Pyramids; The Lateral Area of a Regular Pyramid (11th grade)

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  Pyramids
  The diagram shows the pentagonal pyramid V-ABCDE. Point V is the vertex of the pyramid and pentagon ABCDE is the base.
  The segment from the vertex perpendicular to the base is the altitude and its length is the height, h, of the pyramid.
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  Pyramids
  The five triangular faces with V in common, such as VAB, are lateral faces. These faces intersect in segments called lateral edges.
  The height of a lateral face is called the slant height, l of the pyramid.
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  Most of the pyramids you'll study
  will be regular pyramids.
  All lateral faces are congruent isosceles triangles.
  The altitude meets the base at its center, O.
  These are pyramids with the following properties:
  The base is a regular polygon
  
  All lateral edges are congruent
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  Example 1
  A regular square pyramid has base edges 10 and lateral edges 13. Find its
  slant height
  height.
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  Example 2
  Find the lateral area of the pyramid given in Example 1.
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  The Lateral Area of a Regular Pyramid
  The lateral area of a regular pyramid with n lateral faces is
  (the area of one lateral face x n)
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  The Volume of a Pyramid
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  The prism and pyramid below have congruent bases and equal heights. Since the volume of the prism is Bh, the volume of the pyramid must be less than Bh. In fact, it is exactly
  

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  The Volume of a Pyramid
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  The volume of a pyramid equals one third the area of the base times the height of the pyramid.
  

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  Example 3
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  Suppose the regular hexagonal pyramid shown has base edges 6 and height 12. Find its volume.
  

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  Example 4
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  A regular triangular pyramid has lateral edge 10 and height 6. Find the
  (a) lateral area (b) volume.
  

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  Example 5
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  The shaded pyramid in the diagram is cut from a rectangular solid. How does the volume of the pyramid compare with the volume of the rectangular solid?
  

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