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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.

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