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12.4 Surface Area ofPyramids and ConesGeometryNCSCOS: 1.02; 2.03; 2.04
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Objectives/Assignment• Find the surface area of a pyramid.• Find the surface area of a cone.
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Finding the surface area of apyramid• A pyramid is a polyhedron in which the base is apolygon and the lateral faces are triangles with acommon vertex. The intersection of two lateralfaces is a lateral edge. The intersection of thebase and a lateral face is a base edge. Thealtitude or height of a pyramid is theperpendicular distance between the base andthe vertex.
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More on pyramids• A regular pyramid has aregular polygon for abase and its height meetsthe base at its center.The slant height of aregular pyramid is thealtitude of any lateralface. A nonregularpyramid does not have aslant height.
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Ex. 1: Finding the Area of a LateralFace• Architecture. The lateral faces of thePyramid Arena in Memphis, Tennessee,are covered with steal panels. Use thediagram of the arena to find the area ofeach lateral face of this regular pyramid.
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Hexagonal Pyramids• A regular hexagonalpyramid and its net areshown at the right. Let brepresent the length of abase edge, and let lrepresent the slant heightof the pyramid. The areaof each lateral face is1/2bl and the perimeterof the base if P = 6b. Sothe surface area is asfollows:
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Hexagonal pyramidS = (Area of base) + 6(Area of lateral face)S = B + 6( ½ bl)S = B + (6b)lS = B + PlSubstituteRewrite 6( ½ bl) as ½ (6b)l.Substitute P for 6bSurface Area of a Regular PyramidThe surface area S of a regular pyramid is:S = B + ½ Pl, where B is the area of the base, P isthe perimeter of the base, and l is the slant height.
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Ex. 2: Finding the surface area of apyramid• To find the surface areaof the regular pyramidshown, start by findingthe area of the base.• Use the formula for thearea of a regularpolygon,½ (apothem)(perimeter).A diagram of the baseis shown to the right.
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Ex. 2: Finding the surface area of apyramidAfter substituting, thearea of the base is½ (3 )(6• 6), or1 square meters.√3√3
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Surface area• Now you can find the surface area byusing 54 for the area of the base, B.√3
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Finding the Surface Area of a Cone• A circular cone, or cone,has a circular base and avertex that is NOT in thesame plane as the base.The altitude, or height, is theperpendicular distancebetween the vertex and thebase. In a right cone, theheight meets the base at itscenter and the slant heightis the distance between thevertex and a point on thebase edge.
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Finding the Surface Area of a Cone• The lateral surface of acone consists of allsegments that connect thevertex with points on thebase edge. When you cutalong the slant height andlike the cone flat, you getthe net shown at the right.In the net, the circular basehas an area of πr2and thelateral surface area is thesector of a circle.
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More on cones . . .• You can find the area of this sector byusing a proportion, as shown below.Area of sectorArea of circle=Arc lengthCircumferenceSet up proportionArea of sectorπl2 =2πr2πlSubstituteArea of sector = πl2•2πr2πlMultiply each side by πl2Area of sector = πrl SimplifyThe surface area of a cone is the sum of the basearea and the lateral area, πrl.
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Theorem• Surface Area of a RightConeThe surface area S of a rightcone is S = πr2+ πrl, where ris the radius of the base andl is the slant height
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Ex. 3: Finding the surface area of acone• To find the surface areaof the right cone shown,use the formula for thesurface area.S = πr2+ πrl Write formulaS = π42+ π(4)(6) SubstituteS = 16π + 24π SimplifyS = 40π SimplifyThe surface area is 40π square inches or about 125.7square inches.
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