9.3 Surface Area of Pyramids
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1 Page 1 of Suface Aea of Pyamids and Cones Goa Find the suface aeas of pyamids and cones. Key Wods pyamid height of a pyamid sant height of a pyamid cone height of a cone sant height of a cone The Rainfoest Pyamid on Gaveston Isand in Texas is one exampe of how pyamids ae used in achitectue. The base of a pyamid is a poygon and the atea faces ae tianges with a common vetex. Height This geenhouse is home to pants, buttefies, and bats. Sant height height vetex sant height base The height of a pyamid is the pependicua distance between the vetex and base. The sant height of a pyamid, epesented by the ette, is the height of any of its atea faces. Visuaize It! EXAMPLE 1 Find the Sant Height Find the sant height of the Rainfoest Pyamid. Round you answe to the neaest whoe numbe. 100 ft Fo a squae pyamid, the sant height is the ength of the hypotenuse of a ight tiange fomed by the height of the pyamid and haf of the ength of the base. Soution To find the sant height, use the ight tiange fomed by the height and haf of the base. (sant height) 2 (height) side ,000 10,000 sant height 2 0, ANSWER The sant height is about 141 feet. Use the Pythagoean Theoem. Substitute 100 fo height and 200 fo base side ength. Simpify. 200 ft Take the positive squae oot. Use a cacuato. 200 ft 9.3 Suface Aea of Pyamids and Cones 491
2 Page 2 of 9 Suface Aea of a Pyamid The diagams show the suface aea of a pyamid with a squae base. Suface aea aea of base aea of atea faces B 2 1 P is the height of a tiangua side. s s B s s P 4s Student Hep STUDY TIP A egua pyamid has a egua poygon fo a base. The sant height,, is the same on a of the atea faces of a egua pyamid. SURFACE AREA OF A PYRAMID Wods Suface aea (aea of base) 1 (peimete of base)(sant height) 2 Symbos S B 1 2 P B EXAMPLE 2 Find Suface Aea of a Pyamid Find the suface aea of the pyamid. Student Hep SKILLS REVIEW Fo hep with aea and peimete, see p Soution 4 ft 1 Find the aea of the base. B B 2 Find the peimete of the base. P Find the sant height. (sant height) 2 (height) 2 ( 1 2 side)2 Use the Pythagoean Theoem Substitute. Haf of 6 is Simpify powes. 25 Simpify. sant height Take positive squae oot. 4 Substitute vaues into the fomua fo suface aea of a pyamid. S B 1 2 P (24)(5) 96 Wite the fomua fo suface aea. Substitute. Simpify. ANSWER The suface aea of the pyamid is 96 squae feet. 492 Chapte 9 Suface Aea and Voume
3 Page 3 of 9 Find Suface Aea of a Pyamid TFind the suface aea of the pyamid cm 9 in. 12 ft 7 in. 7 in. 9 cm B 35.1 cm 2 10 ft 10 ft Suface Aea of a Cone A cone has a cicua base and a vetex that is not in the same pane as the base. In a ight cone, the height meets the base at its cente. In this esson, ony ight cones ae shown. vetex height The vetex is diecty above the cente of the cice. sant height base atea suface The height of a cone is the pependicua distance between the vetex and the base. The sant height of a cone is the distance between the vetex and a point on the base edge. Student Hep LOOK BACK To eview how to find the aea of a secto, see p The diagams show the suface aea of a cone. Suface aea aea of base aea of secto B π B SURFACE AREA OF A CONE Wods Suface aea (aea of base) (aea of secto) (aea of base) π(adius of base)(sant height) Symbos S B π π 2 π B π Suface Aea of Pyamids and Cones 493
4 Page 4 of 9 IStudent Hep I C L A S S Z O N E. C O M MORE EXAMPLES Moe exampes at casszone.com EXAMPLE 3 Find Suface Aea of a Cone Find the suface aea of the cone to the neaest whoe numbe. a. b. 4 in. 5 ft 6 in. 12 ft Soution a. The adius of the base is 4 inches and the sant height is 6 inches. S π 2 π Wite the fomua fo suface aea of a cone. π(4) 2 π(4)(6) Substitute 4 fo and 6 fo. 40π Simpify 16π 24π. 126 Mutipy. ANSWER The suface aea is about 126 squae inches. b. Fist find the sant height. (sant height) 2 2 h 2 sant height Use the Pythagoean Theoem. (12) 2 (5) 2 Substitute 12 fo and 5 fo h. 169 Simpify Find the positive squae oot. Simpify. Next substitute 12 fo and 13 fo in the fomua fo suface aea. S π 2 π π(12) 2 π(12)(13) Wite the fomua fo suface aea. Substitute. 300π Simpify 144π 156π. 942 Mutipy. ANSWER The suface aea is about 942 squae feet. Find Suface Aea of Cones Find the suface aea of the cone to the neaest whoe numbe in. 8 ft 4 cm 8 ft 3 cm 494 Chapte 9 Suface Aea and Voume
