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Surface Area

Surface Area. Recall the following definitions related to polyhedra Each surface of a polyhedron is a face of the solid In a prism, the two congruent faces are the bases and the rectangular faces are lateral faces

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Surface Area

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  1. Surface Area • Recall the following definitions related to polyhedra • Each surface of a polyhedron is a face of the solid • In a prism, the two congruent faces are the bases and the rectangular faces are lateral faces • A pyramid has one base and triangular lateral faces that share an edge with each side of the base • The formulas for plane figures can be used to find the surface area of solids • Both figures at right have polygonal bases, and either rectangular or triangular lateral faces • Surface area is the sum of the areas of all faces of the solid • Lateral surfacearea is the sum of the areas of the lateral faces

  2. Surface Area • The surface area of any solid can be calculated by adding up the areas of each face • For example, to find the surface area of a rectangular prism, draw each face and add up the areas • Other solids may not be as obvious, but formulas from earlier in the chapter can still be used • For example, a cylinder has two circles as bases, and the lateral surface can be unrolled into a rectangle • The surface area of the cylinder can be found by adding the areas of the three parts • Surface area = 2(πr2) +  (2πr)h = 2πr (r + h) • For this cylinder, 2(π · 52) + (2 · π · 5 ) · 12 170π or approximately 534 square inches

  3. Surface Area • The surface area of a pyramid is the area of the polygonal base plus the area of each triangular lateral face • The height of each triangular lateral face of the pyramid is called the slant height • Most formulas use the letter l for the slant height to avoid confusion with the height, h, of the pyramid • For the square pyramid at right, the surface area can be calculated as the sum of the area of the base and the four triangles • SA = s2 + 4 (½ sl) or just SA = s2 + 2sl • Some formulas for surface area and volume use B as the area of a base s Side length

  4. Surface Area • The surface area of a pyramid with a regular polygon as its base can be broken up as follows: • The base is a regular polygon • The area of the polygon is ½Pa • Each lateral face is a triangle • Flip over half the triangles to make a parallelogram • The area of the parallelogram is ½Pl • Add together to get SA = ½Pl + ½Pa or SA = ½P(l + a)

  5. Surface Area • As the number of sides in a pyramid increases, it looks more and more like a cone, which can be broken up as follows: • The base of the cone is a circle of radius r • The area of the base is πr2 • The lateral surface is a sector of a circle of radius land arc length 2πr • Flip over half the “triangles” to make a “parallelogram” with base πr and height l • The area of the “parallelogram” is πrl • Add together to get SA = πr2 + πrl or SA = πr(r+l)

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