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2.12-3 Surface Areas of Pyramids and ConesSOLUTION:First, find the lateral area.Find the lateral area and surface area of eachregular pyramid. Round to the nearest tenth ifnecessary.The lateral area L of a regular pyramid is,where is the slant height and P is the perimeter ofthe base.The base is a regular pentagon of side 7 in. and theslant height is 9 in.1.SOLUTION:The lateral area L of a regular pyramid is,where is the slant height and P is the perimeter ofthe base.Here, the base is a square of side 16 cm and theslant height is 12 cm.The surface area S of a regular pyramid is, where L is the lateral area and B is thearea of the base.The area of a regular polygon is half the product ofthe length of the apothem and the perimeter.Find the length of the apothem and the area of thebase.2So, the lateral area of the pyramid is 384 cm .A central angle of the pentagon is, so theangle formed in the triangle below is 36 . Use atrigonometric ratio to find the apothem a.The surface area S of a regularpyramid is, whereL is the lateral area and B is the area of the base.Therefore, the surface area of the pyramid is 6402cm .ANSWER:22L 384 cm ; S 640 cm2.SOLUTION:First, find the lateral area.The Manuallateral- areaL ofbyaCogneroregulareSolutionsPoweredpyramid is,where is the slant height and P is the perimeter ofthe base.Page 1

A central angle of the regular hexagon isso the angle formed in the triangle below is 30.12-3 Surface Areas of Pyramids and Cones,So, the area of the base B is about 84.3 squareinches.Find the surface area of the pyramid.Use a trigonometric ratio to find the length of a sideof the hexagon s.Therefore, the surface area of the pyramid is about2241.8 in .ANSWER:22L 157.5 in ; S 241.8 inFind the lateral area of the pyramid.3.SOLUTION:The segment joining the points where the slant heightand height intersect the base is the apothem.So, the lateral area of the pyramid is about 207.8m2.Find the surface area of the pyramid.Use the Pythagorean Theorem to find the length theapothem of the base.Therefore, the surface area of the pyramid is about2332.6 m .ANSWER:A central angle of the regular hexagon isso the angle formed in the triangle below is 30.eSolutions Manual - Powered by Cognero,22L 207.8 m ; S 332.6 m4. TENTS A conical tent is shown at the right. Roundanswers to the nearest tenth.a. Find the lateral area of the tent and describe whatit represents.b. Find the surface area of the tent and describePage 2what it represents.

L 207.8 m ; S 332.6 mfor the side of the tent.4. TENTS A conical tent is shown at the right. Roundthe nearesttenth. and Cones12-3answersSurfacetoAreasof Pyramidsa. Find the lateral area of the tent and describe whatit represents.b. Find the surface area of the tent and describewhat it represents.b. Find the surface area of the cone.Therefore, the surface area of the tent is about 343.22ft . The surface area represents the area of thecurved surface plus the area of the tent floor.ANSWER:SOLUTION:a. The tent is in the shape of a right circular cone.The radius r isor 6.5 ft and the height is 8 feet.Use the Pythagorean Theorem to find the slantheight .2a. 210.5 ft ; the area of the curved surfaceb. 343.2 ft2 ; the area of the curved surface plus thearea of the tent floorCCSS SENSE-MAKING Find the lateral areaand surface area of each cone. Round to thenearest tenth.5.SOLUTION:The radius is 5 m and the slant height is 12 m.So, the slant height is about 10.3 feet.Find the lateral area of the cone.ANSWER:22L 188.5 m ; S 267.0 mTherefore, the lateral area is about 210.5 ft2. Thelateral area would represent the area of the curvedsurface of the tent or the amount of fabric requiredfor the side of the tent.b. Find the surface area of the cone.6.eSolutions Manual - Powered by CogneroSOLUTION:The radius is 7.5 cm and the height is 15 cm.Page 3