5 Page 5 of Execises Guided Pactice Vocabuay Check Compete the statement using height o sant height. 1. The ed ine segment is the? of the pyamid. Ski Check 2. The bue ine segment is the? of the pyamid. 3. The height of the atea faces of the pyamid is the?. 4. The? of a pyamid is the pependicua distance between the vetex and base. 5. Find the sant height of the pyamid shown at the ight. 12 in. Find the suface aea of the soid m 5 in. 10 ft 3 m B 3.9 m 2 4 in. Pactice and Appications Exta Pactice See p Recognizing Sant Height Te whethe the ed ine segment is the height o the sant height Homewok Hep Exampe 1: Exs Exampe 2: Exs Exampe 3: Exs Finding Sant Height Find the sant height of the soid m 18 mm 12 m 12 mm 12 m 15 in. 8 in. 9.3 Suface Aea of Pyamids and Cones 495
6 Page 6 of 9 Suface Aea of a Pyamid In Execises 15 20, find the suface aea of the pyamid in. 8 m B 21.2 in cm 15 cm 24 cm 5 m 5 m 7 in mm yd B 1.7 yd 2 32 mm 32 mm 24 cm 2 yd 14 cm 14 cm 21. Logica Reasoning Expain why the sant height of a pyamid must be geate than the height of the pyamid. 22. Eo Anaysis Jamie is tying to find the suface aea of the pyamid beow. His soution is shown. What did he do wong in his cacuations? S (160)(15) m 2 40 m 15 m 40 m Suface Aea of a Cone Find the suface aea of the cone. Round you answe to the neaest whoe numbe m 22 m 10 ft 14 m 9 m 4 ft yd cm 10 cm 40 yd 24 in. 496 Chapte 9 Suface Aea and Voume
7 Page 7 of 9 Finding Latea Aea Find the atea aea of the object P 28 cm 14 cm 4.3 in in. d 8 in. 14 in. IStudent Hep I C L A S S Z O N E. C O M HOMEWORK HELP Exta hep with pobem soving in Exs is at casszone.com Visuaize It! Sketch the descibed soid and find its suface aea. If necessay, ound you answe to the neaest whoe numbe. 32. A pyamid has a squae base with conguent edges of 12 metes and a height of 8 metes. 33. A pyamid has a tiangua base with conguent edges of 8 feet and a sant height of 13 feet. The base aea is 27.7 squae feet. 34. A cone has a diamete of 6 yads and a sant height of 7 yads. 35. A cone has a adius of 10 inches and a height of 14 inches. Using Nets Name the soid that can be foded fom the net. Then find its suface aea ft 2 cm 6 cm Caees 5 ft Veteinay Medicine A cone-shaped coa, caed an Eizabethan coa, is used to pevent pets fom aggavating a heaing wound. 12 in. 3 in. 4 in. Diagam of a coa VETERINARIANS povide teatment fo sick and injued animas. They aso offe peventive cae, such as vaccinations, fo heathy animas. Caee Links C L A S S Z O N E. C O M 38. Find the atea aea of the entie cone shown above. 39. Find the atea aea of the sma cone that has a adius of 3 inches and a height of 4 inches. 40. Use you answes to Execises 38 and 39 to find the amount of mateia needed to make the Eizabethan coa shown. 9.3 Suface Aea of Pyamids and Cones 497
8 Page 8 of 9 Lamp Design Lampshades In Execises 41 and 42, efe to the ampshade with a squae base shown at the ight. 41. Use the Pythagoean Theoem to find the sant height of the ampshade. Round you answe to the neaest whoe numbe. 18 cm 42. Estimate the amount of gass needed to make the ampshade by cacuating the atea aea of the pyamid. 28 cm Chaenge Find the suface aea of the combined soids. (Hint : Find the suface aea of each soid and add them togethe. In each cacuation, emembe to omit the suface whee the soids connect.) LAMPSHADES Many ampshades ae shaped ike cones o pyamids. This amp was designed by achitect and designe Fank Loyd Wight Standadized Test Pactice 46. Muti-Step Pobem Compete pats (a) (e). a. Find the suface aea of each cone descibed in the tabe. Round you answe to the neaest whoe numbe. Radius Sant Height Suface Aea Cone A 3 ft? Cone B 3 ft 8 ft? Cone C 3 ft 10 ft? b. What measuement stayed the same in the cones in pat (a)? c. Find the suface aea of each cone descibed in the tabe. Round you answe to the neaest whoe numbe. Radius Sant Height Suface Aea Cone D 2 ft 8 ft? Cone E 4 ft 8 ft? Cone F 8 ft? d. What measuement stayed the same in the cones in pat (c)? e. Compae the measuements you found in the two tabes. Which measuement has a geate infuence on suface aea? Why? 498 Chapte 9 Suface Aea and Voume
9 Page 9 of 9 Mixed Review Evauating Expessions Evauate the expession fo the given vaue of the vaiabe. (Skis Review, p. 670) 47. 3x 2 when x x 2 6 when x x 2 3 when x x 2 10 when x x 2 4x when x x 2 5x when x 5 Finding Aeas of Sectos Find the aea of the geen secto given the aea of the cice. Round you answe to the neaest whoe numbe. (Lesson 8.7) 53. A 180 m A 114 ft A 258 cm Ageba Skis Simpifying Expessions Simpify. (Skis Review, p. 670) 56. 2x 3 6x 57. 4y 5 4y (2x 1) x 59. 7x 2 (3x 2) 60. x 9x 6x c (5 3c) Quiz 1 Identify the shape of the base(s) of the soid and name the soid. Then te if the soid is a poyhedon. If so, count the numbe of faces of the poyhedon. (Lesson 9.1) Find the suface aea of the figue. If necessay, ound you answe to the neaest whoe numbe. (Lessons 9.2, 9.3) 4. 2 ft ft 7 in. 3 in. 8 m 6 m in. 5 in m 9. 8 in. 13 in. 10 m 10 m 5 cm 7 cm 9.3 Suface Aea of Pyamids and Cones 499
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