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Geom12point2and3 97

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Geom12point2and3 97

  1. 1. Chapter 12 - Surface Area & Volume of Solids <ul><li>Objectives: </li></ul><ul><li>Identify types of solids </li></ul><ul><li>Calculate surface area & volume of: </li></ul><ul><li>Prisms </li></ul><ul><li>Pyramids </li></ul><ul><li>Cylinders </li></ul><ul><li>Cones </li></ul><ul><li>Spheres </li></ul>
  2. 2. 12.2 & 3 Surface Area of Prisms & Cylinders & Pyramids & Cones Objectives: Find Surface Area of prisms cylinders pyramids cones
  3. 3. Prisms <ul><li>A prism is a polyhedron with 2 congruent faces, called bases , that lie in parallel planes. </li></ul><ul><li>The other faces, ( lateral faces) , are parallelograms formed by connecting the corresponding vertices of the bases. </li></ul><ul><li>The segments connecting these vertices are lateral edges . </li></ul>
  4. 4. Prisms <ul><li>The altitude or height of each prism is the perpendicular distance between its bases. </li></ul><ul><li>In a right prism , each lateral edge is perpendicular to both bases. </li></ul>
  5. 5. Prisms <ul><li>Prisms that have lateral edges that are not perpendicular to the bases are oblique prisms. </li></ul><ul><li>The length of the oblique lateral edges is the slant height of the prism. </li></ul>http://www.mathwords.com/o/o_assets/o4.gif
  6. 6. Prisms <ul><li>Prisms are classified by the shapes of their bases. For example, the figure above is a triangular prism. </li></ul><ul><li>The surface area of a polyhedron is the sum of the area of its faces. </li></ul><ul><li>The lateral area of a polyhedron is the sum of the area of its lateral faces. </li></ul>
  7. 7. Finding the Surface Area of a Prism <ul><li>Find the surface area of a right rectangular prism with a height of 8 inches, a length of 3 inches, and a width of 5 inches. </li></ul>Faces Dimensions Area of faces Left & right Front & back Top & bottom
  8. 8. Finding the Surface Area of a Prism <ul><li>Find the surface area of a right rectangular prism with a height of 8 inches, a length of 3 inches, and a width of 5 inches. </li></ul>Faces Dimensions Area of faces Left & right 8” x 5” Front & back 8” x 3” Top & bottom 3” x 5”
  9. 9. Finding the Surface Area of a Prism <ul><li>Find the surface area of a right rectangular prism with a height of 8 inches, a length of 3 inches, and a width of 5 inches. </li></ul>Faces Dimensions Area of faces Left & right 8” x 5” 40” *2 Front & back 8” x 3” 24” *2 Top & bottom 3” x 5” 15” *2
  10. 10. Finding the Surface Area of a Prism <ul><li>2*40 + 2*24 + 2*15 = 158” </li></ul>Faces Dimensions Area of faces Left & right 8” x 5” 40” *2 Front & back 8” x 3” 24” *2 Top & bottom 3” x 5” 15” *2
  11. 11. Imagine unfolding a prism … <ul><li>This is called a net: </li></ul>
  12. 12. Surface Area of a Prism Theorem <ul><li>The surface area S of a right prism can be found using the formula S = 2B + Ph, where B is the area of a base, P is the perimeter of a base, and h is the height. </li></ul>
  13. 13. Do example 2, p. 729
  14. 14. Cylinders <ul><li>A cylinder is a solid with congruent circular bases that lie in parallel planes. </li></ul><ul><li>The altitude, or height of a cylinder is the perpendicular distance between its bases. </li></ul><ul><li>The radius of the base is also called the radius of the cylinder. </li></ul>
  15. 15. Cylinders <ul><li>A cylinder is called a right cylinder if the segment joining the centers of the bases is perpendicular to the bases. </li></ul><ul><li>The lateral area of a cylinder is the area of its curved surface. </li></ul><ul><li>The lateral area is equal to 2πr </li></ul>
  16. 16. Cylinders <ul><li>The surface area of a cylinder is equal to the sum of the lateral area and the areas of the 2 bases: </li></ul><ul><li>S = 2B + Ch = 2πr 2 + 2πrh </li></ul>
  17. 17. Do Example 3, p. 730
  18. 18. Pyramids <ul><li>A pyramid is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex. </li></ul><ul><li>The intersection of 2 lateral faces is a lateral edge. </li></ul><ul><li>The intersection of the base and a lateral face is a base edge. </li></ul><ul><li>The altitude, or height, of the pyramid is the perpendicular distance between the base and the vertex. </li></ul>
  19. 19. Pyramids <ul><li>A regular pyramid has a regular polygon for a base and its height meets the base at its center. </li></ul><ul><li>The slant height of a regular pyramid is the altitude of any lateral face. </li></ul><ul><li>A nonregular pyramid does not have a slant height. </li></ul>
  20. 20. Surface Area of a Regular Pyramid <ul><li>S = B + 1/2 P l where </li></ul><ul><ul><li>B is the area of the base </li></ul></ul><ul><ul><li>P is the perimeter of the base </li></ul></ul><ul><ul><li>l is the slant height </li></ul></ul>
  21. 21. Calculating the Base <ul><li>Area of a polygon = 1/2 aP where </li></ul><ul><ul><li>a is the apothem </li></ul></ul><ul><ul><li>P is the perimeter of the base </li></ul></ul><ul><li>The apothem is the distance from the center to any side of the polygon: </li></ul>
  22. 22. Calculating the Base <ul><li>Review: A 30-60-90 triangle has sides in the ration of 1, 2, √3 </li></ul><ul><li>The apothem of a hexagon creates a 30-60-90 triangle. </li></ul><ul><li>If each side is 5, what is the apothem? </li></ul><ul><li>Radius is 5 </li></ul><ul><li>Half side is 2.5 </li></ul><ul><li>Apothem is 2.5 √3 </li></ul>
  23. 23. Do Example 2, p. 736
  24. 24. Cones <ul><li>A circular cone , or cone , has a circular base and a vertex that is not in the same plane as the base. </li></ul><ul><li>The altitude, or height, is the perpendicular distance between the vertex and the base. </li></ul><ul><li>In a right cone, the height meets the base at its center and the slant height is the distance between the vertex & a point on the base edge. </li></ul>
  25. 25. Cones <ul><li>The lateral surface of a cone consists of all segments that connect the vertex with points on the base edge. </li></ul>
  26. 26. Surface Area of a Right Cone <ul><li>S = πr 2 + πr l where </li></ul><ul><ul><li>r is the radius of the base </li></ul></ul><ul><ul><li>l is the slant height </li></ul></ul>
  27. 27. Homework <ul><li>Do worksheets </li></ul><ul><li>Quiz Tomorrow </li></ul>

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