square tiles to form a continuous inner edge. How many tiles will se need to purcase? 35. Once te square tiles are in place around te pool, tere will be extra space between te tiles. Wat sape of tile will best fill tis space? How many tiles of tis sape sould se purcase? 6 in. Look Back To review circle graps, see Lesson 10-. VITION For xercises 36 38, refer to te circle grap. 36. Suppose te radius of te circle on te grap is 1.3 centimeters. Find te area of te circle on te grap. 37. Francesca wants to use tis circle grap for a presentation. Se wants te circle to use as muc space on a by 8 seet of poster board as possible. Find te area of te circle. 38. RITIL THINKIN Make a conjecture about ow you could determine te area of te region representing te pilots wo are certified to fly private airplanes. US TOY Snapsots Wo s licensed to fly Te Federal viation dministration issued 65,581 pilot certificates in 000. More tan 50,000 were private pilot certificates. Most common certificate types issued in 000: Oter 1 3% Students 15% ommercial airlines 19% Private airplanes 40% irline transport 3% 1 Includes recreational, rotocraft, glider, ligter tan air. Source: F; Bureau of Transportation By William Risser and Frank Pompa, US TOY 614 apter 11 reas of Polygons and ircles

6 Find te area of eac saded region. Round to te nearest tent RNS For xercises 45 47, use te following information. Te lizabet Park Rose arden in Hartford, onnecticut, was designed wit a gazebo surrounded by two concentric rose garden plots. Wide pats emanate from te center, dividing te garden into square and circular sections. 175 ft rose plots 60 ft 40 ft 0 ft 175 ft gazebo rcitecture Tis structure is a pagoda. Pagodas are caracterized by aving several exagonal or octagonal stories eac topped wit a curved roof. In tis temple, te sizes of te floors are in te ratio 1 : 3 : 5 : 7. Source: Find te area and perimeter of te entire Rose arden. Round to te nearest tent. 46. Wat is te total of te circumferences of te tree concentric circles formed by te gazebo and te two circular rose garden plots? (Ignore te widt of te rose plots and te widt of te pats.) 47. ac rose plot as a widt of 5 feet. Wat is te area of te pat between te outer two complete circles of rose garden plots? 48. RHITTUR Te nraku-ji Temple in Japan is composed of four octagonal floors of different sizes tat are separated by four octagonal roofs of different sizes. Refer to te information at te left. etermine weter te areas of eac of te four floors are in te same ratio as teir sizes. xplain. SIMILR FIURS For xercises 49 54, use te following information. Polygons FHJK and VWXUZ are similar regular pentagons. X 49. Find te scale factor. H 50. Find te perimeter of eac pentagon. W 51. ompare te ratio of te perimeters of te pentagons to te scale factor. J 5. Find te area of eac pentagon. F 4. cm K V 6.3 cm 53. ompare te ratio of te areas of te pentagons to te scale factor. 54. ompare te ratio of te areas of te pentagons to te ratio of te perimeters of te pentagons. Lesson 11-3 reas of Regular Polygons and ircles 615 Z U

7 55. RITIL THINKIN circle inscribes one regular exagon and circumscribes anoter. If te radius of te circle is 10 units long, find te ratio of te area of te smaller exagon to te area of te larger exagon. 56. WRITIN IN MTH nswer te question tat was posed at te beginning of te lesson. How can you find te area of a polygon? Include te following in your answer: information needed about te gazebo floor to find te area, and ow to find te number of tiles needed to cover te floor. Standardized Test Practice 57. square is inscribed in a circle of area 18 square units. Find te lengt of a side of te square. 3 units B 6 units 3 units 6 units 58. LBR Te average of x numbers is 15. If te sum of te x numbers is 90, wat is te value of x? 5 B Maintain Your Skills Mixed Review Find te area of eac quadrilateral. (Lesson 11-) cm 10 cm 13 cm 10 cm 7 m 16 m 6 m 70 yd 70 yd OORINT OMTRY iven te coordinates of te vertices of a quadrilateral, determine weter it is a square, a rectangle, or a parallelogram. Ten find te area of te quadrilateral. (Lesson 11-1) 6. ( 3, ), B(4, ), (, 1), ( 5, 1) 63. F(4, 1), (4, 5), H(, 5), J(, 1) 64. K( 1, 3), L(, 5), M(1, 5), N(, 3) 65. P(5, 7), Q( 1, 7), R( 1, ), S(5, ) Refer to trapezoid F wit median H. (Lesson 8-6) 66. Find F. 67. Let W X be te median of H. Find WX. 68. Let Y Z be te median of HF. Find YZ. H F etting Ready for te Next Lesson PRRQUISIT SKILL Find. (To review special rigt triangles, see Lesson 7-3.) apter 11 reas of Polygons and ircles