ANSWER:22 and Cones12-3 Surface Areasof PyramidsL 188.5 m ; S 267.0 mANSWER:22L 395.1 cm ; S 571.9 cmFind the lateral area and surface area of eachregular pyramid. Round to the nearest tenth ifnecessary.6.7.SOLUTION:The radius is 7.5 cm and the height is 15 cm.SOLUTION:The lateral area L of a regular pyramid is,where is the slant height and P is the perimeter ofthe base.Here, the base is a square of side 2 m and the slantheight is 5 m.Use the Pythagorean Theorem to find the slantheight.So, the lateral area of the pyramid is 20 m2.The surface area S of a regularpyramid is, whereL is the lateral area and B is the area of the base.Find the lateral and surface area.2Therefore, the surface area of the pyramid is 24 m .ANSWER:22L 20 m ; S 24 mANSWER:22L 395.1 cm ; S 571.9 cmFind the lateral area and surface area of eachregular pyramid. Round to the nearest tenth ifnecessary.8.eSolutions Manual - Powered by Cognero7.SOLUTION:Page 4The base of the pyramid is a square with side lengthof 8 feet and the height is 10 feet. Use thePythagorean Theorem to find the slant height of the

Therefore, the surface area of the pyramid is about2Therefore, the surface area of the pyramid is 24 m .2236.3 ft .ANSWER:212-3 Surface2Areas of Pyramidsand ConesL 20 m ; S 24 mANSWER:22L 172.3 ft ; S 236.3 ft8.SOLUTION:The base of the pyramid is a square with side lengthof 8 feet and the height is 10 feet. Use thePythagorean Theorem to find the slant height of thepyramid.9.SOLUTION:Use the right triangle formed by the slant height of 7,the height of 5, and the apothem and the PythagoreanTheorem to find the length of the apothem of thebase.Find the lateral area and surface area of thepyramid.The base of the pyramid is an equilateral triangle.The measure of each central angle is, sothe angle formed in the triangle below is 60 .So, the lateral area of the pyramid is about 172.3ft2.Therefore, the surface area of the pyramid is about2236.3 ft .Use a trigonometric ratio to find the length s of eachside of the triangle.ANSWER:22L 172.3 ft ; S 236.3 fteSolutions Manual - Powered by CogneroPage 5

Therefore, the surface area of the pyramid is about2302.9 cm .ANSWER:a trigonometricratio to findtheConeslength s of each12-3UseSurfaceAreas of Pyramidsandside of the triangle.22L 178.2 cm ; S 302.9 cm10.SOLUTION:First, find the lateral area.Use the formulas for regular polygons to find theperimeter and area of the base.The lateral area L of a regular pyramid is,where is the slant height and P is the perimeter ofthe base.2So, the lateral area of the pyramid is 150 ft .Next, find the length of the apothem and the area ofthe base.Find the lateral area and surface area of thepyramid.A central angle of the regular pentagon is,so the angle formed in the triangle below is 36 . Usea trigonometric ratio to find the apothem a.So, the lateral area of the pyramid is about 178.2cm2.Therefore, the surface area of the pyramid is about2302.9 cm .ANSWER:22L 178.2 cm ; S 302.9 cmSo, the area of the base B is about 61.9 square feet.10.eSolutionsManual - Powered by CogneroSOLUTION:First, find the lateral area.Find the surface area of the pyramid.Page 6

12-3 Surface Areas of Pyramids and ConesSo, the area of the base B is about 61.9 square feet.Use a trigonometric ratio to find the length of eachside of the square.Find the surface area of the pyramid.Therefore, the surface area of the pyramid is about2211.9 ft .Find the perimeter and area of the base.ANSWER:22L 150 ft ; S 211.9 ft11. square pyramid with an altitude of 12 inches and aslant height of 18 inchesSOLUTION:The base of the pyramid is square. Use thePythagorean Theorem to find the length the apothemof the base.Find the lateral area and surface area of thepyramid.So, the lateral area of the pyramid is about 966 in2.Therefore, the surface area of the pyramid is about21686.0 in .A central angle of the square is, so theangle formed in the triangle below is 45 .ANSWER:2L 966.0 in ; S 1686.0 in212. hexagonal pyramid with a base edge of 6 millimetersand a slant height of 9 millimetersSOLUTION:The base of the pyramid is a regular hexagon. Theperimeter of the hexagon is P 6 6 or 36 mm.Use a trigonometric ratio to find the length of eachside of the square.eSolutions Manual - Powered by CogneroFind the lateral area L of a regular pyramid.Page 7

