PRISMS Lateral Area and Total Area of a Prism (11th grade)

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  1. PRISMS
  The two shaded faces of the prism shown are its bases.
  The bases are congruent polygons lying in parallel planes.
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  The faces of a prism that are not its bases are called lateral faces.
  The lateral faces of a prism are parallelograms.
  If they are rectangles, the prism is a right prism. Otherwise the prism is an oblique prism.
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  The diagrams below show that a prism is also classified by the shape of its bases.
  Right triangular
  prism
  Right rectangular prism
  (Rectangular solid)
  Oblique pentagonal prism
  
  Lateral edge is not an altitude
  Lateral edges are altitudes
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  Lateral Area and Total Area of a Prism
  The lateral area (L.A.) of a prism is the sum of the areas of its lateral faces.
  
  The total area (T.A.) is the sum of the areas of all its faces.
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  Lateral Area and Total Area of a Prism
  L.A. = ah+bh+ch+dh+eh = (a+b+c+d+e)h
  = perimeter • h
  = ph
  
  The right pentagonal prism can be used to illustrate the development of the lateral area formula:
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  Lateral Area and Total Area of a Prism
  The Method applies to any right prism.
  
  The lateral area equals the perimeter of a base times the height of the prism. (L.A. = ph)
  
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  Prisms have volume as well as area. A rectangular solid with square faces is a cube.
  Since each edge of the blue cube shown is 1 unit long, the cube is said to have a volume of 1 cubic unit.
  The larger rectangular solid has 3 layers of cubes, each layer containing (4•2) cubes. Hence its volume is (4•2)•3, or 24 cubic units.
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  Volume = Base area x height
  

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  The Volume of a Right Prism
  The same sort of reasoning is used to find the volume of any right prism.
  
  The volume of a right prism equals the area of a base times the height of the prism. (V = Bh)
  
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  Example 1
  A right trapezoidal prism is shown. Find the
  lateral area,
  total area,
  volume.
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  Example 2
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  A right triangular prism is shown. The volume is 315. Find the total area.
  

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  Example 3
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  The container shown has the shape of a rectangular solid.
  When a rock is submerged, the water level rises 0.5 cm. Find the volume of the rock.
  

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  Example 4
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  A right prism has square bases with edges that are three times as long as the lateral edges. The prism's total area is 750 m2. Find the volume.
  

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  The Volume of an
  Oblique Prism
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  The volume of an oblique prism equals the area of a base times the height of
  the prism or right cross-section area times the length of the lateral edge.
  
  

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  The Volume of an
  Oblique Prism
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  Volume = Base Area x Altitude = Right Cross-section Area x Length of lateral edge
  

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  Example 5
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  The oblique square prism shown has base edge 3. A lateral edge that is 15 makes a 60° angle with the plane containing the base. Find the exact volume.
  

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  Example 6
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  In the parallelepiped ABCD, right cross-section area is a square with sides 1 cm. If =60° and cm, find the volume of the prism.
  

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  Note
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  A diagonal of a right rectangular prism joins two vertices not in the same face. The length of the diagonals is
  

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  Example 7
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  If the diagonals of a cube are cm long, what is the volume?