12-3 Surface Areas of Pyramids and ConesTherefore, the surface area of the pyramid is about2255.5 mm .Find the lateral area L of a regular pyramid.ANSWER:22L 162 mm ; S 255.5 mm13. ARCHITECTURE Find the lateral area of apyramid-shaped building that has a slant height of 210feet and a square base 332 feet by 332 feet.SOLUTION:So, the lateral area of the pyramid is 162 mm2.The lateral area L of a regular pyramid isA central angle of the hexagon isangle formed in the triangle below is 30 ., so the,where is the slant height and P is the perimeter ofthe base.Here, the base is a square of side 332 ft and the slantheight is 210 ft.So, the lateral area of the pyramid is 139,440 ft2.ANSWER:2139,440 ftUse a trigonometric ratio to find the apothem a andthen find the area of the base.Find the lateral area and surface area of eachcone. Round to the nearest tenth.14.SOLUTION:The lateral area L of a right circular cone iswhere r is the radius of the base and isthe slant height.Here, the radius is 4 in and the slant height is 11 in.So, the area of the base B is about 93.5 mm2.Find the surface area of the pyramid.So, the lateral area of the cone is about 138.2 in2.The surface area S of a right circular cone is, where r is the radius of the base andis the slant height.Therefore, the surface area of the pyramid is about2255.5 mm .eSolutions Manual - Powered by CogneroPage 8ANSWER:22L 162 mm ; S 255.5 mm

2188.5 in .So, the lateral area of the pyramid is 139,440 ft2.ANSWER:ANSWER:12-3 Surface Areas of Pyramids and Cones2139,440 ft22L 138.2 in ; S 188.5 inFind the lateral area and surface area of eachcone. Round to the nearest tenth.15.SOLUTION:14.SOLUTION:The lateral area L of a right circular cone iswhere r is the radius of the base and isthe slant height.Here, the radius is 4 in and the slant height is 11 in.The radius of the cone isand the heightis 18 cm. Use the Pythagorean Theorem to find theslant height .So, the lateral area of the cone is about 138.2 in2.The surface area S of a right circular cone is, where r is the radius of the base andis the slant height.Find the lateral and surface areas of the cone.Therefore, the surface area of the cone is about2188.5 in .ANSWER:22L 138.2 in ; S 188.5 in15.SOLUTION:The radius of the cone isand the heightis 18 cm. Use the Pythagorean Theorem to find theslant height .So, the lateral area of the cone is about 357.6 cm2.Therefore, the surface area of the cone is about2470.7 cm .ANSWER:22L 357.6 cm ; S 470.7 cm16. The diameter is 3.4 centimeters, and the slant heightis 6.5 centimeters.SOLUTION:eSolutions Manual - Powered by CogneroThe radius of the cone is.Page 9

Therefore, the surface area of the cone is about2470.7 cm .ANSWER:12-3 Surface Areas of Pyramids and Cones22L 357.6 cm ; S 470.7 cmcm .ANSWER:22L 34.7 cm ; S 43.8 cm16. The diameter is 3.4 centimeters, and the slant heightis 6.5 centimeters.SOLUTION:The radius of the cone isthe surface area of the cone is about 43.8217. The altitude is 5 feet, and the slant height isfeet.SOLUTION:The altitude of the cone is 5 feet and the slant height.isor 9.5 feet. Use the Pythagorean Theorem tofind the radius.Find the lateral and surface areas.So, the lateral area of the cone is about 34.7 cm2.Find the lateral and surface areas of the cone.Therefore, the surface area of the cone is about 43.82cm .ANSWER:2So, the lateral area of the cone is about 241.1 ft2.2L 34.7 cm ; S 43.8 cm17. The altitude is 5 feet, and the slant height isfeet.SOLUTION:The altitude of the cone is 5 feet and the slant heightisor 9.5 feet. Use the Pythagorean Theorem tofind the radius.Therefore, the surface area of the cone is about446.1 ft2.ANSWER:22L 241.1 ft ; S 446.1 ft18. MOUNTAINS A conical mountain has a radius of1.6 kilometers and a height of 0.5 kilometer. What isthe lateral area of the mountain?eSolutions Manual - Powered by CogneroFind the lateral and surface areas of the cone.SOLUTION:The radius of the conical mountain is 1.6 kilometersand the height is 0.5 kilometers. Use the PythagoreanTheorem to find the slant height.Page 10

2446.1 ft2.Therefore, the lateral area is about 8.4 km .ANSWER:12-3 Surface Areas of Pyramids and Cones22L 241.1 ft ; S 446.1 ft18. MOUNTAINS A conical mountain has a radius of1.6 kilometers and a height of 0.5 kilometer. What isthe lateral area of the mountain?SOLUTION:The radius of the conical mountain is 1.6 kilometersand the height is 0.5 kilometers. Use the PythagoreanTheorem to find the slant height.ANSWER:28.4 km19. HISTORY Archaeologists recently discovered a1500-year-old pyramid in Mexico City. The squarepyramid measures 165 yards on each side and oncestood 20 yards tall. What was the original lateral areaof the pyramid?SOLUTION:The pyramid has a square base with sides havinglengths of 165 yards and a height of 20 yards. Usethe Pythagorean Theorem to find the slant height.Find the lateral area L of the conical mountain.Find the lateral area L of a regular pyramid.2Therefore, the lateral area is about 8.4 km .ANSWER:28.4 km19. HISTORY Archaeologists recently discovered a1500-year-old pyramid in Mexico City. The squarepyramid measures 165 yards on each side and oncestood 20 yards tall. What was the original lateral areaof the pyramid?SOLUTION:The pyramid has a square base with sides havinglengths of 165 yards and a height of 20 yards. Usethe Pythagorean Theorem to find the slant height.Therefore, the lateral area of the pyramid is about28,013.6 yd2.ANSWER:228,013.6 yd20. Describe two polyhedrons that have 7 faces.SOLUTION:A hexagonal pyramid has 6 triangular lateral facesand one hexagonal base. A pentagonal prism has 5rectangular lateral faces and two pentagonal bases.ANSWER:hexagonal pyramid, pentagonal prism21. What is the sum of the number of faces, vertices,and edges of an octagonal pyramid?eSolutions Manual - Powered by CogneroSOLUTION:Page 11In any pyramid, the number of faces is n 1 wheren is the number of sides of the base, the number of

and one hexagonal base. A pentagonal prism has 5rectangular lateral faces and two pentagonal bases.12-3ANSWER:Surface Areas of Pyramids and Coneshexagonal pyramid, pentagonal prism21. What is the sum of the number of faces, vertices,and edges of an octagonal pyramid?SOLUTION:In any pyramid, the number of faces is n 1 wheren is the number of sides of the base, the number ofedges is 2n and the number of vertices is n 1. In anoctagonal prism, the base has 8 sides. Therefore, thesum of the number of faces, vertices, and edges is (8 1) (2(8)) (8 1) 34.ANSWER:3422. TEPEES The dimensions of two canvas tepees areshown in the table at the right. Not including the floorapproximately how much more canvas is used to maTepee B than Tepee A?SOLUTION:The tepees are in the shape of a right cone. To findthe amount of canvas used for each tepee, we need torfind its lateral area. The radius of Tepee A isfeet and the radius of Tepee B isor 10 feetTo find how much more canvas is used to makeTepee B than Tepee A, subtract the lateral areas.Tepee B will use about 219.9 ft2 more canvas thanTepee A.ANSWER:2about 219.9 ft23. The surface area of a square pyramid is 24 squaremillimeters and the base area is 4 square millimeters.What is the slant height of the pyramid?SOLUTION:Let x be the side of the square base. The area of thebase is 4, so x is 2. Solve for l.Tepee ATepee BUse the Pythagorean Theorem to find the slant heighof each tepee and then find the lateral area for eacone.eSolutions Manual - Powered by CogneroANSWER:5 mmPage 1224. The surface area of a cone is 18π square inches andthe radius of the base is 3 inches. What is the slant

Tepee B will use about 219.9 ft more canvas thanTepee A.ANSWER:12-3 Surface Areas of Pyramids and Cones2about 219.9 ft23. The surface area of a square pyramid is 24 squaremillimeters and the base area is 4 square millimeters.What is the slant height of the pyramid?SOLUTION:Let x be the side of the square base. The area of thebase is 4, so x is 2. Solve for l.ANSWER:5 mmsquare centimeters, and the base is 24 centimeterswide with a hypotenuse of 25 centimeters. What isthe slant height of the pyramid?SOLUTION:The base of the triangular pyramid is a right triangle(since the measure of the hypotenuse is 25centimeters). Use the Pythagorean Theorem to finda, the length of the third side of the base. Then findthe perimeter and area of the base.P 7 24 25 or 56 cm24. The surface area of a cone is 18π square inches andthe radius of the base is 3 inches. What is the slantheight of the cone?SOLUTION:The surface area S of a right circular cone is, where r is the radius of the base andis the slant height.Solve for.Replace S with 532, P with 56, and B with 84 in theformula for the surface area of a pyramid. Thensolve for the slant height .Therefore, the slant height of the cone is 3 in.ANSWER:3 in.25. The surface area of a triangular pyramid is 532square centimeters, and the base is 24 centimeterswide with a hypotenuse of 25 centimeters. What isthe slant height of the pyramid?SOLUTION:The base of the triangular pyramid is a right triangle(sincethe measurethe hypotenuse is 25eSolutionsManual- Powered ofby Cognerocentimeters). Use the Pythagorean Theorem to finda, the length of the third side of the base. Then findTherefore, the slant height of the pyramid is 16 cm.ANSWER:16 cm26. Find the lateral area of the tent to the nearest tenth.Page 13

Therefore, the slant height of the pyramid is 16 cm.ANSWER:12-316Surfacecm Areas of Pyramids and ConesTherefore, the lateral area of the tent is about 311.22ft .ANSWER:311.2 ft226. Find the lateral area of the tent to the nearest tenth.27. Find the surface area of the tank. Write in terms ofπ.SOLUTION:The tent is a combination of a cylinder and a cone.The cone has a radius of 5 feet and a height of 6feet. Use the Pythagorean Theorem to find the slantheight of the cone.SOLUTION:The tank is a combination of a cylinder and a cone.The cylinder has a height of 11 feet and a radius ofor 7 feet. The cone has a radius ofor 7 feetand a slant height of 9 feet. The surface area of thetank is the sum of the lateral area of a cylinder, thelateral area of the cone, and the area of the one baseof the cylinder.Therefore, the surface area of the tank isThe cylinder has a radius of 5 feet and a height of 12– 6 or 6 feet.Find the sum of the lateral areas of the cone andcylinder to find the lateral area of the tent.Therefore, the lateral area of the tent is about 311.22ft .2ft .ANSWER:2266π ft28. CHANGING DIMENSIONS A cone has a radiusof 6 centimeters and a slant height of 12 centimeters.Describe how each change affects the surface areaof the cone.a. The radius and the slant height are doubled.b. The radius and the slant height are divided by 3.SOLUTION:a. The surface area S of a right circular cone is, where r is the radius of the base andis the slant height.When the radius and the slant height are doubled thesurface area S’ is given by the formulaANSWER:311.2 ft227. Find the surface area of the tank. Write in terms ofπ.eSolutions Manual - Powered by CogneroPage 14

Therefore, the surface area of the tank is2divided by 3, the surface area is divided by 9.ft .ANSWER:12-3 Surface Areas of Pyramids and Cones2266π ft28. CHANGING DIMENSIONS A cone has a radiusof 6 centimeters and a slant height of 12 centimeters.Describe how each change affects the surface areaof the cone.a. The radius and the slant height are doubled.b. The radius and the slant height are divided by 3.SOLUTION:a. The surface area S of a right circular cone is, where r is the radius of the base andis the slant height.When the radius and the slant height are doubled thesurface area S’ is given by the formulaANSWER:a. The surface area is multiplied by 4.b. The surfa

So, the area of the base B is about 61.9 square feet. Find the surface area of the pyramid. Therefore, the surface area of the pyramid is about 211.9 ft 2. $16:(5 L = 150 ft 2; S§ IW 2 square pyramid with an altitude of 12 inches and a slant height of 18 inches 62/87,21 The base of the pyramid is square. Use the